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Line 39 $$\hbox{%

}$$

typing in \kbd{foo(6,4)} would give

you \kbd{foo(6,4,3)}. In the rare case when you want to set some far away

flag, and leave the defaults in between as they stand, you can use the

argument, and leave the defaults in between as they stand, you can use the

``empty arg'' trick alluded to above: \kbd{foo(6,,1)} would yield

\kbd{foo(6,2,1)}. By the way, \kbd{foo()} by itself yields

\kbd{foo(1,2,3)} as was to be expected. In this rather special case of a

Line 50 to put optional arguments at the end of the argument l

they would not make sense otherwise), and in order of decreasing usefulness

so that, most of the time, you will be able to ignore them.

\misctitle{Binary Flags}.\sidx{binary flag} For some of these optional

\misctitle{Flags}. A \tev{flag} is an argument which, rather than conveying

flags, we adopted the customary binary notation as a compact way to

actual information to the routine, intructs it to change its default

represent many toggles with just one number. Letting $(p_0,\dots,p_n)$ be a

behaviour, e.g.~return more or less information. All such

list of switches (i.e.~of properties which can be assumed to take either

flags are optional, and will be called \fl\ in the function descriptions to

the value $0$ or~$1$), the number $2^3 + 2^5=40$ means that $p_3$ and $p_5$

follow. There are two different kind of flags

$\bullet$ generic: all valid values for the flag are individually

described (``If \fl\ is equal to $1$, then\dots'').

$\bullet$ binary:\sidx{binary flag} use customary binary notation as a

compact way to represent many toggles with just one number. Letting

$(p_0,\dots,p_n)$ be a list of switches (i.e.~of properties which take either

the value $0$ or~$1$), the number $2^3 + 2^5 = 40$ means that $p_3$ and $p_5$

have been set (that is, set to $1$), and none of the others were (that is,

they were set to 0). This will usually be announced as ``The binary digits

they were set to $0$). This is announced as ``The binary digits of $\fl$ mean

of $\fl$ mean 1: $p_0$, 2: $p_1$, 4: $p_2$'', and so on, using the

1: $p_0$, 2: $p_1$, 4: $p_2$'', and so on, using the available consecutive

available consecutive powers of~$2$.

powers of~$2$.

\misctitle{Mnemonics for flags}. Numeric flags as mentionned above are

obscure, error-prone, and quite rigid: should the authors

want to adopt a new flag numbering scheme (for instance when noticing

flags with the same meaning but different numeric values), it would break

backward compatibility. The only advantage of explicit numeric values is that

they are fast to type, so their use is only advised when using the calculator

GP.

As an alternative, one can replace a numeric flag by a character string

containing symbolic identifiers. For a generic flag, the mnemonic

corresponding to the numeric identifier is given after it as in (taken from

description of $\tet{log}(x,\{\fl=0\})$):

\centerline{If \fl\ is equal to $1 = \kbd{AGM}$, use an agm formula\dots}

\noindent which means that one can use indifferently \kbd{log($x$, 1)} or

\kbd{log($x$, AGM)}.

For a binary flag, mnemonics corresponding to the various toggles are given

after each of them. They can be negated by prepending \kbd{no\_} to the

mnemonic, or by removing such a prefix. These toggles are grouped together

using any punctuation character (such as ',' or ';'). For instance (taken

from description of $\tet{ploth}(X=a,b,\var{expr},\{\fl=0\},\{n=0\})$)

\centerline{Binary digits of flags mean: $1=\kbd{Parametric}$,

$2=\kbd{Recursive}$, \dots}

\noindent so that, instead of $1$, one could use the mnemonic

\kbd{"Parametric; no\_Recursive"}, or simply \kbd{"Parametric"} since

\kbd{Recursive} is unset by default (default value of $\fl$ is $0$,

i.e.~everything unset).

\misctitle{Pointers}.\sidx{pointer} If a parameter in the function

prototype is prefixed with a \& sign, as in

Line 71 to be typed in explicitly. As of version \vers, this \

Line 111 to be typed in explicitly. As of version \vers, this \

is optional for all documented functions, hence the \& will always appear

between brackets as in \kbd{issquare}$(x,\{\&e\})$.

\misctitle{About library programming}. To finish with our generic

\misctitle{About library programming}.

simple-minded example, the \var{library} function \kbd{foo}, as defined

the \var{library} function \kbd{foo}, as defined

above, is seen to have two mandatory arguments, $x$ and \fl (no PARI

at the beginning of this section, is seen to have two mandatory arguments,

mathematical function has been implemented so as to accept a variable

$x$ and \fl: no PARI mathematical function has been implemented so as to

number of arguments). When not mentioned otherwise, the result and

accept a variable number of arguments, so all arguments are mandatory when

arguments of a function are assumed implicitly to be of type \kbd{GEN}.

programming with the library (often, variants are provided corresponding to

Most other functions return an object of type \kbd{long} integer in C (see

the various flag values). When not mentioned otherwise, the result and

Chapter~4). The variable or parameter names \var{prec} and \fl\ always

arguments of a function are assumed implicitly to be of type \kbd{GEN}. Most

denote \kbd{long} integers.

other functions return an object of type \kbd{long} integer in C (see

Chapter~4). The variable or parameter names \var{prec} and \fl\ always denote

\kbd{long} integers.

The \tet{entree} type is used by the library to implement iterators (loops,

sums, integrals, etc.) when a formal variable has to successively assume a

Line 137 integers, add \kbd{O(p\pow k)} to the result; finally,

Line 179 integers, add \kbd{O(p\pow k)} to the result; finally,

\syn{gdiv}{x,y} for $x$ \kbd{/} $y$.

\subseckbd{\bs}: The expression $x$ \kbd{\bs} $y$ is the

\subseckbd{\bs}: The expression \kbd{$x$ \bs\ $y$} is the \idx{Euclidean

% keep "Euclidean" and "quotient" on same line for gphelp

quotient} of $x$ and $y$. If $y$ is a real scalar, this is defined as

\idx{Euclidean quotient} of $x$ and $y$. The types must be either both

\kbd{floor($x$/$y$)} if $y > 0$, and \kbd{ceil($x$/$y$)} if $y < 0$ and

integer or both polynomials. The result is the Euclidean quotient. In the

the division is not exact. Hence the remainder \kbd{$x$ - ($x$\bs$y$)*$y$}

case of integer division, the quotient is such that the corresponding

is in $[0, |y|[$.

remainder is non-negative.

Note that when $y$ is an integer and $x$ a polynomial, $y$ is first promoted

to a polynomial of degree $0$. When $x$ is a vector or matrix, the operator

is applied componentwise.

\syn{gdivent}{x,y} for $x$ \kbd{\bs} $y$.

\subseckbd{\bs/}: The expression $x$ \b{/} $y$ is the Euclidean

\subseckbd{\bs/}: The expression $x$ \b{/} $y$ evaluates to the rounded

quotient of $x$ and $y$. The types must be either both integer or both

\idx{Euclidean quotient} of $x$ and $y$. This is the same as \kbd{$x$ \bs\ $y$}

polynomials. The result is the rounded Euclidean quotient. In the case of

except for scalar division: the quotient is such that the corresponding

integer division, the quotient is such that the corresponding remainder is

remainder is smallest in absolute value and in case of a tie the quotient

smallest in absolute value and in case of a tie the quotient closest to

closest to $+\infty$ is chosen (hence the remainder would belong to

$+\infty$ is chosen.

$]-|y|/2, |y|/2]$).

When $x$ is a vector or matrix, the operator is applied componentwise.

\syn{gdivround}{x,y} for $x$ \b{/} $y$.

\subseckbd{\%}: The expression $x$ \kbd{\%} $y$ is the

\subseckbd{\%}: The expression \kbd{$x$ \% $y$} evaluates to the modular

% keep "Euclidean" and "remainder" on same line for gphelp

\idx{Euclidean remainder} of $x$ and $y$, which we now define. If $y$ is an

\idx{Euclidean remainder} of $x$ and $y$. The modulus $y$ must be of type

integer, this is the smallest non-negative integer congruent to $x$ modulo

integer or polynomial. The result is the remainder, always non-negative in

$y$. If $y$ is a polynomial, this is the polynomial of smallest degree

the case of integers. Allowed dividend types are scalar exact types when

congruent to $x$ modulo $y$. When $y$ is a non-integral real number,

the modulus is an integer, and polynomials, polmods and rational functions

\kbd{$x$\%$y$} is defined as \kbd{$x$ - ($x$\bs$y$)*$y$}. This

when the modulus is a polynomial.

coincides with the definition for $y$ integer if and only if $x$ is an

integer, but still belongs to $[0, |y|[$. For instance:

\bprog

? (1/2) % 3

%1 = 2

? 0.5 % 3

*** forbidden division t_REAL % t_INT.

? (1/2) % 3.0

%2 = 1/2

@eprog

Note that when $y$ is an integer and $x$ a polynomial, $y$ is first promoted

to a polynomial of degree $0$. When $x$ is a vector or matrix, the operator

is applied componentwise.

\syn{gmod}{x,y} for $x$ \kbd{\%} $y$.

\subsecidx{divrem}$(x,y)$: creates a column vector with two components,

\subsecidx{divrem}$(x,y,\{v\})$: creates a column vector with two components,

the first being the Euclidean quotient, the second the Euclidean remainder,

the first being the Euclidean quotient (\kbd{$x$ \bs\ $y$}), the second the

of the division of $x$ by $y$. This avoids the need to do two divisions if

Euclidean remainder (\kbd{$x$ - ($x$\bs$y$)*$y$}), of the division of $x$ by

one needs both the quotient and the remainder. The arguments must be both

$y$. This avoids the need to do two divisions if one needs both the quotient

integers or both polynomials; in the case of integers, the remainder is

and the remainder. If $v$ is present, and $x$, $y$ are multivariate

non-negative.

polynomials, divide with respect to the variable $v$.

\syn{gdiventres}{x,y}.

Beware that \kbd{divrem($x$,$y$)[2]} is in general not the same as

\kbd{$x$ \% $y$}; there is no operator to obtain it in GP:

\bprog

? divrem(1/2, 3)[2]

%1 = 1/2

? (1/2) % 3

%2 = 2

? divrem(Mod(2,9), 3)[2]

*** forbidden division t_INTMOD \ t_INT.

? Mod(2,9) % 6

%3 = Mod(2,3)

@eprog

\syn{divrem}{x,y,v},where $v$ is a \kbd{long}. Also available as

$\teb{gdiventres}(x,y)$ when $v$ is not needed.

\subseckbd{\pow}: The expression $x\hbox{\kbd{\pow}}n$ is \idx{powering}.

If the exponent is an integer, then exact operations are performed using

binary (left-shift) powering techniques. In particular, in this case $x$

Line 208 Beware that

Line 281 Beware that

\syn{gpow}{x,n,\var{prec}} for $x\hbox{\kbd{\pow}}n$.

\subsecidx{shift}$(x,n)$ or $x$ \kbd{<<} $n$ (= $x$ \kbd{>>} $(-n)$): shifts

\subsecidx{shift}$(x,n,\{\fl=0\})$ or $x$ \kbd{<<} $n$ (= $x$ \kbd{>>} $(-n)$):

$x$ componentwise left by $n$ bits if $n\ge0$ and right by $|n|$ bits if

shifts $x$ componentwise left by $n$ bits if $n\ge0$ and right by $|n|$ bits

$n<0$. A left shift by $n$ corresponds to multiplication by $2^n$. A right

if $n<0$. A left shift by $n$ corresponds to multiplication by $2^n$. A right

shift of an integer $x$ by $|n|$ corresponds to a Euclidean division of

shift of an integer $x$ by $|n|$ corresponds to a Euclidean division of $x$

$x$ by $2^{|n|}$ with a

by $2^{|n|}$ with a remainder of the same sign as $x$, hence is not the same

remainder of the same sign as $x$, hence is not the same (in general) as

(when $x < 0$) as $x \kbd{\bs} 2^{|n|}$. If $\fl$ is non-zero, this behaviour

$x \kbd{\bs} 2^n$.

is modified and right shift of a negative $x$ \var{is} the same as $x

\kbd{\bs} 2^{|n|}$ (which is consistent with $2$-complement semantic of

negative numbers).

\syn{gshift}{x,n} where $n$ is a \kbd{long}.

\syn{gshift3}{x,n,\fl} where $n$ is a \kbd{long}. Also available is

$\teb{gshift}(x,n)$ for the case $\fl=0$.

\subsecidx{shiftmul}$(x,n)$: multiplies $x$ by $2^n$. The difference with

\kbd{shift} is that when $n<0$, ordinary division takes place, hence for

Line 266 Also, \kbd{<>} is accepted as a synonym for \kbd{!=}.

Line 342 Also, \kbd{<>} is accepted as a synonym for \kbd{!=}.

assignment statement.

\subsecidx{lex}$(x,y)$: gives the result of a lexicographic comparison

between $x$ and $y$. This is to be interpreted in quite a wide sense. For

between $x$ and $y$ (as $-1$, $0$ or $1$). This is to be interpreted in quite

example, the vector $[1,3]$ will be considered smaller than the longer

a wide sense: It is admissible to compare objects of different types

vector $[1,3,-1]$ (but of course larger than $[1,2,5]$),

(scalars, vectors, matrices), provided the scalars can be compared, as well

i.e.~\kbd{lex([1,3], [1,3,-1])} will return $-1$.

as vectors/matrices of different lengths. The comparison is recursive.

In case all components are equal up to the smallest length of the operands,

the more complex is considered to be larger. More precisely, the longest is

the largest; when lengths are equal, we have matrix $>$ vector $>$ scalar.

For example:

\bprog

? lex([1,3], [1,2,5])

%1 = 1

? lex([1,3], [1,3,-1])

%2 = -1

? lex([1], [[1]])

%3 = -1

? lex([1], [1]~)

%4 = 0

@eprog

\syn{lexcmp}{x,y}.

\subsecidx{sign}$(x)$: \idx{sign} ($0$, $1$ or $-1$) of $x$, which must be of

Line 359 i.e.~it chops off the $O(X^k)$. If $x$ is a vector, th

Line 450 i.e.~it chops off the $O(X^k)$. If $x$ is a vector, th

the polynomial whose coefficients are given in $x$, with $x[1]$ being the

leading coefficient (which can be zero).

Warning: this is \var{not} a substitution function. It is intended to be

\misctitle{Warning:} this is \var{not} a substitution function. It is

quick and dirty. So if you try \kbd{Pol(a,y)} on the polynomial \kbd{a = x+y},

intended to be quick and dirty. So if you try \kbd{Pol(a,y)} on the

you will get \kbd{y+y}, which is not a valid PARI object.

polynomial \kbd{a = x+y}, you will get \kbd{y+y}, which is not a valid PARI

object.

\syn{gtopoly}{x,v}, where $v$ is a variable number.

Line 479 Negative numbers behave as if modulo a huge power of $

Line 571 Negative numbers behave as if modulo a huge power of $

\syn{gbitnegimply}{x,y}.

\subsecidx{bitor}$(x,y)$: bitwise (inclusive) \tet{or}\sidx{bitwise

\subsecidx{bitor}$(x,y)$: \sidx{bitwise inclusive or}bitwise (inclusive)

inclusive or} of two integers $x$ and $y$, that is the integer

\tet{or} of two integers $x$ and $y$, that is the integer

$$\sum (x_i~\kbd{or}~y_i) 2^i$$

Negative numbers behave as if modulo a huge power of $2$.

\syn{gbitor}{x,y}.

\subsecidx{bittest}$(x,n)$: outputs the $n^{\text{th}}$ bit of $|x|$ starting

\subsecidx{bittest}$(x,n,\{c=1\})$: extracts $|c|$ bits starting from

from the right (i.e.~the coefficient of $2^n$ in the binary expansion of $x$).

number $n$ from the right in the development of $|x|$ (i.e.~the coefficient

The result is 0 or 1. To extract several bits at once as a vector, pass a

of $2^n$ in the binary expansion of $x$), returning the bits as an integer

vector for $n$.

bitmap. That is, if $x = \sum x_i 2^i$ with the $x_i$ in $\{0,1\}$, this

routine returns the integer

$$ \sum_{0\leq i < |c|} x_{n + i} 2^i $$

Bits at negative offsets are 0. A negative value of $c$ means that negative

values of $x$ are treated in the spirit of $2$-complement arithmetic (i.e

modulo a big power of $2$). To extract several bits (or groups of bits if

$|c|>1$) separately at once as a vector, pass a vector for $n$.

\syn{bittest}{x,n}, where $n$ and the result are \kbd{long}s.

\syn{gbittest3}{x,n,c}. Also available are $\teb{gbittest}(x,n)$

(default case $c=1$) and for simple cases $\teb{bittest}(x,n)$, where $n$

and the result are \kbd{long}s.

