Annotation of OpenXM_contrib/pari-2.2/doc/refcard.tex, Revision 1.2
1.2 ! noro 1: % $Id: refcard.tex,v 1.10 2002/09/07 21:09:29 karim Exp $
1.1 noro 2: % This file is intended to be processed by plain TeX (TeX82).
3: % Reference Card for PARI-GP version 2.1
4:
5: % Copyright (c) 1997-2000 Karim Belabas.
6: % Permission is granted to copy, distribute and/or modify this document
7: % under the terms of the GNU Free Documentation License
8:
9: % Based on an earlier version by Joseph H. Silverman who kindly let me
10: % use his original file.
11: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
12: % The original copyright notice read:
13: %
14: %% Copyright (c) 1993,1994 Joseph H. Silverman. May be freely distributed.
15: %% Created Tuesday, July 27, 1993
16: %% Thanks to Stephen Gildea for the multicolumn macro package
17: %% which I modified from his GNU emacs reference card
18: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
19: \def\TITLE{Pari-GP reference card}
1.2 ! noro 20: \def\versionnumber{2.13}% Version of this reference card
! 21: \def\PARIversion{2.2.5}% Version of PARI described on this reference card
! 22: \def\year{2002}
! 23: \def\month{August}
! 24:
1.1 noro 25: % ignore parimacro.tex's \magnification setting
26: \let\oldmagnification\magnification
27: \catcode`@=11
28: \def\magnification{\count@}%
29: \catcode`@=12
30: \input parimacro.tex
31: \let\magnification\oldmagnification
32: \ifPDF
33: \input pdfmacs.tex
34: \pdfpagewidth=11.69in
35: \pdfpageheight=8.26in
36: \fi
37: %**start of header
38: \newcount\columnsperpage
39: % The final reference card has six columns, three on each side.
40: % This file can be used to produce it in any of three ways:
41: % 1 column per page
42: % produces six separate pages, each of which needs to be reduced to 80%.
43: % This gives the best resolution.
44: % 2 columns per page
45: % produces three already-reduced pages.
46: % You will still need to cut and paste.
47: % 3 columns per page
48: % produces two pages which must be printed sideways to make a
49: % ready-to-use 8.5 x 11 inch reference card.
50: % For this you need a dvi device driver that can print sideways.
51: % [For 2 or 3 columns, you'll need 6 and 8 point fonts.]
52: % Which mode to use is controlled by setting \columnsperpage above.
53: %
54: % Specify how many columns per page you want here:
55: \columnsperpage=3
56:
57: % You shouldn't need to modify anything below this line.
58: %
59: % Author:
60: % Karim Belabas
61: % Universite Paris Sud
62: % Departement de Mathematiques (bat. 425)
63: % F-91405 Orsay
1.2 ! noro 64: % email: Karim.Belabas@math.u-psud.fr
1.1 noro 65: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
66: % (original reference card by Joseph H. Silverman)
67: % (original reference card macros due to Stephen Gildea)
68: % Original Thanks and History:
69: %
70: %% Thanks:
71: %% I would like to thank Jim Delaney, Kevin Buzzard, Dan Lieman,
72: %% and Jaap Top for sending me corrections.
73: %%
74: %% History:
75: %% Version 1.0 - July 1993, first general distribution
76: %% Version 1.1 - April 1994, corrected six typos
77: %% Version 1.2 - January 1995, minor corrections and additions
78: %% Version 1.3 - January 1996, minor corrections and additions
79: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
80: % Version 2.0 - November 1997, general distribution for GP 2.0
81: % Version 2.1 - January 1998, set nf,bnf,etc in a sensible font, updated default
82: % Version 2.2 - March 1998, some new functions (modpr, bnrstark), updated
83: % concat, removed spurious tabs.
84: % Version 2.3 - May 1998, added write1, corrected my email address.
85: % Version 2.4 - July 1998, removed vecindexsort, added ellrootno, updated
86: % elllseries
87: % Version 2.5 - October 1998, updated elliptic functions. Added quadray and
88: % user member functions
89: % Version 2.6 - December 1998, added local() keyword
90: % Version 2.7 - February 1999, added some pointer '&' arguments. Removed
91: % rounderror
92: % Version 2.8 - April 1999, removed \k, added \l filename
93: % Version 2.9 - April 2000, added \o¸ updated control statements
94: % Version 2.10 - June 2000, updated polinterpolate
95: % Version 2.11 - November 2000, changed Copyright
1.2 ! noro 96: % From version 2.12 onward, History will be catered for by CVS.