\subsecidx{bitxor}$(x,y)$: bitwise (exclusive) \tet{or}\sidx{bitwise

exclusive or} of two integers $x$ and $y$, that is the integer

Line 633 the degree of a polynomial is equal to its length minu

Line 733 the degree of a polynomial is equal to its length minu

\subsecidx{lift}$(x,\{v\})$: lifts an element $x=a \bmod n$ of $\Z/n\Z$ to

$a$ in $\Z$, and similarly lifts a polmod to a polynomial if $v$ is omitted.

Otherwise, lifts only polmods with main variable $v$ (if $v$ does not occur

Otherwise, lifts only polmods whose modulus has main variable $v$ (if $v$

in $x$, lifts only intmods). If $x$ is of type fraction, complex, quadratic,

does not occur in $x$, lifts only intmods). If $x$ is of recursive (non

polynomial, power series, rational function, vector or matrix, the lift is

modular) type, the lift is done coefficientwise. For $p$-adics, this routine

done for each coefficient. For $p$-adics, this routine acts as

acts as \tet{truncate}. It is not allowed to have $x$ of type \typ{REAL}.

\tet{truncate}. It is not allowed to have $x$ of type \typ{REAL}.

\bprog

? lift(Mod(5,3))

%1 = 2

? lift(3 + O(3^9))

%2 = 3

? lift(Mod(x,x^2+1))

%3 = x

? lift(x * Mod(1,3) + Mod(2,3))

%4 = x + 2

? lift(x * Mod(y,y^2+1) + Mod(2,3))

%5 = y*x + Mod(2, 3) \\@com do you understand this one ?

? lift(x * Mod(y,y^2+1) + Mod(2,3), x)

%6 = Mod(y, y^2+1) * x + Mod(2, y^2+1)

@eprog

\syn{lift0}{x,v}, where $v$ is a \kbd{long} and an omitted $v$ is coded as

$-1$. Also available is \teb{lift}$(x)$ = \kbd{lift0($x$,-1)}.

Line 1012 small.

Line 1126 small.

\subsecidx{besselk}$(\var{nu},x,\{\fl=0\})$: $K$-Bessel function of index

\var{nu} (which can be complex) and argument $x$. Only real and positive

arguments

arguments $x$ are allowed in the present version \vers. If $\fl$ is equal to

$x$ are allowed in the present version \vers. If $\fl$ is equal to 1,

1, uses another implementation of this function which is often faster. In the

uses another implementation of this function which is often faster.

present version \vers, this function is not very accurate when $x$ is small.

\syn{kbessel}{\var{nu},x,\var{prec}} and

$\teb{kbessel2}(\var{nu},x,\var{prec})$ respectively.

Line 1047 repeatedly calling \kbd{eint1($i$ * x)}.

Line 1161 repeatedly calling \kbd{eint1($i$ * x)}.

$\teb{eint1}(x,\var{prec})$.

\subsecidx{erfc}$(x)$: complementary error function

$(2/\sqrt\pi)\int_x^\infty e^{-t^2}\,dt$.

$(2/\sqrt\pi)\int_x^\infty e^{-t^2}\,dt$ ($x\in\R$).

\syn{erfc}{x,\var{prec}}.

Line 1111 $\ln(p)=0$. Hence in particular $\exp(\ln(x))/x$ will

Line 1225 $\ln(p)=0$. Hence in particular $\exp(\ln(x))/x$ will

equal to 1 but to a $(p-1)$-th root of unity (or $\pm1$ if $p=2$)

times a power of $p$.

If $\fl$ is equal to 1, use an agm formula suggested by Mestre, when $x$ is

If $\fl$ is equal to $1 = \kbd{AGM}$, use an agm formula suggested by Mestre,

real, otherwise identical to \kbd{log}.

when $x$ is real, otherwise identical to \kbd{log}.

\syn{glog}{x,\var{prec}} or $\teb{glogagm}(x,\var{prec})$.

Line 1370 Here $y$ must be an integer, but $x$ can be any PARI o

Line 1484 Here $y$ must be an integer, but $x$ can be any PARI o

\syn{binome}{x,y}, where $y$ must be a \kbd{long}.

\subsecidx{chinese}$(x,y)$: if $x$ and $y$ are both integermods or both

\subsecidx{chinese}$(x,\{y\})$: if $x$ and $y$ are both integermods or both

polmods, creates (with the same type) a $z$ in the same residue class

as $x$ and in the same residue class as $y$, if it is possible.

This function also allows vector and matrix arguments, in which case the

operation is recursively applied to each component of the vector or matrix.

For polynomial arguments, it is applied to each coefficient. Finally

For polynomial arguments, it is applied to each coefficient.

$\kbd{chinese}(x,x) = x$ regardless of the type of $x$; this allows vector

arguments to contain other data, so long as they are identical in both

If $y$ is omitted, and $x$ is a vector, \kbd{chinese} is applied recursively

to the components of $x$, yielding a residue belonging to the same class as all

components of $x$.

Finally $\kbd{chinese}(x,x) = x$ regardless of the type of $x$; this allows

vector arguments to contain other data, so long as they are identical in both

vectors.

\syn{chinois}{x,y}.

\subsecidx{content}$(x)$: computes the gcd of all the coefficients of $x$,

when this gcd makes sense. If $x$ is a scalar, this simply returns $x$. If $x$

when this gcd makes sense. If $x$ is a scalar, this simply returns the

is a polynomial (and by extension a power series), it gives the usual content

absolute value of $x$ if $x$ is rationnal (\typ{INT}, \typ{FRAC} or

of $x$. If $x$ is a rational function, it gives the ratio of the contents of

\typ{FRACN}), and $x$ otherwise. If $x$ is a polynomial (and by extension a

the numerator and the denominator. Finally, if $x$ is a vector or a matrix,

power series), it gives the usual content of $x$. If $x$ is a rational

it gives the gcd of all the entries.

function, it gives the ratio of the contents of the numerator and the

denominator. Finally, if $x$ is a vector or a matrix, it gives the gcd of all

the entries.

\syn{content}{x}.

\subsecidx{contfrac}$(x,\{b\},\{lmax\})$: creates the row vector whose

\subsecidx{contfrac}$(x,\{b\},\{nmax\})$: creates the row vector whose

components are the partial quotients of the \idx{continued fraction}

expansion of $x$, the number of partial quotients being limited to $lmax$.

expansion of $x$. That is a result $[a_0,\dots,a_n]$ means that

If $x$ is a real number, the expansion stops at the last significant partial

$x \approx a_0+1/(a_1+\dots+1/a_n)\dots)$. The output is normalized so that

quotient if $lmax$ is omitted. $x$ can also be a rational function or a power

$a_n \neq 1$ (unless we also have $n = 0$).

series.

The number of partial quotients $n$ is limited to $nmax$. If $x$ is a real

number, the expansion stops at the last significant partial quotient if $nmax$

is omitted. $x$ can also be a rational function or a power series.

If a vector $b$ is supplied, the numerators will be equal to the coefficients

of $b$. The length of the result is then equal to the length of $b$, unless a

of $b$ (instead of all equal to $1$ as above). The length of the result is

partial remainder is encountered which is equal to zero. In which case the

then equal to the length of $b$, unless a partial remainder is encountered

expansion stops. In the case of real numbers, the stopping criterion is thus

which is equal to zero, in which case the expansion stops. In the case of real

different from the one mentioned above since, if $b$ is too long, some partial

numbers, the stopping criterion is thus different from the one mentioned above

quotients may not be significant.

since, if $b$ is too long, some partial quotients may not be significant.

If $b$ is an integer, the command is understood as \kbd{contfrac($x,lmax$)}.

If $b$ is an integer, the command is understood as \kbd{contfrac($x,nmax$)}.

\syn{contfrac0}{x,b,lmax}. Also available are

\syn{contfrac0}{x,b,nmax}. Also available are

$\teb{gboundcf}(x,lmax)$, $\teb{gcf}(x)$, or $\teb{gcf2}(b,x)$, where $lmax$

$\teb{gboundcf}(x,nmax)$, $\teb{gcf}(x)$, or $\teb{gcf2}(b,x)$, where $nmax$

is a C integer.

\subsecidx{contfracpnqn}$(x)$: when $x$ is a vector or a one-row matrix, $x$

Line 1522 Also available are

Line 1646 Also available are

$\teb{factor}(x)$ (= $\teb{factor0}(x,-1)$),

$\teb{smallfact}(x)$ (= $\teb{factor0}(x,0)$).

\subsecidx{factorback}$(f,\{nf\})$: $f$ being any factorization, gives back

\subsecidx{factorback}$(f,\{e\},\{nf\})$: gives back the factored object

the factored object. If a second argument $\var{nf}$ is supplied, $f$ is

corresponding to a factorization. If the last argument is of number field

assumed to be a prime ideal factorization in the number field $\var{nf}$.

type (e.g.~created by \kbd{nfinit} or \kbd{bnfinit}), assume we are dealing

The resulting ideal is given in HNF\sidx{Hermite normal form} form.

with an ideal factorization in the number field. The resulting ideal product is

given in HNF\sidx{Hermite normal form} form.

\syn{factorback}{f,\var{nf}}, where an omitted

If $e$ is present, $e$ and $f$ must be vectors of the same length ($e$ being

integral), and the corresponding factorization is the product of the

$f[i]^{e[i]}$.

If not, and $f$ is vector, it is understood as in the preceding case with

$e$ a vector of 1 (the product of the $f[i]$ is returned). Finally,

$f$ can be a regular factorization, as produced with any \kbd{factor}

command. A few examples:

\bprog

? factorback([2,2; 3,1])

%1 = 12

? factorback([2,2], [3,1])

%2 = 12

? factorback([5,2,3])

%3 = 30

? factorback([2,2], [3,1], nfinit(x^3+2))

%4 =

[16 0 0]

[0 16 0]

[0 0 16]

? nf = nfinit(x^2+1); fa = idealfactor(nf, 10)

%5 =

[[2, [1, 1]~, 2, 1, [1, 1]~] 2]

[[5, [-2, 1]~, 1, 1, [2, 1]~] 1]

[[5, [2, 1]~, 1, 1, [-2, 1]~] 1]

? factorback(fa)

*** forbidden multiplication t_VEC * t_VEC.

? factorback(fa, nf)

%6 =

[10 0]

[0 10]

@eprog

In the fourth example, $2$ and $3$ are interpreted as principal ideals in a

cubic field. In the fifth one, \kbd{factorback(fa)} is meaningless since we

forgot to indicate the number field, and the entries in the first column of

\kbd{fa} can't be multiplied.

\syn{factorback0}{f,e,\var{nf}}, where an omitted

$\var{nf}$ or $e$ is entered as \kbd{NULL}. Also available is

\tet{factorback}$(f,\var{nf})$ (case $e = \kbd{NULL}$) where an omitted

$\var{nf}$ is entered as \kbd{NULL}.

\subsecidx{factorcantor}$(x,p)$: factors the polynomial $x$ modulo the

Line 1607 $\teb{simplefactmod}(x,p)$ (= $\teb{factormod}(x,p,1)$

Line 1777 $\teb{simplefactmod}(x,p)$ (= $\teb{factormod}(x,p,1)$

\subsecidx{ffinit}$(p,n,\{v=x\})$: computes a monic polynomial of degree

$n$ which is irreducible over $\F_p$. For instance if

\kbd{P = ffinit(3,2,y)}, you can represent elements in $\F_{3^2}$ as polmods

modulo \kbd{P}.

modulo \kbd{P}. Starting with version 2.2.3 this function use a fast variant

of Adleman--Lenstra algorithm, and is much faster than in earlier versions.

\syn{ffinit}{p,n,v}, where $v$ is a variable number.

\subsecidx{gcd}$(x,y,\{\fl=0\})$: creates the greatest common divisor of $x$

\subsecidx{gcd}$(x,\{y\},\{\fl=0\})$: creates the greatest common divisor of $x$

and $y$. $x$ and $y$ can be of quite general types, for instance both

rational numbers. Vector/matrix types are also accepted, in which case

the GCD is taken recursively on each component. Note that for these

types, \kbd{gcd} is not commutative.

types, \kbd{gcd} is not commutative. \fl\ is obsolete and should not be used.

If $\fl=0$, use \idx{Euclid}'s algorithm.

If $y$ is omitted and $x$ is a vector, return the $\text{gcd}$ of all

components of $x$. The algorithm used is a naive \idx{Euclid} except for the

following inputs:

If $\fl=1$, use the modular gcd algorithm ($x$ and $y$ have to be

$\bullet$ integers: use modified right-shift binary (``plus-minus''

polynomials, with integer coefficients).

variant).

If $\fl=2$, use the \idx{subresultant algorithm}.

$\bullet$ univariate polynomials with coeffients in the same number

field (in particular rational): use modular gcd algorithm.

\syn{gcd0}{x,y,\fl}. Also available are

$\bullet$ general polynomials: use the \idx{subresultant algorithm} if

$\teb{ggcd}(x,y)$, $\teb{modulargcd}(x,y)$, and $\teb{srgcd}(x,y)$

coefficient explosion is likely (exact, non modular, coefficients).

corresponding to $\fl=0$, $1$ and $2$ respectively.

\syn{ggcd}{x,y}. For general polynomial inputs, $\teb{srgcd}(x,y)$ is also

available. For univariate {\it rational} polynomials, one also has

$\teb{modulargcd}(x,y)$.

\subsecidx{hilbert}$(x,y,\{p\})$: \idx{Hilbert symbol} of $x$ and $y$ modulo

$p$. If $x$ and $y$ are of type integer or fraction, an explicit third

parameter $p$ must be supplied, $p=0$ meaning the place at infinity.

Otherwise, $p$ needs not be given, and $x$ and $y$ can be of compatible types

integer, fraction, real, integermod or $p$-adic.

integer, fraction, real, integermod a prime (result is undefined if the

modulus is not prime), or $p$-adic.

\syn{hil}{x,y,p}.

Line 1643 discriminant of a quadratic field, false (0) otherwise

Line 1821 discriminant of a quadratic field, false (0) otherwise

simpler function $\teb{isfundamental}(x)$ which returns a \kbd{long}

should be used if $x$ is known to be of type integer.

\subsecidx{isprime}$(x,\{\fl=0\})$: if $\fl=0$ (default), true (1) if $x$ is a strong pseudo-prime

\subsecidx{isprime}$(x,\{\fl=0\})$: true (1) if $x$ is a (proven) prime

for 10 randomly chosen bases, false (0) otherwise.

number, false (0) otherwise. This can be very slow when $x$ is indeed

prime and has more than $1000$ digits, say. Use \tet{ispseudoprime} to

quickly check for pseudo primality.

If $\fl=1$, use Pocklington-Lehmer ``P-1'' test. true (1) if $x$ is

If $\fl=0$, use a combination of Baillie-PSW pseudo primality test (see

prime, false (0) otherwise.

\tet{ispseudoprime}), Selfridge ``$p-1$'' test if $x-1$ is smooth enough, and

Adleman-Pomerance-Rumely-Cohen-Lenstra (APRCL) for general $x$.

If $\fl=2$, use Pocklington-Lehmer ``P-1'' test and output a primality

If $\fl=1$, use Selfridge-Pocklington-Lehmer ``$p-1$'' test and output a

certificate as follows: return 0 if $x$ is composite, 1 if $x$ is a

primality certificate as follows: return 0 if $x$ is composite, 1 if $x$ is

small prime (currently strictly less than $341 550 071 728 321$), and

small enough that passing Baillie-PSW test guarantees its primality

a matrix if $x$ is a large prime. The matrix has three columns. The

(currently $x < 10^{13}$), $2$ if $x$ is a large prime whose primality could

first contains the prime factors $p$, the second the corresponding

only sensibly be proven (given the algorithms implemented in PARI) using the

elements $a_p$ as in Proposition~8.3.1 in GTM~138, and the third the

APRCL test. Otherwise ($x$ is large and $x-1$ is smooth) output a three

output of isprime(p,2).

column matrix as a primality certificate. The first column contains the prime

factors $p$ of $x-1$, the second the corresponding elements $a_p$ as in

Proposition~8.3.1 in GTM~138, and the third the output of isprime(p,1). The

algorithm fails if one of the pseudo-prime factors is not prime, which is

exceedingly unlikely (and well worth a bug report).

In the two last cases, the algorithm fails if one of the (strong

If $\fl=2$, use APRCL.

pseudo-)prime factors is not prime, but it should be exceedingly rare.

\syn{gisprime}{x,\fl}, but the simpler function $\teb{isprime}(x)$

which returns a \kbd{long} should be used if $x$ is known to be of

type integer. Also available is $\teb{plisprime}(N,\fl)$,

corresponding to $\teb{gisprime}(x,\fl+1)$ if $x$ is known to be of

type integer.

\subsecidx{ispseudoprime}$(x)$: true (1) if $x$ is a strong

\subsecidx{ispseudoprime}$(x,\{\fl\})$: true (1) if $x$ is a strong pseudo

pseudo-prime for a randomly chosen base, false (0) otherwise.

prime (see below), false (0) otherwise. If this function returns false, $x$

is not prime; if, on the other hand it returns true, it is only highly likely

that $x$ is a prime number. Use \tet{isprime} (which is of course much

slower) to prove that $x$ is indeed prime.

\syn{gispsp}{x}, but the

If $\fl = 0$, checks whether $x$ is a Baillie-Pomerance-Selfridge-Wagstaff

simpler function $\teb{ispsp}(x)$ which returns a \kbd{long}

pseudo prime (strong Rabin-Miller pseudo prime for base $2$, with

should be used if $x$ is known to be of type integer.

end-matching to catch square roots of $-1$, followed by strong Lucas test for

the sequence $(P,-1)$, $P$ smallest positive integer such that $P^2 - 4$ is

not a square mod $x$).

There are no known composite numbers passing this test, although it is

expected that infinitely such numbers exist.

If $\fl > 0$, checks whether $x$ is a strong Miller-Rabin pseudo prime (with

end-matching) for $\fl$ randomly chosen bases.

\syn{gispseudoprime}{x,\fl}, but the simpler function $\teb{ispseudoprime}(x)$

which returns a \kbd{long} should be used if $x$ is known to be of type

integer.

\subsecidx{issquare}$(x,\{\&n\})$: true (1) if $x$ is square, false (0) if

not. $x$ can be of any type. If $n$ is given and an exact square root had to

be computed in the checking process, puts that square root in $n$. This is in

Line 1700 must be of type integer.

Line 1896 must be of type integer.

\syn{kronecker}{x,y}, the result ($0$ or $\pm 1$) is a \kbd{long}.

\subsecidx{lcm}$(x,y)$: least common multiple of $x$ and $y$, i.e.~such

\subsecidx{lcm}$(x,\{y\})$: least common multiple of $x$ and $y$, i.e.~such

that $\text{lcm}(x,y)*\text{gcd}(x,y)=\text{abs}(x*y)$.

If $y$ is omitted and $x$ is a vector, return the $\text{lcm}$ of all

components of $x$.

\syn{glcm}{x,y}.

\subsecidx{moebius}$(x)$: \idx{Moebius} $\mu$-function of $|x|$. $x$ must

Line 1717 this function returns $x$ and not the smallest prime s

Line 1916 this function returns $x$ and not the smallest prime s

\syn{nextprime}{x}.

\subsecidx{numdiv}$(x)$: number of divisors of $|x|$. $x$ must be of type

integer, and the result is a \kbd{long}.

integer.

\syn{numbdiv}{x}.

Line 1743 are the first $x$ prime numbers, which must be among t

Line 1942 are the first $x$ prime numbers, which must be among t

\syn{primes}{x}. $x$ must be a \kbd{long}.

\subsecidx{qfbclassno}$(x,\{\fl=0\})$: class number of the quadratic field

\subsecidx{qfbclassno}$(D,\{\fl=0\})$: ordinary class number of the quadratic

of discriminant $x$. In the present version \vers, a simple algorithm is used

order of discriminant $D$. In the present version \vers, a $O(D^{1/2})$

for $x>0$, so $x$ should not be too large (say $x<10^7$) for the time to be

algorithm is used for $D > 0$ (using Euler product and the functional

reasonable. On the other hand, for $x<0$ one can reasonably compute

equation) so $D$ should not be too large, say $D < 10^8$, for the time to be

classno($x$) for $|x|<10^{25}$, since the method used is \idx{Shanks}' method

reasonable. On the other hand, for $D < 0$ one can reasonably compute

which is in $O(|x|^{1/4})$. For larger values of $|D|$, see

\kbd{qfbclassno($D$)} for $|D|<10^{25}$, since the routine uses

\kbd{quadclassunit}.

\idx{Shanks}'method which is in $O(|D|^{1/4})$. For larger values of $|D|$,

see \kbd{quadclassunit}, which unfortunately works reliably only for

fundamental discriminants.

If $\fl=1$, compute the class number using \idx{Euler product}s and the

functional equation. However, it is in $O(|x|^{1/2})$.

functional equation. However, it is in $O(|D|^{1/2})$.

\misctitle{Important warning.} For $D<0$, this function often gives

\misctitle{Important warning.} For $D < 0$, this function often gives

incorrect results when the class group is non-cyclic, because the authors

were too lazy to implement \idx{Shanks}' method completely. It is therefore

strongly recommended to use either the version with $\fl=1$, the function

strongly recommended to use either the version with $\fl = 1$, the function

$\kbd{qfhclassno}(-x)$ if $x$ is known to be a fundamental discriminant, or

$\kbd{qfbhclassno}(-D)$ if $D$ is known to be a fundamental discriminant, or

the function \kbd{quadclassunit}.

\syn{qfbclassno0}{x,\fl}. Also available are

\misctitle{Warning.} contrary to what its name implies, this routine does not

$\teb{classno}(x)$ (= $\teb{qfbclassno}(x)$),

compute the number of classes of binary primitive forms of discriminant $D$,

$\teb{classno2}(x)$ (= $\teb{qfbclassno}(x,1)$), and finally

which is equal to the \var{narrow} class number. The two notions are the same

there exists the function $\teb{hclassno}(x)$ which computes the class

when $D < 0$ or the fundamental unit $\varepsilon$ has negative norm; when $D

number of an imaginary quadratic field by counting reduced forms, an $O(|x|)$

> 0$ and $N\varepsilon > 0$, the number of classes of forms is twice the

ordinary class number. This is a problem which we cannot fix for backward

compatibility reasons. Use the following routine if you are only interested

in the number of classes of form:

\bprog

QFBclassno(D) =

qfbclassno(D) * if (D < 0 || norm(quadunit(D)) < 0, 1, 2)

@eprog

\noindent Here are a few examples, including an erroneous answer:

\bprog

? qfbclassno(400000028)

time = 3,140 ms.

%1 = 1

? quadclassunit(400000028).no

time = 20 ms. \\@com{ much faster}

%2 = 1

? qfbclassno(-400000028)

time = 0 ms.

%3 = 7253 \\@com{ correct, and fast enough}

? quadclassunit(-400000028).no

*** Warning: not a fundamental discriminant in quadclassunit.

time = 0 ms.

%4 = 7253

? qfbclassno(-2878367)

time = 0 ms.

%6 = 1056 \\@com{ This is wrong! Correct answer is $1152$}

(14:40) gp > quadclassunit(-2878367)

time = 10 ms.

%7 = [1152, [48, 24], [Qfb(647, 189, 1126), Qfb(2, 1, 359796)], 1, 1.000250]

@eprog

\syn{qfbclassno0}{D,\fl}. Also available:

$\teb{classno}(D)$ (= $\teb{qfbclassno}(D)$),

$\teb{classno2}(D)$ (= $\teb{qfbclassno}(D,1)$), and finally

there exists the function $\teb{hclassno}(D)$ which computes the class

number of an imaginary quadratic field by counting reduced forms, an $O(|D|)$

algorithm. See also \kbd{qfbhclassno}.

\subsecidx{qfbcompraw}$(x,y)$ \idx{composition} of the binary quadratic forms

Line 1783 non-negative and congruent to 0 or 3 modulo 4. See als

Line 2019 non-negative and congruent to 0 or 3 modulo 4. See als

definite binary quadratic forms $x$ and $y$ using the NUCOMP and NUDUPL

algorithms of \idx{Shanks} (\`a la Atkin). $l$ is any positive constant,

but for optimal speed, one should take $l=|D|^{1/4}$, where $D$ is the common

discriminant of $x$ and $y$.

discriminant of $x$ and $y$. When $x$ and $y$ do not have the same

discriminant, the result is undefined.

\syn{nucomp}{x,y,l}. The auxiliary function

$\teb{nudupl}(x,l)$ should be used instead for speed when $x=y$.

Line 2060 only available for curves defined over $\Q_p$.\smallsk

Line 2297 only available for curves defined over $\Q_p$.\smallsk

Some functions, in particular those relative to height computations (see

\kbd{ellheight}) require also that the curve be in minimal Weierstrass

form. This is achieved by the function \kbd{ellglobalred}.

form. This is achieved by the function \kbd{ellminimalmodel}.

All functions related to elliptic curves share the prefix \kbd{ell}, and the

precise curve we are interested in is always the first argument, in either

Line 2082 known to hold in full generality thanks to the work of

Line 2319 known to hold in full generality thanks to the work of

medium or long vector of the type given by \kbd{ellinit}. For this function

to work for every $n$ and not just those prime to the conductor, $E$ must

be a minimal Weierstrass equation. If this is not the case, use the

function \kbd{ellglobalred} first before using \kbd{ellak}.

function \kbd{ellminimalmodel} first before using \kbd{ellak}.

\syn{akell}{E,n}.

Line 2157 minimal model of $E$ and the global \idx{Tamagawa numb

Line 2394 minimal model of $E$ and the global \idx{Tamagawa numb

elliptic curve given by a medium or long vector of the type given by

\kbd{ellinit}, {\it and is supposed to have all its coefficients $a_i$ in}

$\Q$. The result is a 3 component vector $[N,v,c]$. $N$ is the arithmetic

conductor of the curve, $v$ is itself a vector $[u,r,s,t]$ with rational

conductor of the curve. $v$ gives the coordinate change for $E$ over $\Q$ to

components. It gives a coordinate change for $E$ over $\Q$ such that the

the minimal integral model (see \tet{ellminimalmodel}). Finally $c$ is the

resulting model has integral coefficients, is everywhere minimal, $a_1$ is 0

product of the local Tamagawa numbers $c_p$, a quantity which enters in the

or 1, $a_2$ is 0, 1 or $-1$ and $a_3$ is 0 or 1. Such a model is unique, and

\idx{Birch and Swinnerton-Dyer conjecture}.\sidx{minimal model}

the vector $v$ is unique if we specify that $u$ is positive. To get the new

model, simply type \kbd{ellchangecurve(E,v)}. Finally $c$ is the product of

the local Tamagawa numbers $c_p$, a quantity which enters in the

\idx{Birch and Swinnerton-Dyer conjecture}.

\syn{globalreduction}{E}.

Line 2297 $O(N^{1/2})$ algorithm.

Line 2530 $O(N^{1/2})$ algorithm.

\syn{lseriesell}{E,s,A,\var{prec}} where $\var{prec}$ is a \kbd{long} and an

omitted $A$ is coded as \kbd{NULL}.

\subsecidx{ellminimalmodel}$(E,\{\&v\})$: return the standard minimal

integral model of the rational elliptic curve $E$. If present, sets $v$ to the

corresponding change of variables, which is a vector $[u,r,s,t]$ with

rational components. The return value is identical to that of

\kbd{ellchangecurve(E, v)}.

The resulting model has integral coefficients, is everywhere minimal, $a_1$

is 0 or 1, $a_2$ is 0, 1 or $-1$ and $a_3$ is 0 or 1. Such a model is unique,

and the vector $v$ is unique if we specify that $u$ is positive, which we do.

\sidx{minimal model}

\syn{ellminimalmodel}{E,\&v}, where an omitted $v$ is coded as \kbd{NULL}.

\subsecidx{ellorder}$(E,z)$: gives the order of the point $z$ on the elliptic

curve $E$ if it is a torsion point, zero otherwise. In the present version

\vers, this is implemented only for elliptic curves defined over $\Q$.

Line 2364 $H/\Gamma_0(N)$ where $N$ is the conductor of the curv

Line 2610 $H/\Gamma_0(N)$ where $N$ is the conductor of the curv

the power series for $u$ and $v$ is $x$, which is implicitly understood to be

equal to $\exp(2i\pi z)$. It is assumed that the curve is a \var{strong}

\idx{Weil curve}, and the Manin constant is equal to 1. The equation of

the curve $E$ must be minimal (use \kbd{ellglobalred} to get a minimal

the curve $E$ must be minimal (use \kbd{ellminimalmodel} to get a minimal

equation).

\syn{taniyama}{E}, and the precision of the result is determined by the

Line 2445 associated to the number field: signature, maximal ord

Line 2691 associated to the number field: signature, maximal ord

$\bullet$ $\tev{bnf}$ denotes a big number field, i.e.~a 10-component

vector in the format output by \tet{bnfinit}. This contains $\var{nf}$ and

the deeper invariants of the field: units, class groups, as well as a lot of

technical data necessary for some complex fonctions like \kbd{bnfisprincipal}.

technical data necessary for some complex functions like \kbd{bnfisprincipal}.

$\bullet$ $\tev{bnr}$ denotes a big ``ray number field'', i.e.~some data

structure output by \kbd{bnrinit}, even more complicated than $\var{bnf}$,

Line 2531 determinant of a pseudo-matrix is the determinant of a

Line 2777 determinant of a pseudo-matrix is the determinant of a

module it generates.

Finally, when defining a relative extension, the base field should be

defined by a variable having a lower priority (i.e.~a higher number)

defined by a variable having a lower priority (i.e.~a higher number, see

than the variable defining the extension. For example, under GP you can

\secref{se:priority}) than the variable defining the extension. For example,

use the variable name $y$ (or $t$) to define the base field, and the

under GP you can use the variable name $y$ (or $t$) to define the base field,

variable name $x$ to define the relative extension.

and the variable name $x$ to define the relative extension.

Now a last set of definitions concerning the way big ray number fields

(or \var{bnr}) are input, using class field theory.

Line 2615 fundamental units, $w$ generates the torsion.\cr

Line 2861 fundamental units, $w$ generates the torsion.\cr

\+\tet{reg} &(\var{bnr},& \var{bnf},&&)&: & regulator.\cr

\+\tet{roots}&(\var{bnr},& \var{bnf},& \var{nf}&)&: & roots of the

polnomial generating the field.\cr

polynomial generating the field.\cr

\+\tet{sign} &(\var{bnr},& \var{bnf},& \var{nf}&)&: & $[r_1,r_2]$ the

signature of the field. This means that the field has $r_1$ real \cr

Line 2653 finding class and unit groups under \idx{GRH}, due to

Line 2899 finding class and unit groups under \idx{GRH}, due to

The general call to the functions concerning class groups of general number

fields (i.e.~excluding \kbd{quadclassunit}) involves a polynomial $P$ and a

technical vector

$$\var{tech} = [c,c2,\var{nrel},\var{borne},\var{nrpid},\var{minsfb}],$$

$$\var{tech} = [c,c2,\var{nrel},\var{small\_norm},\var{nrpid},\var{minsfb}],$$

where the parameters are to be understood as follows:

$P$ is the defining polynomial for the number field, which must be in

Line 2670 quadratic case, but then you should use the much faste

Line 2916 quadratic case, but then you should use the much faste

\kbd{quadclassunit}). Reasonable values for $c$ are between $0.1$ and

$2$. (The defaults are $c=c2=0.3$).

$\var{nrel}$ is the number of initial extra relations requested in

$\var{nrel}$ is the number of initial extra relations requested in computing

computing the

the relation matrix. Reasonable values are between 5 and 20. (The default is

relation matrix. Reasonable values are between 5 and 20. (The default is 5).

5).

$\var{borne}$ is a multiplicative coefficient of the Minkowski bound which

$\var{small\_norm}$ is a toggle used to disable the search for small norm

controls

relations. If this parameter is set equal to $0$, the program does not search

the search for small norm relations. If this parameter is set equal to 0, the

for small norm relations (default value is $1$). {\sl This is obsolete and

program does not search for small norm relations. Otherwise reasonable values

should not be used. Set $\var{nrpid} = 0$ instead}.

are between $0.5$ and $2.0$. (The default is $1.0$).

$\var{nrpid}$ is the maximal number of small norm relations associated to each

ideal in the factor base. Irrelevant when $\var{borne}=0$. Otherwise,

ideal in the factor base. Irrelevant when $\var{small\_norm}=0$. Otherwise,

reasonable values are between 4 and 20. (The default is 4).

$\var{minsfb}$ is the minimal number of elements in the ``sub-factorbase''.

If the

If the program does not seem to succeed in finding a full rank matrix (which

program does not seem to succeed in finding a full rank matrix (which you can

you can see in GP by typing \kbd{\bs g 2}), increase this number. Reasonable

see in GP by typing \kbd{\bs g 2}), increase this number. Reasonable values

values are between 2 and 5. (The default is 3).

are between 2 and 5. (The default is 3).