1.1 noro 97: %% Thanks to Bill Allombert, Henri Cohen, Gerhard Niklasch, and Joe
98: %% Silverman for many comments and corrections.
99:
100: \def\version{\month\ \year\ v\versionnumber}
101:
102: \def\shortcopyrightnotice{\vskip .5ex plus 2 fill
103: \centerline{\small \copyright\ \year\ Karim Belabas.
104: Permissions on back. v\versionnumber}}
105:
106: \def\<#1>{$\langle${#1}$\rangle$}
107: \def\copyrightnotice{\vskip 1ex plus 2 fill
108: \begingroup\small
109: \centerline{Based on an earlier version by Joseph H. Silverman}
110: \centerline{\version. Copyright \copyright\ \year\ K. Belabas}
111: \centerline{GP copyright by The PARI Group}
112:
113: Permission is granted to make and distribute copies of this card provided the
114: copyright and this permission notice are preserved on all copies.
115:
116: Send comments and corrections to \<Karim.BELABAS@math.u-psud.fr>
117: \endgroup}
118:
119: % make \bye not \outer so that the \def\bye in the \else clause below
120: % can be scanned without complaint.
121: \def\bye{\par\vfill\supereject\end}
122:
123: \newdimen\intercolumnskip
124: \newbox\columna
125: \newbox\columnb
126:
127: \def\ncolumns{\the\columnsperpage}
128:
129: \message{[\ncolumns\space
130: column\if 1\ncolumns\else s\fi\space per page]}
131:
132: \def\scaledmag#1{ scaled \magstep #1}
133:
134: % This multi-way format was designed by Stephen Gildea
135: % October 1986.
136: \if 1\ncolumns
137: \hsize 4in
138: \vsize 10in
139: \voffset -.7in
140: \font\titlefont=\fontname\tenbf \scaledmag3
141: \font\headingfont=\fontname\tenbf \scaledmag2
142: \font\smallfont=\fontname\sevenrm
143: \font\smallsy=\fontname\sevensy
144:
145: \footline{\hss\folio}
146: \def\makefootline{\baselineskip10pt\hsize6.5in\line{\the\footline}}
147: \else
148: \hsize 3.2in
149: % \vsize 7.95in
150: \vsize 7.90in
151: \hoffset -.75in
152: % \voffset -.745in
153: \voffset -.815in
154: \font\titlefont=cmbx10 \scaledmag2
155: \font\headingfont=cmbx10 \scaledmag1
156: \font\smallfont=cmr6
157: \font\smallsy=cmsy6
158: \font\eightrm=cmr8
159: \font\eightbf=cmbx8
160: \font\eightit=cmti8
161: \font\eighttt=cmtt8
162: \font\eightsy=cmsy8
163: \font\eightsl=cmsl8
164: \font\eighti=cmmi8
165: \font\eightex=cmex10 at 8pt
166: \textfont0=\eightrm
167: \textfont1=\eighti
168: \textfont2=\eightsy
169: \textfont3=\eightex
170: \def\rm{\fam0 \eightrm}
171: \def\bf{\eightbf}
172: \def\it{\eightit}
173: \def\tt{\eighttt}
174: \normalbaselineskip=.8\normalbaselineskip
175: \normallineskip=.8\normallineskip
176: \normallineskiplimit=.8\normallineskiplimit
177: \normalbaselines\rm %make definitions take effect
178:
179: \if 2\ncolumns
180: \let\maxcolumn=b
181: \footline{\hss\rm\folio\hss}
182: \def\makefootline{\vskip 2in \hsize=6.86in\line{\the\footline}}
183: \else \if 3\ncolumns
184: \let\maxcolumn=c
185: \nopagenumbers
186: \else
187: \errhelp{You must set \columnsperpage equal to 1, 2, or 3.}
188: \errmessage{Illegal number of columns per page}
189: \fi\fi
190:
191: \intercolumnskip=.46in
192: \def\abc{a}
193: \output={%
194: % This next line is useful when designing the layout.