\misctitle{Remarks.}

Line 2815 Theory}, Graduate Texts in Maths \key{138}, Springer-V

Line 3059 Theory}, Graduate Texts in Maths \key{138}, Springer-V

$\var{bnf}[1]$ contains the matrix $W$, i.e.~the matrix in Hermite normal

form giving relations for the class group on prime ideal generators

$(\p_i)_{1\le i\le r}$.

$(\wp_i)_{1\le i\le r}$.

$\var{bnf}[2]$ contains the matrix $B$, i.e.~the matrix containing the

expressions of the prime ideal factorbase in terms of the $\p_i$. It is an

expressions of the prime ideal factorbase in terms of the $\wp_i$. It is an

$r\times c$ matrix.

$\var{bnf}[3]$ contains the complex logarithmic embeddings of the system of

Line 2831 relations of the matrix $(W|B)$.

Line 3075 relations of the matrix $(W|B)$.

$\var{bnf}[5]$ contains the prime factor base, i.e.~the list of prime

ideals used in finding the relations.

$\var{bnf}[6]$ contains the permutation of the prime factor base which was

$\var{bnf}[6]$ used to contain a permutation of the prime factor base, but

necessary to reduce the relation matrix to the form explained in subsection

has been obsoleted. It contains a dummy $0$.

6.5.5 of GTM~138 (i.e.~with a big $c\times c$ identity matrix on the lower

right). Note that in the above mentioned book, the need to permute the rows

of the relation matrices which occur was not emphasized.

$\var{bnf}[9]$ is a 3-element row vector used in \tet{bnfisprincipal} only

and obtained as follows. Let $D = U W V$ obtained by applying the

\idx{Smith normal form} algorithm to the matrix $W$ (= $\var{bnf}[1]$) and

let $U_r$ be the reduction of $U$ modulo $D$. The first elements of the

factorbase are given (in terms of \kbd{bnf.gen}) by the columns of $U_r$,

with archimedian component $g_a$; let also $GD_a$ be the archimedian

with Archimedean component $g_a$; let also $GD_a$ be the Archimedean

components of the generators of the (principal) ideals defined by the

\kbd{bnf.gen[i]\pow bnf.cyc[i]}. Then $\var{bnf}[9]=[U_r, g_a, GD_a]$.

Line 2947 simple true/false answer: it gives a row vector $[v_1,

Line 3188 simple true/false answer: it gives a row vector $[v_1,

$v_1$ is the vector of components $c_i$ of the class of the ideal $x$ in the

class group, expressed on the generators $g_i$ given by \kbd{bnfinit}

(specifically \kbd{\var{bnf}.clgp.gen} which is the same as

(specifically \kbd{\var{bnf}.gen}). The $c_i$ are chosen so that $0\le c_i<n_i$

\kbd{\var{bnf}[8][1][3]}). The $c_i$ are chosen so that $0\le c_i<n_i$

where $n_i$ is the order of $g_i$ (the vector of $n_i$ being \kbd{\var{bnf}.cyc}).

where $n_i$ is the order of $g_i$ (the vector of $n_i$ being

\kbd{\var{bnf}.clgp.cyc}, that is \kbd{\var{bnf}[8][1][2]}).

$v_2$ gives on the integral basis the components of $\alpha$ such that

$x=\alpha\prod_ig_i^{c_i}$. In particular, $x$ is principal if and only if

$v_1$ is equal to the zero vector, and if this the case $x=\alpha\Z_K$ where

$v_1$ is equal to the zero vector. In the latter case, $x = \alpha\Z_K$ where

$\alpha$ is given by $v_2$. Note that if $\alpha$ is too large to be given, a

warning message will be printed and $v_2$ will be set equal to the empty

vector.

Line 3021 special case of \kbd{bnrclass}.

Line 3260 special case of \kbd{bnrclass}.

\subsecidx{bnfsignunit}$(\var{bnf})$: $\var{bnf}$ being a big number field

output by \kbd{bnfinit}, this computes an $r_1\times(r_1+r_2-1)$ matrix

having $\pm1$ components, giving the signs of the real embeddings of the

fundamental units.

fundamental units. The following functions compute generators for the totally

positive units:

\bprog

/* exponents of totally positive units generators on bnf.tufu */

tpuexpo(bnf)=

{ local(S,d,K);

S = bnfsignunit(bnf); d = matsize(S);

S = matrix(d[1],d[2], i,j, if (S[i,j] < 0, 1,0));

S = concat(S, vectorv(d[1],i,1)); \\ sign(-1)

K = lift(matker(S * Mod(1,2)));

if (K, mathnfmodid(K, 2), 2*matid(d[1]))

}

/* totally positive units */

tpu(bnf)=

{ local(vu, ex = tpuexpo(bnf));

vu = nfbasistoalg(bnf, bnf.tufu);

vector(length(ex)-1, i, factorback([vu, ex[,i+1]])) \\ ex[,1] is 1

}

@eprog

\syn{signunits}{\var{bnf}}.

\subsecidx{bnfreg}$(\var{bnf})$: $\var{bnf}$ being a big number field

Line 3057 $v[6]$ is a copy of $S$.

Line 3318 $v[6]$ is a copy of $S$.

\syn{bnfsunit}{\var{bnf},S,\var{prec}}.

\subsecidx{bnfunit}$(\var{bnf})$: $\var{bnf}$ being a big number field as

output by

output by \kbd{bnfinit}, outputs a two-component row vector giving in the

\kbd{bnfinit}, outputs a two-component row vector giving in the first

first component the vector of fundamental units of the number field, and in

component the vector of fundamental units of the number field, and in the

the second component the number of bit of accuracy which remained in the

second component the number of bit of accuracy which remained in the

computation (which is always correct, otherwise an error message is printed).

This function is mainly for people who used the wrong flag in \kbd{bnfinit}

and would like to skip part of a lengthy \kbd{bnfinit} computation.

\syn{buchfu}{\var{bnf}}.

\subsecidx{bnrL1}$(\var{bnr},\var{subgroup},\{\fl=0\})$:

\subsecidx{bnrL1}$(\var{bnr},\{\var{subgroup}\},\{\fl=0\})$:

\var{bnr} being the number field data which is output by

\kbd{bnrinit(,,1)} and \var{subgroup} being a square matrix defining a

congruence subgroup of the ray class group corresponding to \var{bnr}

(or $0$ for the trivial congruence subgroup), returns for each

(the trivial congruence subgroup if omitted), returns for each

\idx{character} $\chi$ of the ray class group which is trivial on this

subgroup, the value at $s = 1$ (or $s = 0$) of the abelian

$L$-function associated to $\chi$. For the value at $s = 0$, the

Line 3095 Example:

Line 3355 Example:

\bprog

bnf = bnfinit(x^2 - 229);

bnr = bnrinit(bnf,1,1);

bnrL1(bnr, 0)

bnrL1(bnr)

@eprog\noindent

returns the order and the first non-zero term of the abelian

$L$-functions $L(s, \chi)$ at $s = 0$ where $\chi$ runs through the

characters of the class group of $\Q(\sqrt{229})$. Then

\bprog

bnr2 = bnrinit(bnf,2,1);

bnrL1(bnr2,0,2)

bnrL1(bnr2,,2)

@eprog\noindent

returns the order and the first non-zero terms of the abelian

$L$-functions $L_S(s, \chi)$ at $s = 0$ where $\chi$ runs through the

Line 3112 $2$ (note that the ray class group modulo $2$ is in fa

Line 3372 $2$ (note that the ray class group modulo $2$ is in fa

group, so \kbd{bnrL1(bnr2,0)} returns exactly the same answer as

\kbd{bnrL1(bnr,0)}!).

\syn{bnrL1}{\var{bnr},\var{subgroup},\fl,\var{prec}}

\syn{bnrL1}{\var{bnr},\var{subgroup},\fl,\var{prec}}, where an omitted

\var{subgroup} is coded as \kbd{NULL}.

\subsecidx{bnrclass}$(\var{bnf},\var{ideal},\{\fl=0\})$:

$\var{bnf}$ being a big number field

Line 3170 the conductor of this character as a modulus.

Line 3431 the conductor of this character as a modulus.

\syn{bnrconductorofchar}{\var{bnr},\var{chi}}.

\subsecidx{bnrdisc}$(a1,\{a2\},\{a3\},\{\fl=0\})$: $a1$, $a2$, $a3$

defining a big ray number field $L$ over a groud field $K$ (see \kbd{bnr}

defining a big ray number field $L$ over a ground field $K$ (see \kbd{bnr}

at the beginning of this section for the

meaning of $a1$, $a2$, $a3$), outputs a 3-component row vector $[N,R_1,D]$,

where $N$ is the (absolute) degree of $L$, $R_1$ the number of real places of

Line 3285 field and $\zeta = e^{2i\pi/N}$ where $N$ is the order

Line 3546 field and $\zeta = e^{2i\pi/N}$ where $N$ is the order

\syn{bnrrootnumber}{\var{bnf},\var{chi},\fl}

\subsecidx{bnrstark}${(\var{bnr},\var{subgroup},\{\fl=0\})}$: \var{bnr}

\subsecidx{bnrstark}${(\var{bnr},\{\var{subgroup}\},\{\fl=0\})}$: \var{bnr}

being as output by \kbd{bnrinit(,,1)}, finds a relative equation for the

class field corresponding to the modulus in \var{bnr} and the given

congruence subgroup using \idx{Stark units} (set $\var{subgroup}=0$ if you

congruence subgroup using \idx{Stark units} (omit $\var{subgroup}=0$ if you

want the whole ray class group). The main variable of \var{bnr} must not be

$x$, and the ground field and the class field must be totally real and not

$x$, and the ground field and the class field must be totally real. When the

isomorphic to $\Q$ (over the rationnals, use \tet{polsubcyclo} or

base field is $\Q$, the vastly simpler \tet{galoissubcyclo} is used instead.

\tet{galoissubcyclo}). \fl\ is optional and may be set to 0 to obtain a

reduced relative polynomial, 1 to be satisfied with any relative

polynomial, 2 to obtain an absolute polynomial and 3 to obtain the

irreducible relative polynomial of the Stark unit, 0 being default.

Example:

\fl\ is optional and may be set to 0 (default) to obtain a reduced relative

polynomial, 1 to be satisfied with any relative polynomial, 2 to obtain an

absolute polynomial and 3 to obtain the irreducible relative polynomial of

the Stark unit. Example:

\bprog

bnf = bnfinit(y^2 - 3);

bnr = bnrinit(bnf, 5, 1);

Line 3319 In this case, the corresponding congruence group is a

Line 3579 In this case, the corresponding congruence group is a

groups and, for the time being, the class field has to be obtained by

splitting this group into its cyclic components.

\syn{bnrstark}{\var{bnr},\var{subgroup},\fl}.

\syn{bnrstark}{\var{bnr},\var{subgroup},\fl}, where an omitted \var{subgroup}

is coded by \kbd{NULL}.

\subsecidx{dirzetak}$(\var{nf},b)$: gives as a vector the first $b$

coefficients of the \idx{Dedekind} zeta function of the number field $\var{nf}$

Line 3330 considered as a \idx{Dirichlet series}.

Line 3591 considered as a \idx{Dirichlet series}.

\subsecidx{factornf}$(x,t)$: factorization of the univariate polynomial $x$

over the number field defined by the (univariate) polynomial $t$. $x$ may

have coefficients in $\Q$ or in the number field. The main variable of

$t$ must be of \var{lower} priority than that of $x$ (in other words the

$t$ must be of \var{lower} priority than that of $x$ (see

variable number of $t$ must be \var{greater} than that of $x$). However if

\secref{se:priority}). However if

the coefficients of the number field occur explicitly (as polmods) as

coefficients of $x$, the variable of these polmods \var{must} be the same as

the main variable of $t$. For example

Line 3360 If $\fl=1$ return only the polynomial $P$.

Line 3621 If $\fl=1$ return only the polynomial $P$.

If $\fl=2$ return $[P,x,F]$ where $P$ and $x$ are as above and $F$ is the

factorization of $\var{gal}.pol$ over the field defined by $P$, where

variable $v$ ($y$ by default) stands for a root of $P$. The priority of $v$

must be less than the priority of the variable of $\var{gal}.pol$.

must be less than the priority of the variable of $\var{gal}.pol$ (see

\secref{se:priority}). Example:

Example:

\bprog

G = galoisinit(x^4+1);

galoisfixedfield(G,G.group[2],2)

Line 3374 computes the factorization $x^4+1=(x^2-\sqrt{-2}x-1)(

Line 3634 computes the factorization $x^4+1=(x^2-\sqrt{-2}x-1)(

\syn{galoisfixedfield}{\var{gal},\var{perm},p}.

\subsecidx{galoisinit}$(\var{pol},\{den\})$: computes the Galois group

and all neccessary information for computing the fixed fields of the

and all necessary information for computing the fixed fields of the

Galois extension $K/\Q$ where $K$ is the number field defined by

$\var{pol}$ (monic irreducible polynomial in $\Z[X]$ or

a number field as output by \tet{nfinit}). The extension $K/\Q$ must be

Line 3444 If present $den$ must be a suitable value for $\var{ga

Line 3704 If present $den$ must be a suitable value for $\var{ga

\syn{galoisinit}{\var{gal},\var{den}}.

\subsecidx{galoisisabelian}$(\var{gal},{fl=0})$: \var{gal} being as output by \kbd{galoisinit}, return $0$ if

\var{gal} is not an abelian group, and the HNF matrix of \var{gal} over \kbd{gal.gen} if $fl=0$, $1$ if

$fl=1$.

\syn{galoisisabelian}{\var{gal},\var{fl}} where \var{fl} is a C long integer.

\subsecidx{galoispermtopol}$(\var{gal},\var{perm})$: \var{gal} being a

Galois field as output by \kbd{galoisinit} and \var{perm} a element of

$\var{gal}.group$, return the polynomial defining the Galois

Line 3462 is greater or equal to $2$.

Line 3728 is greater or equal to $2$.

\syn{galoispermtopol}{\var{gal},\var{perm}}.

\subsecidx{galoissubcyclo}$(n,H,\{Z\},\{v\},\{fl=0\})$: If $fl=0$, compute a polynomial

\subsecidx{galoissubcyclo}$(N,H,\{fl=0\},\{v\})$: computes the subextension

defining the subfield of $\Q(\zeta_n)$ fixed by the subgroup \var{H} of

of $\Q(\zeta_n)$ fixed by the subgroup $H \subset (\Z/n\Z)^*$. By the

$(\Z/n\Z)^*$. The subgroup \var{H} can be given by a generator, a set of

Kronecker-Weber theorem, all abelian number fields can be generated in this

generators given by a vector or a HNF matrix. If present \kbd{Z} must be

way (uniquely if $n$ is taken to be minimal).

\kbd{znstar(n)}, and is currently only used when \var{H} is a HNF matrix. If

\var{v} is given, the polynomial is given in the variable \var{v}.

If $fl=1$, compute only the (finite part of) conductor of the abelian extension.

\noindent The pair $(n, H)$ is deduced from the parameters $(N, H)$ as follows

If $fl=2$, output $[pol, f_0]$, where $pol$ is the polynomial as output when $fl=0$ and $f_0$ the conductor as output when $fl=1$.

$\bullet$ $N$ an integer: then $n = N$; $H$ is a generator, i.e. an

integer or an integer modulo $n$; or a vector of generators.

The following function can be used to compute all subfields of

$\bullet$ $N$ the output of \kbd{znstar($n$)}. $H$ as in the first case

$\Q(\zeta_n)$ (of order less than \kbd{d}, if \kbd{d} is set):

above, or a matrix, taken to be a HNF left divisor of the SNF for $(\Z/n\Z)^*$

(of type \kbd{$N$.cyc}), giving the generators of $H$ in terms of \kbd{$N$.gen}.

$\bullet$ $N$ the output of \kbd{bnrinit(bnfinit(y), $m$, 1)} where $m$ is a

module. $H$ as in the first case, or a matrix taken to be a HNF left

divisor of the SNF for the ray class group modulo $m$

(of type \kbd{$N$.cyc}), giving the generators of $H$ in terms of \kbd{$N$.gen}.

In this last case, beware that $H$ is understood relatively to $N$; in

particular, if the infinite place does not divide the module, e.g if $m$ is

an integer, then it is not a subgroup of $(\Z/n\Z)^*$, but of its quotient by

$\{\pm 1\}$.

If $fl=0$, compute a polynomial (in the variable \var{v}) defining the

the subfield of $\Q(\zeta_n)$ fixed by the subgroup \var{H} of $(\Z/n\Z)^*$.

If $fl=1$, compute only the conductor of the abelian extension, as a module.

If $fl=2$, output $[pol, N]$, where $pol$ is the polynomial as output when

$fl=0$ and $N$ the conductor as output when $fl=1$.

The following function can be used to compute all subfields of

$\Q(\zeta_n)$ (of exact degree \kbd{d}, if \kbd{d} is set):

\bprog

subcyclo(n, d = -1)=

{

local(Z,G,S);

local(bnr,L,IndexBound);

if (d < 0, d = n);

IndexBound = if (d < 0, n, [d]);

Z = znstar(n);

bnr = bnrinit(bnfinit(y), [n,[1]], 1);

G = matdiagonal(Z[2]);

L = subgrouplist(bnr, IndexBound, 1);

S = [];

vector(#L,i, galoissubcyclo(bnr,L[i]));

forsubgroup(H = G, d,

S = concat(S, galoissubcyclo(n, mathnf(concat(G,H)),Z));

);

}

@eprog

@eprog\noindent

Setting \kbd{L = subgrouplist(bnr, [d])} would produce subfields of exact

conductor $n\infty$.

\misctitle{Note:} The interface to this function needs to be cleaned up, and

\syn{galoissubcyclo}{N,H,fl,v} where \var{fl} is a C long integer, and

so is subject to change.

\var{v} a variable number.

\syn{galoissubcyclo}{n,H,Z,v} where n is a C long integer.

\subsecidx{galoissubfields}$(G,\{fl=0\},\{v\})$: Output all the subfields of the Galois group \var{G},

as a vector.

This works by applying \kbd{galoisfixedfield} to all subgroups. The meaning of the flag \var{fl} is

the same as for \kbd{galoisfixedfield}.

\syn{galoissubfields}{\var{G},fl,v}, where \var{fl} is a long and \var{v} a variable number.

\subsecidx{galoissubgroups}$(G)$: Output all the subgroups of the Galois group \kbd{G}.

A subgroup is a vector $[gen, orders]$, with the same meaning as for $\var{gal}.gen$

and $\var{gal}.orders$. Hence \var{gen} is a vector of permutations generating the

subgroup, and \var{orders} is the relatives orders of the generators. The cardinal

of a subgroup is the product of the relative orders.

\syn{galoissubgroups}{\var{G}}.

\subsecidx{idealadd}$(\var{nf},x,y)$: sum of the two ideals $x$ and $y$ in the

number field $\var{nf}$. When $x$ and $y$ are given by $\Z$-bases, this does

not depend on $\var{nf}$ and can be used to compute the sum of any two

Line 3517 $1\le i\le k$ and $\sum_{1\le i\le k}e[i]=1$.

Line 3814 $1\le i\le k$ and $\sum_{1\le i\le k}e[i]=1$.

\subsecidx{idealappr}$(\var{nf},x,\{\fl=0\})$: if $x$ is a fractional ideal

(given in any form), gives an element $\alpha$ in $\var{nf}$ such that for

all prime ideals $\p$ such that the valuation of $x$ at $\p$ is non-zero, we

all prime ideals $\wp$ such that the valuation of $x$ at $\wp$ is non-zero, we

have $v_{\p}(\alpha)=v_{\p}(x)$, and. $v_{\p}(\alpha)\ge0$ for all other

have $v_{\wp}(\alpha)=v_{\wp}(x)$, and. $v_{\wp}(\alpha)\ge0$ for all other

${\p}$.

${\wp}$.

If $\fl$ is non-zero, $x$ must be given as a prime ideal factorization, as

output by \kbd{idealfactor}, but possibly with zero or negative exponents.

This yields an element $\alpha$ such that for all prime ideals $\p$ occurring

This yields an element $\alpha$ such that for all prime ideals $\wp$ occurring

in $x$, $v_{\p}(\alpha)$ is equal to the exponent of $\p$ in $x$, and for all

in $x$, $v_{\wp}(\alpha)$ is equal to the exponent of $\wp$ in $x$, and for all

other prime ideals, $v_{\p}(\alpha)\ge0$. This generalizes

other prime ideals, $v_{\wp}(\alpha)\ge0$. This generalizes

$\kbd{idealappr}(\var{nf},x,0)$ since zero exponents are allowed. Note that

the algorithm used is slightly different, so that

\kbd{idealappr(\var{nf},idealfactor(\var{nf},x))} may not be the same as

Line 3538 the algorithm used is slightly different, so that

Line 3835 the algorithm used is slightly different, so that

column integral exponents), $y$ a vector of elements in $\var{nf}$ indexed by

the ideals in $x$, computes an element $b$ such that

$v_\p(b - y_\p) \geq v_\p(x)$ for all prime ideals in $x$ and $v_\p(b)\geq 0$

$v_\wp(b - y_\wp) \geq v_\wp(x)$ for all prime ideals in $x$ and $v_\wp(b)\geq 0$

for all other $\p$.

for all other $\wp$.

\syn{idealchinese}{\var{nf},x,y}.

\subsecidx{idealcoprime}$(\var{nf},x,y)$: given two integral ideals $x$ and $y$

in the number field $\var{nf}$, finds a $\beta$ in the field, expressed on the

integral basis $\var{nf}[7]$, such that $\beta\cdot y$ is an integral ideal

integral basis $\var{nf}[7]$, such that $\beta\cdot x$ is an integral ideal

coprime to $x$.

coprime to $y$.

\syn{idealcoprime}{\var{nf},x}.

\syn{idealcoprime}{\var{nf},x,y}.

\subsecidx{idealdiv}$(\var{nf},x,y,\{\fl=0\})$: quotient $x\cdot y^{-1}$ of the

two ideals $x$ and $y$ in the number field $\var{nf}$. The result is given in

Line 3839 entries are expressed as a column vector on the integr

Line 4136 entries are expressed as a column vector on the integr

\syn{algtobasis}{\var{nf},x}.

\subsecidx{nfbasis}$(x,\{\fl=0\},\{p\})$: \idx{integral basis} of the number

\subsecidx{nfbasis}$(x,\{\fl=0\},\{fa\})$: \idx{integral basis} of the number

field defined by the irreducible, preferably monic, polynomial $x$,

field defined by the irreducible, preferably monic, polynomial $x$, using a

using a modified version of the \idx{round 4} algorithm by

modified version of the \idx{round 4} algorithm by default, due to David

default. The binary digits of $\fl$ have the following meaning:

\idx{Ford}, Sebastian \idx{Pauli} and Xavier \idx{Roblot}. The binary digits

of $\fl$ have the following meaning:

1: assume that no square of a prime greater than the default \kbd{primelimit}

divides the discriminant of $x$, i.e.~that the index of $x$ has only small

prime divisors.

2: use \idx{round 2} algorithm. For small degrees and coefficient size, this is

2: use \idx{round 2} algorithm. For small degrees and coefficient size, this

sometimes a little faster. (This program is the translation into C of a program

is sometimes a little faster. (This program is the translation into C of a

written by David \idx{Ford} in Algeb.)

program written by David \idx{Ford} in Algeb.)

Thus for instance, if $\fl=3$, this uses the round 2 algorithm and outputs

an order which will be maximal at all the small primes.

If $p$ is present, we assume (without checking!) that it is the two-column

If $fa$ is present, we assume (without checking!) that it is the two-column

matrix of the factorization of the discriminant of the polynomial $x$. Note

that it does \var{not} have to be a complete factorization. This is

especially useful if only a local integral basis for some small set of places

is desired: only factors with exponents greater or equal to 2 will be

considered.

\syn{nfbasis0}{x,\fl,p}. An extended version

\syn{nfbasis0}{x,\fl,fa}. An extended version

is $\teb{nfbasis}(x,\&d,\fl,p)$, where $d$ will receive the discriminant of

is $\teb{nfbasis}(x,\&d,\fl,fa)$, where $d$ will receive the discriminant of

the number field (\var{not} of the polynomial $x$), and an omitted $p$ should

the number field (\var{not} of the polynomial $x$), and an omitted $fa$ should

be input as \kbd{gzero}. Also available are $\teb{base}(x,\&d)$ ($\fl=0$),

be input as \kbd{NULL}. Also available are $\teb{base}(x,\&d)$ ($\fl=0$),

$\teb{base2}(x,\&d)$ ($\fl=2$) and $\teb{factoredbase}(x,p,\&d)$.

$\teb{base2}(x,\&d)$ ($\fl=2$) and $\teb{factoredbase}(x,fa,\&d)$.

\subsecidx{nfbasistoalg}$(\var{nf},x)$: this is the inverse function of

\kbd{nfalgtobasis}. Given an object $x$ whose entries are expressed on the

Line 3881 is particularly useful in conjunction with \kbd{nfhnfm

Line 4179 is particularly useful in conjunction with \kbd{nfhnfm

\syn{nfdetint}{\var{nf},x}.

\subsecidx{nfdisc}$(x,\{\fl=0\},\{p\})$: \idx{field discriminant} of the

\subsecidx{nfdisc}$(x,\{\fl=0\},\{fa\})$: \idx{field discriminant} of the

number field defined by the integral, preferably monic, irreducible

polynomial $x$. $\fl$ and $p$ are exactly as in \kbd{nfbasis}. That is, $p$

polynomial $x$. $\fl$ and $fa$ are exactly as in \kbd{nfbasis}. That is, $fa$

provides the matrix of a partial factorization of the discriminant of $x$,

and binary digits of $\fl$ are as follows:

1: assume that no square of a prime greater than \kbd{primelimit}

divides the discriminant.

2: use the round 2 algorithm, instead of the default \idx{round 4}.

2: use the round 2 algorithm, instead of the default \idx{round 4}. This

This should be

should be slower except maybe for polynomials of small degree and

slower except maybe for polynomials of small degree and coefficients.

coefficients.

\syn{nfdiscf0}{x,\fl,p} where, to omit $p$, you should input \kbd{gzero}. You

\syn{nfdiscf0}{x,\fl,fa} where, to omit $fa$, you should input \kbd{NULL}. You

can also use $\teb{discf}(x)$ ($\fl=0$).

\subsecidx{nfeltdiv}$(\var{nf},x,y)$: given two elements $x$ and $y$ in

Line 3967 an element $x$ of the number field $\var{nf}$ and a pr

Line 4265 an element $x$ of the number field $\var{nf}$ and a pr

\kbd{modpr} format compute a canonical representative for the class of $x$

modulo \var{pr}.

\syn{nfreducemodpr2}{\var{nf},x,\var{pr}}.

\syn{nfreducemodpr}{\var{nf},x,\var{pr}}.

\subsecidx{nfeltval}$(\var{nf},x,\var{pr})$: given an element $x$ in

\var{nf} and a prime ideal \var{pr} in the format output by

Line 3982 principal ideal), but it would be less efficient.

Line 4280 principal ideal), but it would be less efficient.

polynomial $x$ over the number field $\var{nf}$ given by \kbd{nfinit}. $x$

has coefficients in $\var{nf}$ (i.e.~either scalar, polmod, polynomial or

column vector). The main variable of $\var{nf}$ must be of \var{lower}

priority than that of $x$ (in other words, the variable number of $\var{nf}$

priority than that of $x$ (see \secref{se:priority}). However if

must be \var{greater} than that of $x$). However if the polynomial defining

the polynomial defining the number field occurs explicitly in the

the number field occurs explicitly in the coefficients of $x$ (as modulus of

coefficients of $x$ (as modulus of a \typ{POLMOD}), its main variable must be

a \typ{POLMOD}), its main variable must be \var{the same} as the main

\var{the same} as the main variable of $x$. For example,

variable of $x$. For example,

\bprog

? nf = nfinit(y^2 + 1);

? nffactor(nf, x^2 + y); \\@com OK

Line 4003 polmod, polynomial, column vector) or modulo the prime

Line 4300 polmod, polynomial, column vector) or modulo the prime

modulo the rational prime under \var{pr}, polmod or polynomial with

integermod coefficients, column vector of integermod). The prime ideal

\var{pr} \var{must} be in the format output by \kbd{idealprimedec}. The

main variable of $\var{nf}$ must be of lower priority than that of $x$ (in

main variable of $\var{nf}$ must be of lower priority than that of $x$

other words the variable number of $\var{nf}$ must be greater than that of

(see \secref{se:priority}). However if the coefficients of the number

$x$). However if the coefficients of the number field occur explicitly (as

field occur explicitly (as polmods) as coefficients of $x$, the variable of

polmods) as coefficients of $x$, the variable of these polmods \var{must}

these polmods \var{must} be the same as the main variable of $t$ (see

be the same as the main variable of $t$ (see \kbd{nffactor}).

\kbd{nffactor}).

\syn{nffactormod}{\var{nf},x,\var{pr}}.

Line 4101 preferably monic, irreducible polynomial in $\Z[X]$, i

Line 4398 preferably monic, irreducible polynomial in $\Z[X]$, i

by \var{pol}. As such, it's a technical object passed as the first argument

to most \kbd{nf}\var{xxx} functions, but it contains some information which

may be directly useful. Access to this information via \var{member

functions} is prefered since the specific data organization specified below

functions} is preferred since the specific data organization specified below

may change in the future. Currently, \kbd{nf} is a row vector with 9

components:

Line 4115 $\var{nf}[3]$ contains the discriminant $d(K)$ (\kbd{\

Line 4412 $\var{nf}[3]$ contains the discriminant $d(K)$ (\kbd{\

$\var{nf}[4]$ contains the index of $\var{nf}[1]$,

i.e.~$[\Z_K : \Z[\theta]]$, where $\theta$ is any root of $\var{nf}[1]$.

$\var{nf}[5]$ is a vector containing 7 matrices $M$, $MC$, $T2$, $T$,

$\var{nf}[5]$ is a vector containing 7 matrices $M$, $G$, $T2$, $T$,

$MD$, $TI$, $MDI$ useful for certain computations in the number field $K$.

\quad$\bullet$ $M$ is the $(r1+r2)\times n$ matrix whose columns represent

the numerical values of the conjugates of the elements of the integral

basis.

\quad$\bullet$ $MC$ is essentially the conjugate of the transpose of $M$,

\quad$\bullet$ $G$ is such that $T2 = {}^t G G$, where $T2$ is the quadratic

except that the last $r2$ columns are also multiplied by 2.

form $T_2(x) = \sum |\sigma(x)|^2$, $\sigma$ running over the embeddings of

$K$ into $\C$.

\quad$\bullet$ $T2$ is an $n\times n$ matrix equal to the real part of the

\quad$\bullet$ The $T2$ component is deprecated and currently unused.

product $MC\cdot M$ (which is a real positive definite symmetric matrix), the

so-called $T_2$-matrix (\kbd{\var{nf}.t2}).

\quad$\bullet$ $T$ is the $n\times n$ matrix whose coefficients are

$\text{Tr}(\omega_i\omega_j)$ where the $\omega_i$ are the elements of the

integral basis. Note that $T=\overline{MC}\cdot M$ and in particular that

integral basis. Note also that $\det(T)$ is equal to the discriminant of the

$T=T_2$ if the field is totally real (in practice $T_2$ will have real

field $K$.

approximate entries and $T$ will have integer entries). Note also that

$\det(T)$ is equal to the discriminant of the field $K$.

\quad$\bullet$ The columns of $MD$ (\kbd{\var{nf}.diff}) express a $\Z$-basis

of the different of $K$ on the integral basis.

Line 4153 $\var{nf}[6]$ is the vector containing the $r1+r2$ roo

Line 4447 $\var{nf}[6]$ is the vector containing the $r1+r2$ roo

embeddings of the number field into $\C$ (the first $r1$ components are real,

the next $r2$ have positive imaginary part).

$\var{nf}[7]$ is an integral basis in Hermite normal form for $\Z_K$

$\var{nf}[7]$ is an integral basis for $\Z_K$ (\kbd{\var{nf}.zk}) expressed

(\kbd{\var{nf}.zk}) expressed on the powers of~$\theta$.

on the powers of~$\theta$. Its first element is guaranteed to be $1$. This

basis is LLL-reduced with respect to $T_2$ (strictly speaking, it is a

permutation of such a basis, due to the condition that the first element be

$1$).

$\var{nf}[8]$ is the $n\times n$ integral matrix expressing the power

basis in terms of the integral basis, and finally

Line 4168 useful given the existence of \tet{nfnewprec}, to inpu

Line 4465 useful given the existence of \tet{nfnewprec}, to inpu

\kbd{bnf} instead of a polynomial.

The special input format $[x,B]$ is also accepted where $x$ is a polynomial

as above and $B$ is the integer basis, as computed by \tet{nfbasis}. This can

as above and $B$ is the integer basis, as would be computed by \tet{nfbasis}.

be useful since \kbd{nfinit} uses the round 4 algorithm by default, which can

This can be useful if the integer basis is known in advance.

be very slow in pathological cases where round 2 (\kbd{nfbasis(x,2)}) would

succeed very quickly.