195: %\immediate\write16{Column \folio\abc\space starts with \firstmark}
196: \if \maxcolumn\abc \multicolumnformat \global\def\abc{a}
197: \else\if a\abc
198: \global\setbox\columna\columnbox \global\def\abc{b}
199: %% in case we never use \columnb (two-column mode)
200: \global\setbox\columnb\hbox to -\intercolumnskip{}
201: \else
202: \global\setbox\columnb\columnbox \global\def\abc{c}\fi\fi}
203: \def\multicolumnformat{\shipout\vbox{\makeheadline
204: \hbox{\box\columna\hskip\intercolumnskip
205: \box\columnb\hskip\intercolumnskip\columnbox}
206: \makefootline}\advancepageno}
207: \def\columnbox{\leftline{\pagebody}}
208:
209: \def\bye{\par\vfill\supereject
210: \if a\abc \else\null\vfill\eject\fi
211: \if a\abc \else\null\vfill\eject\fi
212: \end}
213: \fi
214:
215: % we won't be using math mode much, so redefine some of the characters
216: % we might want to talk about
217: %\catcode`\^=12
218: %\catcode`\_=12
219: %\catcode`\~=12
220:
221: \chardef\\=`\\
222: \chardef\{=`\{
223: \chardef\}=`\}
224:
225: \hyphenation{}
226:
227: \parindent 0pt
228: \parskip 0pt
229:
230: \def\small{\smallfont\textfont2=\smallsy\baselineskip=.8\baselineskip}
231:
232: \outer\def\newcolumn{\vfill\eject}
233:
234: \outer\def\title#1{{\titlefont\centerline{#1}}}
235:
236: \outer\def\section#1{\par\filbreak
237: \vskip 1.4ex plus .4ex minus .5ex
238: {\headingfont #1}\mark{#1}%
239: \vskip .7ex plus .3ex minus .5ex
240: }
241:
242: \outer\def\subsec#1{\filbreak
243: \vskip 0.1ex plus 0.05ex
244: {\bf #1}
245: \vskip 0.04ex plus 0.05ex
246: }
247:
248: \newdimen\keyindent
249: \def\beginindentedkeys{\keyindent=1em}
250: \def\endindentedkeys{\keyindent=0em}
251: \def\begindoubleindentedkeys{\keyindent=2em}
252: \def\enddoubleindentedkeys{\keyindent=1em}
253: \endindentedkeys
254:
255: \def\kbd#1{{\tt#1}\null} %\null so not an abbrev even if period follows
256: \def\var#1{\hbox{\it #1}}
257: \def\fl{\{\var{f{}l\/}\}}
258:
259: \def\key#1#2{\leavevmode\hbox to \hsize{\vtop
260: {\hsize=.75\hsize\rightskip=1em
261: \hskip\keyindent\relax#1}\kbd{#2}\hfil}}
262:
263: \newbox\libox
264: \setbox\libox\hbox{\kbd{M-x }}
265: \newdimen\liwidth
266: \liwidth=\wd\libox
267:
268: \def\li#1#2{\leavevmode\hbox to \hsize{\hbox to .75\hsize
269: {\hskip\keyindent\relax#1\hfil}%
270: \hskip -\liwidth minus 1fil
271: \kbd{#2}\hfil}}
272:
273: \def\threecol#1#2#3{\hskip\keyindent\relax#1\hfil&\kbd{#2}\quad
274: &\kbd{#3}\quad\cr}
275:
276: \def\mod{\;\hbox{\rm mod}\;}
277: \def\expr{\hbox{\it expr}}
278: \def\seq{\hbox{\it seq}}
279: \def\args{\hbox{\it args}}
280: \def\file{\hbox{\it file}}
281: \def\QQ{\hbox{\bf Q}}
282: \def\ZZ{\hbox{\bf Z}}
283: \def\RR{\hbox{\bf R}}
284: \def\FF{\hbox{\bf F}}
285: \def\CC{\hbox{\bf C}}
286: \def\deg{\mathop{\rm deg}}
287: \def\bs{\char'134}
288: \def\pow{\^{}\hskip0pt}
289: \def\til{\raise-0.3em\hbox{\~{}}}
290: \def\typ#1{\kbd{t\_#1}}
291: %**end of header
292:
293: \title{PARI-GP Reference Card}
294: \centerline{(PARI-GP version \PARIversion)}
295: Note: optional arguments are surrounded by braces {\tt \{\}}.