If $\fl=2$: \var{pol} is changed into another polynomial $P$ defining the same

number field, which is as simple as can easily be found using the

Line 4246 $d$ of the number field defined by the (monic, integra

Line 4541 $d$ of the number field defined by the (monic, integra

\var{pol} (all subfields if $d$ is null or omitted). The result is a vector

of subfields, each being given by $[g,h]$, where $g$ is an absolute equation

and $h$ expresses one of the roots of $g$ in terms of the root $x$ of the

polynomial defining $\var{nf}$. This routine uses J.~Kl\"uners's algorithm,

polynomial defining $\var{nf}$. This routine uses J.~Kl\"uners's algorithm

our na�ve implementation would be very slow when no sufficiently inert primes

in the general case, and B.~Allombert's \tet{galoissubfields} when \var{nf}

can be found, e.g.~when \var{nf} is a compositum of many quadratic fields. So

is Galois (with weakly supersolvable Galois group).\sidx{Galois}\sidx{subfield}

it may abort with an error message (\kbd{too many block systems}) if it looks

like the computation is hopeless. This shall eventually be

corrected, using the ideas from \kbd{nfgaloisconj}.\sidx{Galois}\sidx{subfield}

%If the field is abelian, you can circonvene the problem by using

%\tet{galoisinit} and \tet{galoisfixedfield}, as in

%\bprog

% subf(pol) =

% {

% local(gal,G,H,h,l,gen, L = []);

% gl = galoisinit(pol);

% G = matdiagonal(Vec(g.orders));

% l = length(G); gen = Mat(vectorv(l,i, g.gen[i]));

% forsubgroup(H = G,,

% h = mathnf(concat(H,G));

% perm = vector(l,i, factorback(concat(gen, h[,i])));

% L = concat(L, [galoisfixedfield(g, perm)])

% ); L

% }

%@eprog

%which returns the same output as \kbd{nfsubfields(pol)}. For non-abelian

%Galois fields, you can still use \tet{galoisinit} and \teb{galoisfixedfield},

%but you will have to use another package than PARI to compute the subgroup

%lattice, since \tet{forsubgroup} is restricted to the abelian case.

\syn{subfields}{\var{nf},d}.

\subsecidx{nfroots}$(\var{nf},x)$: roots of the polynomial $x$ in the number

field $\var{nf}$ given by \kbd{nfinit} without multiplicity. $x$ has

coefficients in the number field (scalar, polmod, polynomial, column

vector). The main variable of $\var{nf}$ must be of lower priority than that

of $x$ (in other words the variable number of $\var{nf}$ must be greater than

of $x$ (see \secref{se:priority}). However if the coefficients of the

that of $x$). However if the coefficients of the number field occur

number field occur explicitly (as polmods) as coefficients of $x$, the

explicitly (as polmods) as coefficients of $x$, the variable of these

variable of these polmods \var{must} be the same as the main variable of $t$

polmods \var{must} be the same as the main variable of $t$ (see

(see \kbd{nffactor}).

\kbd{nffactor}).

\syn{nfroots}{\var{nf},x}.

Line 4364 are also implemented.

Line 4634 are also implemented.

The output is a 3-component vector $[n,s,k]$ with the following meaning: $n$

is the cardinality of the group, $s$ is its signature ($s=1$ if the group is

a subgroup of the alternating group $A_n$, $s=-1$ otherwise), and $k$ is the

a subgroup of the alternating group $A_n$, $s=-1$ otherwise).

number of the group corresponding to a given pair $(n,s)$ ($k=1$ except in 2

cases). Specifically, the groups are coded as follows, using standard

$k$ is more arbitrary and the choice made up to version~2.2.3 of PARI is rather

notations (see GTM 138, quoted at the beginning of this section; see also

unfortunate: for $n > 7$, $k$ is the numbering of the group among all

``The transitive groups of degree up to eleven'', by G.~Butler and J.~McKay

transitive subgroups of $S_n$, as given in ``The transitive groups of degree up

in Communications in Algebra, vol.~11, 1983, pp.~863--911):

to eleven'', G.~Butler and J.~McKay, Communications in Algebra, vol.~11, 1983,

pp.~863--911 (group $k$ is denoted $T_k$ there). And for $n \leq 7$, it was ad

hoc, so as to ensure that a given triple would design a unique group.

Specifically, for polynomials of degree $\leq 7$, the groups are coded as

follows, using standard notations

\smallskip

In degree 1: $S_1=[1,-1,1]$.

In degree 1: $S_1=[1,1,1]$.

\smallskip

In degree 2: $S_2=[2,-1,1]$.

\smallskip

Line 4393 In degree 7: $C_7=[7,1,1]$, $D_7=[14,-1,1]$, $M_{21}=[

Line 4667 In degree 7: $C_7=[7,1,1]$, $D_7=[14,-1,1]$, $M_{21}=[

$M_{42}=[42,-1,1]$, $PSL_2(7)=PSL_3(2)=[168,1,1]$, $A_7=[2520,1,1]$,

$S_7=[5040,-1,1]$.

\smallskip

The method used is that of resolvent polynomials and is sensitive to the

This is deprecated and obsolete, but for reasons of backward compatibility,

current precision. The precision is updated internally but, in very rare

we cannot change this behaviour yet. So you can use the default

cases, a wrong result may be returned if the initial precision was not

\tet{newgaloisformat} to switch to a consistent naming scheme, namely $k$ is

sufficient.

always the standard numbering of the group among all transitive subgroups of

$S_n$. If this default is in effect, the above groups will be coded as:

\smallskip

In degree 1: $S_1=[1,1,1]$.

\smallskip

In degree 2: $S_2=[2,-1,1]$.

\smallskip

In degree 3: $A_3=C_3=[3,1,1]$, $S_3=[6,-1,2]$.

\smallskip

In degree 4: $C_4=[4,-1,1]$, $V_4=[4,1,2]$, $D_4=[8,-1,3]$, $A_4=[12,1,4]$,

$S_4=[24,-1,5]$.

\smallskip

In degree 5: $C_5=[5,1,1]$, $D_5=[10,1,2]$, $M_{20}=[20,-1,3]$,

$A_5=[60,1,4]$, $S_5=[120,-1,5]$.

\smallskip

In degree 6: $C_6=[6,-1,1]$, $S_3=[6,-1,2]$, $D_6=[12,-1,3]$, $A_4=[12,1,4]$,

$G_{18}=[18,-1,5]$, $A_4\times C_2=[24,-1,6]$, $S_4^+=[24,1,7]$,

$S_4^-=[24,-1,8]$, $G_{36}^-=[36,-1,9]$, $G_{36}^+=[36,1,10]$,

$S_4\times C_2=[48,-1,11]$, $A_5=PSL_2(5)=[60,1,12]$, $G_{72}=[72,-1,13]$,

$S_5=PGL_2(5)=[120,-1,14]$, $A_6=[360,1,15]$, $S_6=[720,-1,16]$.

\smallskip

In degree 7: $C_7=[7,1,1]$, $D_7=[14,-1,2]$, $M_{21}=[21,1,3]$,

$M_{42}=[42,-1,4]$, $PSL_2(7)=PSL_3(2)=[168,1,5]$, $A_7=[2520,1,6]$,

$S_7=[5040,-1,7]$.

\smallskip

\syn{galois}{x,\var{prec}}.

\misctitle{Warning} The method used is that of resolvent polynomials and is

sensitive to the current precision. The precision is updated internally but,

in very rare cases, a wrong result may be returned if the initial precision

was not sufficient.

\subsecidx{polred}$(x,\{\fl=0\},\{p\})$: finds polynomials with reasonably

\syn{galois}{x,\var{prec}}. To enable the new format in library mode, set the

global variable \tet{new_galois_format} to $1$.

\subsecidx{polred}$(x,\{\fl=0\},\{fa\})$: finds polynomials with reasonably

small coefficients defining subfields of the number field defined by $x$.

One of the polynomials always defines $\Q$ (hence is equal to $x-1$),

and another always defines the same number field as $x$ if $x$ is irreducible.

Line 4409 polynomials, \kbd{nf}, \kbd{bnf}, \kbd{[x,Z\_K\_basis]

Line 4713 polynomials, \kbd{nf}, \kbd{bnf}, \kbd{[x,Z\_K\_basis]

The following binary digits of $\fl$ are significant:

1: does a partial reduction only. This means that only a suborder of the

1: possibly use a suborder of the maximal order. The primes dividing the

maximal order may be used.

index of the order chosen are larger than \tet{primelimit} or divide integers

stored in the \tet{addprimes} table.

2: gives also elements. The result is a two-column matrix, the first column

giving the elements defining these subfields, the second giving the

corresponding minimal polynomials.

If $p$ is given, it is assumed that it is the two-column matrix of the

If $fa$ is given, it is assumed that it is the two-column matrix of the

factorization of the discriminant of the polynomial $x$.

\syn{polred0}{x,\fl,p,\var{prec}}, where an omitted $p$ is

\syn{polred0}{x,\fl,fa}, where an omitted $fa$ is coded by \kbd{NULL}. Also

coded by $gzero$. Also available are $\teb{polred}(x,\var{prec})$ and

available are $\teb{polred}(x)$ and $\teb{factoredpolred}(x,fa)$, both

$\teb{factoredpolred}(x,p,\var{prec})$, both corresponding to $\fl=0$.

corresponding to $\fl=0$.

\subsecidx{polredabs}$(x,\{\fl=0\})$: finds one of the polynomial defining

the same number field as the one defined by $x$, and such that the sum of the

Line 4429 squares of the modulus of the roots (i.e.~the $T_2$-no

Line 4734 squares of the modulus of the roots (i.e.~the $T_2$-no

All $x$ accepted by \tet{nfinit} are also allowed here (e.g. non-monic

polynomials, \kbd{nf}, \kbd{bnf}, \kbd{[x,Z\_K\_basis]}).

\misctitle{Warning:} this routine uses an exponential-time algorithm to

enumerate all potential generators, and may be exceedingly slow when the

number field has many subfields, hence a lot of elements of small $T_2$-norm.

E.g. do not try it on the compositum of many quadratic fields, use

\tet{polred} instead.

The binary digits of $\fl$ mean

1: outputs a two-component row vector $[P,a]$, where $P$ is the default

Line 4438 whose minimal polynomial is equal to $x$.

Line 4749 whose minimal polynomial is equal to $x$.

4: gives \var{all} polynomials of minimal $T_2$ norm (of the two polynomials

$P(x)$ and $P(-x)$, only one is given).

\syn{polredabs0}{x,\fl,\var{prec}}.

16: possibly use a suborder of the maximal order. The primes dividing the

index of the order chosen are larger than \tet{primelimit} or divide integers

stored in the \tet{addprimes} table. In that case it may happen that the

output polynomial does not have minimal $T_2$ norm.\label{se:polredabs}

\syn{polredabs0}{x,\fl}.

\subsecidx{polredord}$(x)$: finds polynomials with reasonably small

coefficients and of the same degree as that of $x$ defining suborders of the

order defined by $x$. One of the polynomials always defines $\Q$ (hence

Line 4496 where \var{conductor} is the conductor of the extensio

Line 4812 where \var{conductor} is the conductor of the extensio

class group corresponding to the conductor given as a 3-component

vector [h,cyc,gen] as usual for a group, and \var{subgroup} is a

matrix in HNF defining the subgroup of the ray class group on the

given generators gen. If $\fl$ is non-zero, check under GRH that

given generators gen. If $\fl$ is non-zero, check that \var{pol} indeed

\var{pol} indeed defines an Abelian extension, return 0 if it does not.

defines an Abelian extension, return 0 if it does not.

\syn{rnfconductor}{\var{rnf},\var{pol},\fl}.

Line 4526 the relative

Line 4842 the relative

discriminant of $L$. This is a two-element row vector $[D,d]$, where $D$ is

the relative ideal discriminant and $d$ is the relative discriminant

considered as an element of $\var{nf}^*/{\var{nf}^*}^2$. The main variable of

$\var{nf}$ \var{must} be of lower priority than that of \var{pol}.

$\var{nf}$ \var{must} be of lower priority than that of \var{pol}, see

\secref{se:priority}.

Note: As usual, $\var{nf}$ can be a $\var{bnf}$ as output by \kbd{nfinit}.

Line 4587 $\theta = \beta+k\alpha$ where $\theta$ is a root of $

Line 4904 $\theta = \beta+k\alpha$ where $\theta$ is a root of $

of $\var{pol}$.

The main variable of $\var{nf}$ \var{must} be of lower priority than that

of \var{pol}. Note that for efficiency, this does not check whether the

of \var{pol} (see \secref{se:priority}). Note that for efficiency, this does

relative equation is irreducible over $\var{nf}$, but only if it is

not check whether the relative equation is irreducible over $\var{nf}$, but

squarefree. If it is reducible but squarefree, the result will be the

only if it is squarefree. If it is reducible but squarefree, the result will

absolute equation of the \'etale algebra defined by \var{pol}. If \var{pol}

be the absolute equation of the \'etale algebra defined by \var{pol}. If

is not squarefree, an error message will be issued.

\var{pol} is not squarefree, an error message will be issued.

\syn{rnfequation0}{\var{nf},\var{pol},\fl}.

Line 4690 the ideal list starts with $x$, all the other ideals b

Line 5007 the ideal list starts with $x$, all the other ideals b

format considered as base field, and \var{pol} a polynomial defining a relative

extension over $\var{nf}$, this computes all the necessary data to work in the

relative extension. The main variable of \var{pol} must be of higher priority

(i.e.~lower number) than that of $\var{nf}$, and the coefficients of \var{pol}

(i.e.~lower number, see \secref{se:priority}) than that of $\var{nf}$, and

must be in $\var{nf}$.

the coefficients of \var{pol} must be in $\var{nf}$.

The result is an 11-component row vector as follows (most of the components

are technical), the numbering being very close to that of \kbd{nfinit}. In

Line 4708 are the number of real and complex places of $L$ above

Line 5025 are the number of real and complex places of $L$ above

$K$ so that $r_{j,1}=0$ and $r_{j,2}=n$ if $j$ is a complex place, while if

$j$ is a real place we have $r_{j,1}+2r_{j,2}=n$.

$\var{rnf}[3]$ is a two-component row vector $[\d(L/K),s]$ where $\d(L/K)$

$\var{rnf}[3]$ is a two-component row vector $[\goth{d}(L/K),s]$ where

is the relative ideal discriminant of $L/K$ and $s$ is the discriminant of

$\goth{d}(L/K)$ is the relative ideal discriminant of $L/K$ and $s$ is the

$L/K$ viewed as an element of $K^*/(K^*)^2$, in other words it is the output

discriminant of $L/K$ viewed as an element of $K^*/(K^*)^2$, in other words it is

of \kbd{rnfdisc}.

the output of \kbd{rnfdisc}.

$\var{rnf}[4]$ is the ideal index $\f$, i.e.~such that

$\var{rnf}[4]$ is the ideal index $\goth{f}$, i.e.~such that

$d(pol)\Z_K=\f^2\d(L/K)$.

$d(pol)\Z_K=\goth{f}^2\goth{d}(L/K)$.

$\var{rnf}[5]$ is a vector \var{vm} with 7 entries useful for certain

computations in the relative extension $L/K$. $\var{vm}[1]$ is a vector of

Line 4786 free, false (0) if not.

Line 5103 free, false (0) if not.

\syn{rnfisfree}{\var{bnf},x}, and the result is a \kbd{long}.

\subsecidx{rnfisnorm}$(\var{bnf},\var{ext},\var{el},\{\fl=1\})$: similar to

\subsecidx{rnfisnorm}$(\var{T},\var{a},\{\fl=0\})$: similar to

\kbd{bnfisnorm} but in the relative case. This tries to decide whether the

\kbd{bnfisnorm} but in the relative case. $T$ is as output by

element \var{el} in \var{bnf} is the norm of some $y$ in \var{ext}.

\tet{rnfisnorminit} applied to the extension $L/K$. This tries to decide

$\var{bnf}$ is as output by \kbd{bnfinit}.

whether the element $a$ in $K$ is the norm of some $x$ in the extension

$L/K$.

$\var{ext}$ is a relative extension which has to be a row vector whose

The output is a vector $[x,q]$, where $\var{a}=\var{Norm}(x)*q$. The

components are:

algorithm looks for a solution $x$ which is an $S$-integer, with $S$ a list

of places of $K$ containing at least the ramified primes, the generators of

the class group of $L$, as well as those primes dividing \var{a}. If $L/K$ is

Galois, then this is enough; otherwise, $\fl$ is used to add more primes to

$S$: all the places above the primes $p \leq \fl$ (resp.~$p|\fl$) if $\fl>0$

(resp.~$\fl<0$).

$\var{ext}[1]$: a relative equation of the number field \var{ext} over

The answer is guaranteed (i.e.~\var{a} is a norm iff $q = 1$) if the field is

\var{bnf}. As usual, the priority of the variable of the polynomial

Galois, or, under \idx{GRH}, if $S$ contains all primes less than

defining the ground field \var{bnf} (say $y$) must be lower than the

$12\log^2\left|\text{disc}(\var{M})\right|$, where \var{M} is the normal

main variable of $\var{ext}[1]$, say $x$.

closure of $L/K$.

$\var{ext}[2]$: the generator $y$ of the base field as a polynomial in $x$ (as

If \tet{rnfisnorminit} has determined (or was told) that $L/K$ is

given by \kbd{rnfequation} with $\fl = 1$).

\idx{Galois}, and $\fl \neq 0$, a Warning is issued (so that you can set

$\fl = 1$ to check whether $L/K$ is known to be Galois, according to $T$).

$\var{ext}[3]$: is the \kbd{bnfinit} of the absolute extension $\var{ext}/\Q$.

Example:

This returns a vector $[a,b]$, where $\var{el}=\var{Norm}(a)*b$. It looks for a

solution which is an $S$-integer, with $S$ a list of places (of \var{bnf})

containing the ramified primes, the generators of the class group of

\var{ext}, as well as those primes dividing \var{el}. If $\var{ext}/\var{bnf}$

is known to be \idx{Galois}, set $\fl=0$ (here \var{el} is a norm iff $b=1$).

If $\fl$ is non zero add to $S$ all the places above the primes which: divide

$\fl$ if $\fl<0$, or are less than $\fl$ if $\fl>0$. The answer is guaranteed

(i.e.~\var{el} is a norm iff $b=1$) under \idx{GRH}, if $S$ contains all

primes less than $12\log^2\left|\text{disc}(\var{Ext})\right|$, where

\var{Ext} is the normal closure of $\var{ext} / \var{bnf}$. Example:

\bprog

bnf = bnfinit(y^3 + y^2 - 2*y - 1);

p = x^2 + Mod(y^2 + 2*y + 1, bnf.pol);

rnf = rnfequation(bnf,p,1);

T = rnfisnorminit(bnf, p);

ext = [p, rnf[2], bnfinit(rnf[1])];

rnfisnorm(T, 17)

rnfisnorm(bnf,ext,17, 1)

@eprog

\noindent checks whether $17$ is a norm in the Galois extension $\Q(\beta) /

\Q(\alpha)$, where $\alpha^3 + \alpha^2 - 2\alpha - 1 = 0$ and $\beta^2 +

\alpha^2 + 2*\alpha + 1 = 0$ (it is).

\alpha^2 + 2\alpha + 1 = 0$ (it is).

\syn{rnfisnorm}{\var{bnf},ext,x,\fl,\var{prec}}.

\syn{rnfisnorm}{\var{T},x,\fl}.

\subsecidx{rnfkummer}$(\var{bnr},\var{subgroup},\{deg=0\})$: \var{bnr}

\subsecidx{rnfisnorminit}$(\var{pol},\var{polrel},\{\fl=2\})$:

let $K$ be defined by a root of \var{pol}, and $L/K$ the extension defined by

the polynomial \var{polrel}. As usual, \var{pol} can in fact be an \var{nf},

or \var{bnf}, etc; if \var{pol} has degree $1$ (the base field is $\Q$),

polrel is also allowed to be an \var{nf}, etc. Computes technical data needed

by \tet{rnfisnorm} to solve norm equations $Nx = a$, for $x$ in $L$, and $a$

in $K$.

If $\fl = 0$, do not care whether $L/K$ is Galois or not.

If $\fl = 1$, $L/K$ is assumed to be Galois (unchecked), which speeds up

\tet{rnfisnorm}.

If $\fl = 2$, let the routine determine whether $L/K$ is Galois.

\syn{rnfisnorminit}{\var{pol},\var{polrel},\fl}.

\subsecidx{rnfkummer}$(\var{bnr},\{\var{subgroup}\},\{deg=0\})$: \var{bnr}

being as output by \kbd{bnrinit}, finds a relative equation for the

class field corresponding to the module in \var{bnr} and the given

congruence subgroup. If \var{deg} is positive, outputs the list of all

congruence subgroup (the full ray class field if \var{subgroup} is omitted).

relative equations of degree \var{deg} contained in the ray class field

If \var{deg} is positive, outputs the list of all relative equations of

defined by \var{bnr}.

degree \var{deg} contained in the ray class field defined by \var{bnr}, with

the same conductor as $(\var{bnr}, \var{subgroup})$.

(THIS PROGRAM IS STILL IN DEVELOPMENT STAGE)

\misctitle{Warning:} this routine only works for subgroups of prime index. It

uses Kummer theory, adjoining necessary roots of unity (it needs to compute a

tough \kbd{bnfinit} here), and finds a generator via Hecke's characterization

of ramification in Kummer extensions of prime degree. If your extension does

not have prime degree, for the time being, you have to split it by hand as a

tower of such extensions.

\syn{rnfkummer}{\var{bnr},\var{subgroup},\var{deg},\var{prec}},

\syn{rnfkummer}{\var{bnr},\var{subgroup},\var{deg},\var{prec}}, where

where \var{deg} is a \kbd{long}.

\var{deg} is a \kbd{long} and an omitted \var{subgroup} is coded as

\kbd{NULL}

\subsecidx{rnflllgram}$(\var{nf},\var{pol},\var{order})$: given a polynomial

\var{pol} with coefficients in \var{nf} defining a relative extension $L$ and

Line 4854 class field as output by \kbd{bnrinit} and \var{pol} a

Line 5190 class field as output by \kbd{bnrinit} and \var{pol} a

defining an \idx{Abelian extension}, computes the norm group (alias Artin

or Takagi group) corresponding to the Abelian extension of $\var{bnf}=bnr[1]$

defined by \var{pol}, where the module corresponding to \var{bnr} is assumed

to be a multiple of the conductor (i.e.~polrel defines a subextension of

to be a multiple of the conductor (i.e.~\var{pol} defines a subextension of

bnr). The result is the HNF defining the norm group on the given generators

of $\var{bnr}[5][3]$. Note that neither the fact that \var{pol} defines an

Abelian extension nor the fact that the module is a multiple of the conductor

Line 4866 is checked. The result is undefined if the assumption

Line 5202 is checked. The result is undefined if the assumption

Given a monic polynomial \var{pol} with coefficients in $\var{nf}$, finds a

list of relative polynomials defining some subfields, hopefully simpler and

containing the original field. In the present version \vers, this is slower

than \kbd{rnfpolredabs}.

and less efficient than \kbd{rnfpolredabs}.

\syn{rnfpolred}{\var{nf},\var{pol},\var{prec}}.

\subsecidx{rnfpolredabs}$(\var{nf},\var{pol},\{\fl=0\})$: relative version of

\kbd{polredabs}. Given a monic polynomial \var{pol} with coefficients in

$\var{nf}$, finds a simpler relative polynomial defining the same field. If

$\var{nf}$, finds a simpler relative polynomial defining the same field. The

$\fl=1$, returns $[P,a]$ where $P$ is the default output and $a$ is an

binary digits of $\fl$ mean

1: returns $[P,a]$ where $P$ is the default output and $a$ is an

element expressed on a root of $P$ whose characteristic polynomial is

\var{pol}, if $\fl=2$, returns an absolute polynomial (same as

\var{pol}

2: returns an absolute polynomial (same as

{\tt rnfequation(\var{nf},rnfpolredabs(\var{nf},\var{pol}))}

but faster).

\noindent but faster).

16: possibly use a suborder of the maximal order. This is slower than the

default when the relative discriminant is smooth, and much faster otherwise.

See \secref{se:polredabs}.

\misctitle{Remark.} In the present implementation, this is both faster and

much more efficient than \kbd{rnfpolred}, the difference being more

Line 4899 a four-element row vector $[A,I,D,d]$, where $D$ is th

Line 5241 a four-element row vector $[A,I,D,d]$, where $D$ is th

discriminant and $d$ is the relative discriminant considered as an element of

$\var{nf}^*/{\var{nf}^*}^2$.

Note: As usual, $\var{nf}$ can be a $\var{bnf}$ as output by \kbd{bnfinit}.

\syn{rnfpseudobasis}{\var{nf},\var{pol}}.

\subsecidx{rnfsteinitz}$(\var{nf},x)$: given a number field $\var{nf}$ as

Line 4914 as in \kbd{rnfpseudobasis}. The name of this function

Line 5254 as in \kbd{rnfpseudobasis}. The name of this function

that the ideal class of the last ideal of $I$ (which is well defined) is

called the \idx{Steinitz class} of the module $\Z_L$.

Note: $\var{nf}$ can be a $\var{bnf}$ as output by \kbd{bnfinit}.

\syn{rnfsteinitz}{\var{nf},x}.

\subsecidx{subgrouplist}$(\var{bnr},\{\var{bound}\},\{\fl=0\})$:

\var{bnr} being as output by \kbd{bnrinit} or a list of cyclic components

of a finite Abelian group $G$, outputs the list of subgroups of $G$

of a finite Abelian group $G$, outputs the list of subgroups of $G$. Subgroups

(of index bounded by \var{bound}, if not omitted). Subgroups are given

are given as HNF\sidx{Hermite normal form} left divisors of the

as HNF\sidx{Hermite normal form} left divisors of the

SNF\sidx{Smith normal form} matrix corresponding to $G$.

SNF\sidx{Smith normal form} matrix corresponding to $G$. If $\fl=0$

(default) and \var{bnr} is as output by

\kbd{bnrinit}, gives only the subgroups whose modulus is the conductor.

\syn{subgrouplist0}{\var{bnr},\var{bound},\fl}, where \var{bound} and $\fl$

\misctitle{Warning:} the present implementation cannot treat a group $G$

are long integers.

where any cyclic factor has more than $2^{31}$, resp.~$2^{63}$ elements on a

$32$-bit, resp.~$64$-bit architecture. \tet{forsubgroup} is a bit more

general and can handle $G$ if all $p$-Sylow subgroups of $G$ satisfy the

condition above.

If $\fl=0$ (default) and \var{bnr} is as output by $\kbd{bnrinit}(,,1)$, gives

only the subgroups whose modulus is the conductor. Otherwise, the modulus is

not taken into account.

If \var{bound} is present, and is a positive integer, restrict the output to

subgroups of index less than \var{bound}. If \var{bound} is a vector

containing a single positive integer $B$, then only subgroups of index

exactly equal to $B$ are computed. For instance

\bprog

? subgrouplist([6,2])

%1 = [[6, 0; 0, 2], [2, 0; 0, 2], [6, 3; 0, 1], [2, 1; 0, 1], [3, 0; 0, 2],

[1, 0; 0, 2], [6, 0; 0, 1], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]]

? subgrouplist([6,2],3) \\@com index less than 3

%2 = [[2, 1; 0, 1], [1, 0; 0, 2], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]]

? subgrouplist([6,2],[3]) \\@com index 3

%3 = [[3, 0; 0, 1]]

? bnr = bnrinit(bnfinit(x), [120,[1]], 1);

? L = subgrouplist(bnr, [8]);

@eprog\noindent

In the last example, $L$ corresponds to the 24 subfields of

$\Q(\zeta_{120})$, of degree $8$ and conductor $120\infty$ (by setting \fl,

we see there are a total of $43$ subgroups of degree $8$).

\bprog

? vector(#L, i, galoissubcyclo(bnr, L[i]))

@eprog\noindent

will produce their equations. (For a general base field, you would

have to rely on \tet{bnrstark}, or \tet{rnfkummer}.)

\syn{subgrouplist0}{\var{bnr},\var{bound},\fl}, where $\fl$

is a long integer, and an omitted \var{bound} is coded by \kbd{NULL}.

\subsecidx{zetak}$(\var{znf},x,\{\fl=0\})$: \var{znf} being a number

field initialized by \kbd{zetakinit} (\var{not} by \kbd{nfinit}),

computes the value of the \idx{Dedekind} zeta function of the number

Line 5037 the same extension of $\Q_p$ as $a$.

Line 5406 the same extension of $\Q_p$ as $a$.

\subsecidx{polcoeff}$(x,s,\{v\})$: coefficient of degree $s$ of the

polynomial $x$, with respect to the main variable if $v$ is omitted, with

respect to $v$ otherwise.

respect to $v$ otherwise. Also applies to power series, scalars (polynomial

of degree $0$), and to rational functions provided the denominator is a

monomial.

\syn{polcoeff0}{x,s,v}, where $v$ is a \kbd{long} and an omitted $v$ is coded

as $-1$. Also available is \teb{truecoeff}$(x,v)$.

Line 5141 GP it is kept in the variable \kbd{realprecision} and

Line 5512 GP it is kept in the variable \kbd{realprecision} and

user, but it must be explicitly given as a second argument in library mode.

The algorithm used is a modification of A.~Sch\"onhage\sidx{Sch\"onage}'s

remarkable root-finding algorithm, due to and implemented by X.~Gourdon.

root-finding algorithm, due to and implemented by X.~Gourdon. Barring bugs, it

Barring bugs, it is guaranteed to converge and to give the roots to the

is guaranteed to converge and to give the roots to the required accuracy.

required accuracy.

If $\fl=1$, use a variant of the Newton-Raphson method, which is \var{not}

guaranteed to converge, but is rather fast. If you get the messages ``too

many iterations in roots'' or ``INTERNAL ERROR: incorrect result in roots'',

use the default function (i.e.~no flag or $\fl=0$). This used to be the

use the default algorithm. This used to be the default root-finding function in

default root-finding function in PARI until version 1.39.06.

PARI until version 1.39.06.

\syn{roots}{\var{pol},\var{prec}} or $\teb{rootsold}(\var{pol},\var{prec})$.

Line 5166 $\teb{rootmod2}(\var{pol},p)$ ($\fl=1$).

Line 5536 $\teb{rootmod2}(\var{pol},p)$ ($\fl=1$).

\subsecidx{polrootspadic}$(\var{pol},p,r)$: row vector of $p$-adic roots of the

polynomial \var{pol} with $p$-adic precision equal to $r$. Multiple roots are

\var{not} repeated. $p$ is assumed to be a prime.

\var{not} repeated. $p$ is assumed to be a prime, and \var{pol} to be

non-zero modulo $p$.

\syn{rootpadic}{\var{pol},p,r}, where $r$ is a \kbd{long}.

Line 5178 polynomial \var{pol} in the interval $]a,b]$, using St

Line 5549 polynomial \var{pol} in the interval $]a,b]$, using St

\teb{sturm}\kbd{(\var{pol})} is equivalent to

\key{sturmpart}\kbd{(\var{pol},NULL,NULL)}. The result is a \kbd{long}.

\subsecidx{polsubcyclo}$(n,d,\{v=x\})$: gives a polynomial (in variable

\subsecidx{polsubcyclo}$(n,d,\{v=x\})$: gives polynomials (in variable

$v$) defining the sub-Abelian extension of degree $d$ of the cyclotomic

$v$) defining the sub-Abelian extensions of degree $d$ of the cyclotomic

field $\Q(\zeta_n)$, where $d\mid \phi(n)$. $(\Z/n\Z)^*$ has to be cyclic

field $\Q(\zeta_n)$, where $d\mid \phi(n)$.

(i.e.~$n=2$, $4$, $p^k$ or $2p^k$ for an odd prime $p$). The function

\tet{galoissubcyclo} covers the general case.

\syn{subcyclo}{n,d,v}, where $v$ is a variable number.

If there is exactly one such extension the output is a polynomial, else it is

a vector of polynomials, eventually empty.

To be sure to get a vector, you can use \kbd{concat([],polsubcyclo(n,d))}

The function \tet{galoissubcyclo} allows to specify more closely which sub-Abelian extension should be computed.

\syn{polsubcyclo}{n,d,v}, where $n$, $d$ and $v$ are \kbd{long} and $v$ is a

variable number. When $(\Z/n\Z)^*$ is cyclic, you can use

\teb{subcyclo}$(n,d,v)$, where $n$, $d$ and $v$ are \kbd{long} and $v$ is a

variable number.

\subsecidx{polsylvestermatrix}$(x,y)$: forms the Sylvester matrix

corresponding to the two polynomials $x$ and $y$, where the coefficients of

the polynomials are put in the columns of the matrix (which is the natural

Line 5290 as \kbd{NULL}.

Line 5669 as \kbd{NULL}.