296:
297: \section{Starting \& Stopping GP}
298: \key{to enter GP, just type its name:}{gp}
299: \key{to exit GP, type}{\\q {\rm or }quit}
300:
301: \section{Help}
302: \li{describe function}{?{\rm function}}
303: \li{extended description}{??{\rm keyword}}
304: \li{list of relevant help topics}{???{\rm pattern}}
305:
306: \section{Input/Output \& Defaults}
307: \li{output previous line, the lines before}
308: {\%{\rm, }\%`{\rm, }\%``{\rm, etc.}}
309: \key{output from line $n$}{\%$n$}
310: \key{separate multiple statements on line}{;}
311: \key{extend statement on additional lines}{\\}
312: \key{extend statements on several lines}{\{\seq1; \seq2;\}}
313: \key{comment}{/* \dots */}
314: \key{one-line comment, rest of line ignored}{\\\\ \dots}
315: \li{set default $d$ to \var{val}} {default$(\{d\},\{\var{val}\},\fl)$}
316: \li{mimic behaviour of GP 1.39} {default(compatible,3)}
317:
318: \section{Metacommands}
319: \key{toggle timer on/off}{\#}
320: \key{print time for last result}{\#\#}
321: \key{print \%$n$ in raw format}{\\a $n$}
322: \key{print \%$n$ in pretty format}{\\b $n$}
323: \key{print defaults}{\\d}
324: \key{set debug level to $n$}{\\g $n$}
325: \key{set memory debug level to $n$}{\\gm $n$}
326: \key{enable/disable logfile}{\\l \{filename\}}
327: \key{print \%$n$ in pretty matrix format}{\\m}
328: \key{set output mode (raw, default, prettyprint)}{\\o $n$}
329: \key{set $n$ significant digits}{\\p $n$}
330: \key{set $n$ terms in series}{\\ps $n$}
331: \key{quit GP}{\\q}
332: \key{print the list of PARI types}{\\t}
333: \key{print the list of user-defined functions}{\\u}
334: \li{read file into GP}{\\r {\rm filename}}
335: \li{write \%$n$ to file}{\\w $n$ {\rm filename}}
336:
337: \section{GP Within Emacs}
338: \li{to enter GP from within Emacs:}{M-x gp{\rm,} C-u M-x gp}
339: \li{word completion}{<TAB>}
340: \li{help menu window}{M-\\c}
341: \li{describe function}{M-?}
342: \li{display \TeX'd PARI manual}{M-x gpman}
343: \li{set prompt string}{M-\\p}
344: \li{break line at column 100, insert \kbd{\\}}{M-\\\\}
345: \li{PARI metacommand \kbd{\\}{\it letter}}{M-\\\hbox{\it letter}}
346:
347: \section{Reserved Variable Names}
348: \li{$\pi=3.14159\cdots$}{Pi}
349: \li{Euler's constant ${}=.57721\cdots$}{Euler}
350: \li{square root of $-1$}{I}
351: \li{big-oh notation}{O}
352:
353: % ****************************************
354: % This goes at the bottom of page 1
355: \shortcopyrightnotice
356: \newcolumn
357:
358: \section{PARI Types \& Input Formats}
359: \li{\typ{INT}. Integers}{$\pm n$}
360: \li{\typ{REAL}. Real Numbers}{$\pm n.ddd$}
361: \li{\typ{INTMOD}. Integers modulo $m$}{Mod$(n,m)$}
362: \li{\typ{FRAC}. Rational Numbers}{$n/m$}
363: \li{\typ{COMPLEX}. Complex Numbers}{$x+\kbd{I}*y$}
364: \li{\typ{PADIC}. $p$-adic Numbers}{$x+O(p$\pow$k)$}
365: \li{\typ{QUAD}. Quadratic Numbers}{$x + y\,*\;$quadgen$(D)$}
366: \li{\typ{POLMOD}. Polynomials modulo $g$}{Mod$(f,g)$}
367: \li{\typ{POL}. Polynomials}{$a*x$\pow$n+\cdots+b$}
368: \li{\typ{SER}. Power Series}{$f+O(x$\pow$k)$}
369: \li{\typ{QFI}/\typ{QFR}. Imag/Real bin.\ quad.\ forms}
370: {Qfb$(a,b,c,\{d\})$}
371: \li{\typ{RFRAC}. Rational Functions}{$f/g$}
372: \li{\typ{VEC}/\typ{COL}. Row/Column Vectors}
373: {$[x,y,z]${\rm,} $[x,y,z]$\til}
374: %\li{\typ{COL}. Column Vectors}{$[x,y,z]$\til}
375: \li{\typ{MAT}. Matrices}{$[x,y;z,t;u,v]$}
376: \li{\typ{LIST}. Lists}{List$([x,y,z])$}
377: \li{\typ{STR}. Strings}{"aaa"}
378:
379: \section{Standard Operators}
380: \li{basic operations}{+{\rm,} - {\rm,} *{\rm,} /{\rm,} \pow}
381: \li{\kbd{i=i+1}, \kbd{i=i-1}, \kbd{i=i*j}, \dots}
382: {i++{\rm,} i--{\rm,} i*=j{\rm,}\dots}
383: \li{euclidean quotient, remainder}{$x$\bs/$y${\rm,} $x$\bs$y${\rm,}
384: $x$\%$y${\rm,} divrem$(x,y)$}
385: \li{shift $x$ left or right $n$ bits}{ $x$<<$n$, $x$>>$n$
386: {\rm or} shift$(x,n)$}
387: \li{comparison operators}{<={\rm, }<{\rm, }>={\rm, }>{\rm, }=={\rm, }!=}
388: \li{boolean operators (or, and, not)}{||{\rm, } \&\&{\rm ,} !}
389: \li{sign of $x=-1,0,1$}{sign$(x)$}
390: \li{maximum/minimum of $x$ and $y$}{max{\rm,} min$(x,y)$}
391: \li{integer or real factorial of $x$}{$x$!~{\rm or} fact$(x)$}
392:
393: \section{Conversions}
394: %
395: \subsec{Change Objects}
396: \li{make $x$ a vector, matrix, set, list, string}
397: {Vec{\rm,}Mat{\rm,}Set{\rm,}List{\rm,}Str}
398: \li{create PARI object $(x\mod y)$}{Mod$(x,y)$}
399: \li{make $x$ a polynomial of $v$}{Pol$(x,\{v\})$}
400: \li{as above, starting with constant term}{Polrev$(x,\{v\})$}
401: \li{make $x$ a power series of $v$}{Ser$(x,\{v\})$}
402: \li{PARI type of object $x$}{type$(x, \{t\})$}
403: \li{object $x$ with precision $n$}{prec$(x,\{n\})$}
404: \li{evaluate $f$ replacing vars by their value}{eval$(f)$}
405: %
406: \subsec{Select Pieces of an Object}
407: \li{length of $x$}{length$(x)$}
408: \li{$n$-th component of $x$}{component$(x,n)$}
409: \li{$n$-th component of vector/list $x$}{$x$[n]}
410: \li{$(m,n)$-th component of matrix $x$}{$x$[m,n]}
411: \li{row $m$ or column $n$ of matrix $x$}{$x$[m,]{\rm,} $x$[,n]}
412: \li{numerator of $x$}{numerator$(x)$}
413: \li{lowest denominator of $x$}{denominator$(x)$}
414: %
415: \subsec{Conjugates and Lifts}
416: \li{conjugate of a number $x$}{conj$(x)$}
417: \li{conjugate vector of algebraic number $x$}{conjvec$(x)$}
418: \li{norm of $x$, product with conjugate}{norm$(x)$}
419: \li{square of $L^2$ norm of vector $x$}{norml2$(x)$}
420: \li{lift of $x$ from Mods}{lift{\rm,} centerlift$(x)$}
421:
422: \section{Random Numbers}
423: \li{random integer between $0$ and $N-1$}{random$(\{N\})$}
424: \li{get random seed}{getrand$()$}
425: \li{set random seed to $s$}{setrand$(s)$}
426:
427: \begingroup
428: \outer\def\subsec#1{\filbreak
429: \vskip 0.05ex plus 0.05ex
430: {\bf #1}
431: \vskip 0.05ex plus 0.05ex
432: }
433:
434: \section{Lists, Sets \& Sorting}
435: \li{sort $x$ by $k$th component}{vecsort$(x,\{k\},\{\fl=0\})$}
436: {\bf Sets} (= row vector of strings with strictly increasing entries)\hfill\break
437: %
438: \li{intersection of sets $x$ and $y$}{setintersect$(x,y)$}
439: \li{set of elements in $x$ not belonging to $y$}{setminus$(x,y)$}