\subsecidx{thueinit}$(P,\{\fl=0\})$: initializes the \var{tnf}

corresponding to $P$. It is meant to be used in conjunction with \tet{thue}

to solve Thue equations $P(x,y) = a$, where $a$ is an integer. If $\fl$ is

non-zero, certify the result unconditionnaly, Otherwise, assume \idx{GRH},

non-zero, certify the result unconditionnally, Otherwise, assume \idx{GRH},

this being much faster of course.

\syn{thueinit}{P,\fl,\var{prec}}.

Line 5311 guaranteed to be irreducible!). One can check the clos

Line 5690 guaranteed to be irreducible!). One can check the clos

polynomial evaluation or substitution, or by computing the roots of the

polynomial given by algdep.

If $x$ is padic, $\fl$ is meaningless and the algorithm LLL-reduces the

If $x$ is $p$-adic, $\fl$ is meaningless and the algorithm LLL-reduces the

``dual lattice'' corresponding to the powers of $x$.

Otherwise, if $\fl$ is zero, the algorithm used is a variant of the \idx{LLL}

Line 5414 If $\fl>0$, uses the LLL algorithm. $\fl$ is a paramet

Line 5793 If $\fl>0$, uses the LLL algorithm. $\fl$ is a paramet

between one half the number of decimal digits of precision and that number

(see \kbd{algdep}).

If $\fl<0$, returns as soon as one relation has been found.

If $\fl<0$, $x$ is allowed to be (and in any case interpreted as) a matrix.

Returns a non trivial element of the kernel of $x$, or $0$ if $x$ has trivial

kernel.

\syn{lindep0}{x,\fl,\var{prec}}. Also available is

$\teb{lindep}(x,\var{prec})$ ($\fl=0$).

Line 5623 any type.

Line 6004 any type.

If $x$ is known to have integral entries, set $\fl=1$.

\noindent Note: The library function $\tet{ker_mod_p}(x, p)$, where $x$ has

\noindent Note: The library function $\tet{FpM_ker}(x, p)$, where $x$ has

integer entries and $p$ is prime, which is equivalent to but many orders of

magnitude faster than \kbd{matker(x*Mod(1,p))} and needs much less stack

space. To use it under GP, type \kbd{install(ker\_mod\_p, GG)} first.

space. To use it under GP, type \kbd{install(FpM\_ker, GG)} first.

\syn{matker0}{x,\fl}. Also available are $\teb{ker}(x)$ ($\fl=0$),

$\teb{keri}(x)$ ($\fl=1$) and $\kbd{ker\_mod\_p}(x,p)$.

Line 5640 If $\fl=0$, uses a modified integer LLL algorithm.

Line 6021 If $\fl=0$, uses a modified integer LLL algorithm.

If $\fl=1$, uses $\kbd{matrixqz}(x,-2)$. If LLL reduction of the final result

is not desired, you can save time using \kbd{matrixqz(matker(x),-2)} instead.

If $\fl=2$, uses another modified LLL. In the present version \vers, only

independent rows are allowed in this case.

\syn{matkerint0}{x,\fl}. Also available is

$\teb{kerint}(x)$ ($\fl=0$).

Line 5788 result is a transformation matrix $T$ such that $x\cdo

Line 6166 result is a transformation matrix $T$ such that $x\cdo

basis of the lattice generated by the column vectors of $x$.

If $\fl=0$ (default), the computations are done with real numbers (i.e.~not

with rational numbers) hence are fast but as presently programmed (version

with rational numbers), using Householder matrices for orthogonalization

\vers) are numerically unstable.

(as presently programmed: slow but stable).

If $\fl=1$, it is assumed that the corresponding Gram matrix is integral.

The computation is done entirely with integers and the algorithm is both

Line 5805 two distinct basis vectors $v_i, \, v_j$.]

Line 6183 two distinct basis vectors $v_i, \, v_j$.]

This can be significantly faster than $\fl=1$ when one row is huge compared

to the other rows.

If $\fl=3$, all computations are done in rational numbers. This does not

incur numerical instability, but is extremely slow. This function is

essentially superseded by case 1, so will soon disappear.

If $\fl=4$, $x$ is assumed to have integral entries, but needs not be of

maximal rank. The result is a two-component vector of matrices~: the

columns of the first matrix represent a basis of the integer kernel of $x$

Line 5818 of the matrix $x$.

Line 6192 of the matrix $x$.

If $\fl=5$, case as case $4$, but $x$ may have polynomial coefficients.

If $\fl=7$, uses an older version of case $0$ above.

If $\fl=8$, same as case $0$, where $x$ may have polynomial coefficients.

If $\fl=9$, variation on case $1$, using content.

\syn{qflll0}{x,\fl,\var{prec}}. Also available are

$\teb{lll}(x,\var{prec})$ ($\fl=0$), $\teb{lllint}(x)$ ($\fl=1$), and

$\teb{lllkerim}(x)$ ($\fl=4$).

Line 5843 $\fl=4$: $x$ has integer entries, gives the kernel and

Line 6213 $\fl=4$: $x$ has integer entries, gives the kernel and

$\fl=5$: same as $4$ for generic $x$.

$\fl=7$: an older version of case $0$.

\syn{qflllgram0}{x,\fl,\var{prec}}. Also available are

$\teb{lllgram}(x,\var{prec})$ ($\fl=0$), $\teb{lllgramint}(x)$ ($\fl=1$), and

$\teb{lllgramkerim}(x)$ ($\fl=4$).

Line 6006 For example, \kbd{vecextract(x, vecsort(x,,1))} is equ

Line 6374 For example, \kbd{vecextract(x, vecsort(x,,1))} is equ

$\bullet$ 2: sorts $x$ by ascending lexicographic order (as per the

\kbd{lex} comparison function).

$\bullet$ 4: use decreasing instead of ascending order.

$\bullet$ 4: use descending instead of ascending order.

\syn{vecsort0}{x,k,flag}. To omit $k$, use \kbd{NULL} instead. You can also

use the simpler functions

Line 6030 last two arguments is omitted, fill the vector with ze

Line 6398 last two arguments is omitted, fill the vector with ze

\synt{vecteur}{GEN nmax, entree *ep, char *expr}.

\subsecidx{vectorsmall}$(n,\{X\},\{\var{expr}=0\})$: creates a row vector of small integers (type

\typ{VECSMALL}) with $n$ components whose components are the expression

\var{expr} evaluated at the integer points between 1 and $n$. If one of the

last two arguments is omitted, fill the vector with zeroes.

\synt{vecteursmall}{GEN nmax, entree *ep, char *expr}.

\subsecidx{vectorv}$(n,X,\var{expr})$: as \tet{vector}, but returns a

column vector (type \typ{COL}).

Line 6207 the positive divisors of $n$.

Line 6582 the positive divisors of $n$.

Arithmetic functions like \tet{sigma} use the multiplicativity of the

underlying expression to speed up the computation. In the present version

\vers, there is no way to indicate that \var{expr} is multiplicative in

$n$, hence specialized functions should be prefered whenever possible.

$n$, hence specialized functions should be preferred whenever possible.

\synt{divsum}{entree *ep, GEN num, char *expr}.

Line 6363 Overlapping regions will thus be drawn twice, and the

Line 6738 Overlapping regions will thus be drawn twice, and the

transparent. Then display the whole drawing in a special window on your

screen.

\subsecidx{plotfile}$(s)$: set the output file for plotting output. Special

\subsecidx{plotfile}$(s)$: set the output file for plotting output. The

filename \kbd{-} redirects to the same place as PARI output.

special filename \kbd{"-"} redirects to the same place as PARI output. This

is only taken into account by the \kbd{gnuplot} interface.

\subsecidx{ploth}$(X=a,b,\var{expr},\{\fl=0\},\{n=0\})$: high precision

plot of the function $y=f(x)$ represented by the expression \var{expr}, $x$

Line 6388 the same window.

Line 6764 the same window.

\noindent The binary digits of $\fl$ mean:

$\bullet$ 1: \tev{parametric plot}. Here \var{expr} must be a vector with

$\bullet$ $1 = \kbd{Parametric}$: \tev{parametric plot}. Here \var{expr} must

an even number of components. Successive pairs are then understood as the

be a vector with an even number of components. Successive pairs are then

parametric coordinates of a plane curve. Each of these are then drawn.

understood as the parametric coordinates of a plane curve. Each of these are

then drawn.

For instance:

Line 6403 curves.

Line 6780 curves.

$y=x$.

$\bullet$ 2: \tev{recursive plot}. If this flag is set, only \var{one}

$\bullet$ $2 = \kbd{Recursive}$: \tev{recursive plot}. If this flag is set,

curve can be drawn at time, i.e.~\var{expr} must be either a two-component

only \var{one} curve can be drawn at time, i.e.~\var{expr} must be either a

vector (for a single parametric curve, and the parametric flag \var{has} to

two-component vector (for a single parametric curve, and the parametric flag

be set), or a scalar function. The idea is to choose pairs of successive

\var{has} to be set), or a scalar function. The idea is to choose pairs of

reference points, and if their middle point is not too far away from the

successive reference points, and if their middle point is not too far away

segment joining them, draw this as a local approximation to the curve.

from the segment joining them, draw this as a local approximation to the

Otherwise, add the middle point to the reference points. This is very fast,

curve. Otherwise, add the middle point to the reference points. This is very

and usually more precise than usual plot. Compare the results of

fast, and usually more precise than usual plot. Compare the results of

$$\kbd{ploth(X=-1,1,sin(1/X),2)}\quad

\text{and}\quad\kbd{ploth(X=-1,1,sin(1/X))}$$

for instance. But beware that if you are extremely unlucky, or choose too few

Line 6425 curve with slightly different parameters.

Line 6802 curve with slightly different parameters.

The other values toggle various display options:

$\bullet$ 4: do not rescale plot according to the computed extrema. This is

$\bullet$ $4 = \kbd{no\_Rescale}$: do not rescale plot according to the

meant to be used when graphing multiple functions on a rectwindow (as a

computed extrema. This is meant to be used when graphing multiple functions

\tet{plotrecth} call), in conjuction with \tet{plotscale}.

on a rectwindow (as a \tet{plotrecth} call), in conjunction with

\tet{plotscale}.

$\bullet$ 8: do not print the $x$-axis.

$\bullet$ $8 = \kbd{no\_X\_axis}$: do not print the $x$-axis.

$\bullet$ 16: do not print the $y$-axis.

$\bullet$ $16 = \kbd{no\_Y\_axis}$: do not print the $y$-axis.

$\bullet$ 32: do not print frame.

$\bullet$ $32 = \kbd{no\_Frame}$: do not print frame.

$\bullet$ 64: only plot reference points, do not join them.

$\bullet$ $64 = \kbd{no\_Lines}$: only plot reference points, do not join them.

$\bullet$ 256: use splines to interpolate the points.

$\bullet$ $128 = \kbd{Points_too}$: plot both lines and points.

$\bullet$ 512: plot no $x$-ticks.

$\bullet$ $256 = \kbd{Splines}$: use splines to interpolate the points.

$\bullet$ 1024: plot no $y$-ticks.

$\bullet$ $512 = \kbd{no\_X\_ticks}$: plot no $x$-ticks.

$\bullet$ 2048: plot all ticks with the same length.

$\bullet$ $1024 = \kbd{no\_Y\_ticks}$: plot no $y$-ticks.

$\bullet$ $2048 = \kbd{Same\_ticks}$: plot all ticks with the same length.

\subsecidx{plothraw}$(\var{listx},\var{listy},\{\fl=0\})$: given

\var{listx} and \var{listy} two vectors of equal length, plots (in high

precision) the points whose $(x,y)$-coordinates are given in \var{listx}

Line 6468 must be between $0$ and $1$) and internally converted

Line 6848 must be between $0$ and $1$) and internally converted

The plotting device imposes an upper bound for $x$ and $y$, for instance the

number of pixels for screen output. These bounds are available through the

\tet{plothsizes} function. The following sequence initializes in a portable

way (i.e independant of the output device) a window of maximal size, accessed

way (i.e independent of the output device) a window of maximal size, accessed

through coordinates in the $[0,1000] \times [0,1000]$ range~:

\bprog

Line 6479 plotscale(0, 0,1000, 0,1000);

Line 6859 plotscale(0, 0,1000, 0,1000);

\subsecidx{plotkill}$(w)$: erase rectwindow $w$ and free the corresponding

memory. Note that if you want to use the rectwindow $w$ again, you have to

use \kbd{initrect} first to specify the new size. So it's better in this case

use \kbd{plotinit} first to specify the new size. So it's better in this case

to use \kbd{initrect} directly as this throws away any previous work in the

to use \kbd{plotinit} directly as this throws away any previous work in the

given rectwindow.

\subsecidx{plotlines}$(w,X,Y,\{\fl=0\})$: draw on the rectwindow $w$

Line 6544 rectwindow $w$ the curve output of \kbd{ploth}$(w,X=a,

Line 6924 rectwindow $w$ the curve output of \kbd{ploth}$(w,X=a,

\var{data} is a vector of vectors, each corresponding to a list a coordinates.

If parametric plot is set, there must be an even number of vectors, each

successive pair corresponding to a curve. Otherwise, the first one containe

successive pair corresponding to a curve. Otherwise, the first one contains

the $x$ coordinates, and the other ones contain the $y$-coordinates

of curves to plot.

Line 6688 HNF\sidx{Hermite normal form} form), one can for insta

Line 7068 HNF\sidx{Hermite normal form} form), one can for insta

[1, 0; 0, 2]

[2, 0; 0, 1]

[1, 0; 0, 1]

@eprog

@eprog\noindent

Note that in this last representation, the index $[G:H]$ is given by the

determinant. See \tet{galoissubcyclo} and \tet{galoisfixedfield} for

\tet{nfsubfields} applications to \idx{Galois} theory.

\misctitle{Warning:} the present implementation cannot treat a group $G$, if

one of its $p$-Sylow subgroups has a cyclic factor has more than $2^{31}$,

resp.~$2^{63}$ elements on a $32$-bit, resp.~$64$-bit architecture.

\subsubsecidx{forvec}$(X=v,\var{seq},\{\fl=0\})$: $v$ being an $n$-component

vector (where $n$ is arbitrary) of two-component vectors $[a_i,b_i]$

for $1\le i\le n$, the \var{seq} is evaluated with the formal variable

Line 6725 statement is \var{not} a loop (obviously!).

Line 7109 statement is \var{not} a loop (obviously!).

\subsubsecidx{next}$(\{n=1\})$: interrupts execution of current $seq$,

resume the next iteration of the innermost enclosing loop, within the

current fonction call (or top level loop). If $n$ is specified, resume at

current function call (or top level loop). If $n$ is specified, resume at

the $n$-th enclosing loop. If $n$ is bigger than the number of enclosing

loops, all enclosing loops are exited.

\subsubsecidx{return}$(\{x=0\})$: returns from current subroutine, with

result $x$.

result $x$. If $x$ is omitted, return the \kbd{(void)} value (return no

result, like \kbd{print}).

\subsubsecidx{until}$(a,\var{seq})$: evaluates expression sequence \var{seq}

until $a$ is not equal to 0 (i.e.~until $a$ is true). If $a$ is initially

Line 7030 result of the evaluation of \var{rec}, and the control

Line 7415 result of the evaluation of \var{rec}, and the control

GP prompt. In particular, current computation is then lost.

The following error handler prints the list of all user variables, then

stores in a file their name and their values:

stores in a file their name and their values:\sidx{writebin}

\bprog

? { trap( ,

print(reorder);

write("crash", reorder);

writebin("crash")) }

write("crash", eval(reorder))) }

@eprog

If no recovery code is given (\var{rec} is omitted) a so-called

Line 7134 the corresponding user variable to the saved value. E.

Line 7518 the corresponding user variable to the saved value. E.

x = 1; writebin("log")

@eprog

\noindent reading \kbd{log} into a clean session will set \kbd{x} to $1$.

The relative variables priorities of new variables set in this way remain the

The relative variables priorities (see \secref{se:priority}) of new variables

same (preset variables retain their former priority, but are set to the new

set in this way remain the same (preset variables retain their former

value). In particular, reading such a session log into a clean session will

priority, but are set to the new value). In particular, reading such a

restore all variables exactly as they were in the original one.

session log into a clean session will restore all variables exactly as they

were in the original one.

User functions, installed functions and history objects can not be saved via

this function. Just as a regular input file, a binary file can be compressed

using \tet{gzip}, provided the file name has the standard \kbd{.gz}

extension. \label{se:writebin}\sidx{binary file}

In the present implementation, the binary files are architecture dependant

and compatibility with future versions of GP is not guaranteed. Hence

binary files should not be used for long term storage (also, they are

larger and harder to compress than text files).

\subsubsecidx{writetex}$(\var{filename},\{\var{str}*\})$: as \kbd{write},

in \TeX\ format.\label{se:writetex}

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