440: \li{union of sets $x$ and $y$}{setunion$(x,y)$}
441: \li{look if $y$ belongs to the set $x$}{setsearch$(x,y,\fl)$}
442: %
443: \subsec{Lists}
444: \li{create empty list of maximal length $n$}{listcreate$(n)$}
445: \li{delete all components of list $l$}{listkill$(l)$}
446: \li{append $x$ to list $l$}{listput$(l,x,\{i\})$}
447: \li{insert $x$ in list $l$ at position $i$}{listinsert$(l,x,i)$}
448: \li{sort the list $l$}{listsort$(l,\fl)$}
449:
450: \section{Programming \& User Functions}
451: \subsec{Control Statements {\rm ($X$: formal parameter in expression \seq)}}
452: \li{eval.\ \seq\ for $a\le X\le b$}{for$(X=a,b,\seq)$}
453: \li{eval.\ \seq\ for $X$ dividing $n$}{fordiv$(n,X,\seq)$}
454: \li{eval.\ \seq\ for primes $a\le X\le b$}{forprime$(X=a,b,\seq)$}
455: \li{eval.\ \seq\ for $a\le X\le b$ stepping $s$}{forstep$(X=a,b,s,\seq)$}
456: \li{multivariable {\tt for}}{forvec$(X=v,\seq)$}
457: \li{if $a\ne0$, evaluate \seq1, else \seq2}{if$(a,\{\seq1\},\{\seq2\})$}
458: \li{evaluate \seq\ until $a\ne0$}{until$(a,\seq)$}
459: \li{while $a\ne0$, evaluate \seq}{while$(a,\seq)$}
460: \li{exit $n$ innermost enclosing loops}{break$(\{n\})$}
461: \li{start new iteration of $n$th enclosing loop}{next$(\{n\})$}
462: \li{return $x$ from current subroutine}{return$(x)$}
463: \li{error recovery (try \seq1)}{trap$(\{err\},\{\seq2\},\{\seq1\})$}
464: %
465: \subsec{Input/Output}
466: \li{prettyprint args with/without newline}{printp(){\rm,} printp1()}
467: \li{print args with/without newline}{print(){\rm,} print1()}
468: \li{read a string from keyboard}{input$()$}
469: \li{reorder priority of variables $[x,y,z]$}{reorder$(\{[x,y,z]\})$}
470: \li{output \args\ in \TeX\ format}{printtex$(\args)$}
471: \li{write \args\ to file}{write{\rm,} write1{\rm,} writetex$(\file,\args)$}
472: \li{read file into GP}{read(\{\file\})}
473: %
474: \subsec{Interface with User and System}
475: \li{allocates a new stack of $s$ bytes}{allocatemem$(\{s\})$}
476: \li{execute system command $a$}{system$(a)$}
477: \li{as above, feed result to GP}{extern$(a)$}
478: \li{install function from library}{install$(f,code,\{\var{gpf\/}\},\{\var{lib}\})$}
479: \li{alias \var{old}\ to \var{new}}{alias$(\var{new},\var{old})$}
480: \li{new name of function $f$ in GP 2.0}{whatnow$(f)$}
481: %
482: \subsec{User Defined Functions}
483: \leavevmode
484: {\tt name(formal vars) = local(local vars); \var{seq}}\hfill\break
485: {\tt struct.member = \var{seq}}\hfill\break
486: \li{kill value of variable or function $x$}{kill$(x)$}
487: \li{declare global variables}{global$(x,...)$}
488:
489: \section{Iterations, Sums \& Products}
490: \li{numerical integration}{intnum$(X=a,b,\expr,\fl)$}
491: \li{sum \expr\ over divisors of $n$}{sumdiv$(n,X,\expr)$}
492: \li{sum $X=a$ to $X=b$, initialized at $x$}{sum$(X=a,b,\expr,\{x\})$}
493: \li{sum of series \expr}{suminf$(X=a,\expr)$}
494: \li{sum of alternating/positive series}{sumalt{\rm,} sumpos}
495: \li{product $a\le X\le b$, initialized at $x$}{prod$(X=a,b,\expr,\{x\})$}
496: \li{product over primes $a\le X\le b$}{prodeuler$(X=a,b,\expr)$}
497: \li{infinite product $a\le X\le\infty$}{prodinf$(X=a,\expr)$}
498: \li{real root of \expr\ between $a$ and $b$}{solve$(X=a,b,\expr)$}
499: \endgroup
500:
501: % This goes at the top of page 4 (=1st column on back of reference card)
502:
503: \section{Vectors \& Matrices}
504: %
505: \li{dimensions of matrix $x$}{matsize$(x)$}
506: \li{concatenation of $x$ and $y$}{concat$(x,\{y\})$}
507: \li{extract components of $x$}{vecextract$(x,y,\{z\})$}
508: \li{transpose of vector or matrix $x$}{mattranspose$(x)$ {\rm or} $x$\til}
509: \li{adjoint of the matrix $x$}{matadj$(x)$}
510: \li{eigenvectors of matrix $x$}{mateigen$(x)$}
511: \li{characteristic polynomial of $x$}{charpoly$(x,\{v\},\fl)$}
512: \li{trace of matrix $x$}{trace$(x)$}
513: %
514: \subsec{Constructors \& Special Matrices}
515: \li{row vec.\ of \expr\ eval'ed at $1\le X\le n$}{vector$(n,\{X\},\{\expr\})$}
516: \li{col.\ vec.\ of \expr\ eval'ed at $1\le X\le n$}{vectorv$(n,\{X\},\{\expr\})$}
517: \li{matrix $1\le X\le m$, $1\le Y\le n$}{matrix$(m,n,\{X\},\{Y\},\{\expr\})$}
518: \li{diagonal matrix whose diag. is $x$}{matdiagonal$(x)$}
519: \li{$n\times n$ identity matrix}{matid$(n)$}
520: \li{Hessenberg form of square matrix $x$}{mathess$(x)$}
521: \li{$n\times n$ Hilbert matrix $H_{ij}=(i+j-1)^{-1}$}{mathilbert$(n)$}
522: \li{$n\times n$ Pascal triangle $P_{ij}={i\choose j}$}{matpascal$(n-1)$}
523: \li{companion matrix to polynomial $x$}{matcompanion$(x)$}
524: %
525: \subsec{Gaussian elimination}
526: \li{determinant of matrix $x$}{matdet$(x,\fl)$}
527: \li{kernel of matrix $x$}{matker$(x,\fl)$}
528: \li{intersection of column spaces of $x$ and $y$}{matintersect$(x,y)$}
529: \li{solve $M*X = B$ ($M$ invertible)}{matsolve$(M,B)$}
530: \li{as solve, modulo $D$ (col. vector)}{matsolvemod$(M,D,B)$}
531: \li{one sol of $M*X = B$}{matinverseimage$(M,B)$}
532: \li{basis for image of matrix $x$}{matimage$(x)$}
533: \li{supplement columns of $x$ to get basis}{matsupplement$(x)$}
534: \li{rows, cols to extract invertible matrix}{matindexrank$(x)$}
535: \li{rank of the matrix $x$}{matrank$(x)$}
536:
537: \section{Lattices \& Quadratic Forms}
538: \li{upper triangular Hermite Normal Form}{mathnf$(x)$}
539: \li{HNF of $x$ where $d$ is a multiple of det$(x)$}{mathnfmod$(x,d)$}
540: \li{vector of elementary divisors of $x$}{matsnf$(x)$}
541: \li{LLL-algorithm applied to columns of $x$}{qflll$(x,\fl)$}
542: \li{like \kbd{qflll}, $x$ is Gram matrix of lattice}
543: {qflllgram$(x,\fl)$}
544: \li{LLL-reduced basis for kernel of $x$}{matkerint$(x)$}
545: \li{$\ZZ$-lattice $\longleftrightarrow$ $\QQ$-vector space}{matrixqz$(x,p)$}
546: %
547: \subsec{Quadratic Forms}
548: \li{signature of quad form $^ty*x*y$}{qfsign$(x)$}
549: \li{decomp into squares of $^ty*x*y$}{qfgaussred$(x)$}
550: \li{find up to $m$ sols of $^ty*x*y\le b$}{qfminim$(x,b,m)$}
551: %\li{perfection rank of $x$}{qfperfection$(x)$}
552: \li{eigenvals/eigenvecs for real symmetric $x$}{qfjacobi$(x)$}
553:
554: \section{Formal \& p-adic Series}
555: \li{truncate power series or $p$-adic number}{truncate$(x)$}
556: \li{valuation of $x$ at $p$}{valuation$(x,p)$}
557: \subsec{Dirichlet and Power Series}
558: \li{Taylor expansion around $0$ of $f$ w.r.t. $x$}{taylor$(f,x)$}
559: \li{$\sum a_kb_kt^k$ from $\sum a_kt^k$ and $\sum b_kt^k$}{serconvol$(x,y)$}
560: \li{$f=\sum a_k*t^k$ from $\sum (a_k/k!)*t^k$}{serlaplace$(f)$}
561: \li{reverse power series $F$ so $F(f(x))=x$}{serreverse$(f)$}
562: \li{Dirichlet series multiplication / division}{dirmul{\rm,} dirdiv$(x,y)$}
563: \li{Dirichlet Euler product ($b$ terms)}{direuler$(p=a,b,\expr)$}
564: \subsec{$p$-adic Functions}
565: \li{square of $x$, good for $2$-adics}{sqr$(x)$}
566: \li{Teichmuller character of $x$}{teichmuller$(x)$}
567: \li{Newton polygon of $f$ for prime $p$}{newtonpoly$(f,p)$}
568:
569: \newcolumn
570: \title{PARI-GP Reference Card}
571: \centerline{(PARI-GP version \PARIversion)}
572:
573: \section{Polynomials \& Rational Functions}
574: %
575: \li{degree of $f$}{poldegree$(f)$}
576: \li{coefficient of degree $n$ of $f$}{polcoeff$(f,n)$}
577: \li{round coeffs of $f$ to nearest integer}{round$(f,\{\&e\})$}
578: \li{gcd of coefficients of $f$}{content$(f)$}
579: \li{replace $x$ by $y$ in $f$}{subst$(f,x,y)$}
580: \li{discriminant of polynomial $f$}{poldisc$(f)$}
581: %\li{elementary divisors of Z[a]/f'(a)Z[a]}{poldiscreduced$(f)$}
582: \li{resultant of $f$ and $g$}{polresultant$(f,g,\fl)$}
583: \li{as above, give $[u,v,d]$, $xu + yv = d$}{bezoutres$(x,y)$}
584: \li{derivative of $f$ w.r.t. $x$}{deriv$(f,x)$}
585: \li{formal integral of $f$ w.r.t. $x$}{intformal$(f,x)$}
586: \li{reciprocal poly $x^{\deg f}f(1/x)$}{polrecip$(f)$}
587: \li{interpolating poly evaluated at $a$}{polinterpolate$(X,\{Y\},\{a\},\{\&e\})$}
588: \li{initialize $t$ for Thue equation solver}{thueinit(f)}
589: \li{solve Thue equation $f(x,y)=a$}{thue$(t,a,\{sol\})$}
590: %
591: \subsec{Roots and Factorization}
592: \li{number of real roots of $f$, $a < x\le b$}{polsturm$(f,\{a\},\{b\})$}
593: \li{complex roots of $f$}{polroots$(f)$}
594: \li{symmetric powers of roots of $f$ up to $n$}{polsym$(f,n)$}
595: \li{roots of $f \mod p$}{polrootsmod$(f,p,\fl)$}
596: \li{factor $f$}{factor$(f,\{lim\})$}
597: \li{factorization of $f\mod p$}{factormod$(f,p,\fl)$}
598: \li{factorization of $f$ over $\FF_{p^a}$}{factorff$(f,p,a)$}
599: \li{$p$-adic fact. of $f$ to prec. $r$}{factorpadic$(f,p,r,\fl)$}
600: \li{$p$-adic roots of $f$ to prec. $r$}{polrootspadic$(f,p,r)$}
601: \li{$p$-adic root of $f$ cong. to $a\mod p$}{padicappr$(f,a)$}
602: \li{Newton polygon of $f$ for prime $p$}{newtonpoly$(f,p)$}
603: %
604: \subsec{Special Polynomials}
605: \li{$n$th cyclotomic polynomial in var. $v$}{polcyclo$(n,\{v\})$}
606: \li{$d$-th degree subfield of $\QQ(\zeta_n)$} {polsubcyclo$(n,d,\{v\})$}
607: \li{$n$-th Legendre polynomial}{pollegendre$(n)$}
608: \li{$n$-th Tchebicheff polynomial}{poltchebi$(n)$}
609: \li{Zagier's polynomial of index $n$,$m$}{polzagier$(n,m)$}
610:
611: \section{Transcendental Functions}
612: \li{real, imaginary part of $x$}{real$(x)$, imag$(x)$}
613: \li{absolute value, argument of $x$}{abs$(x)$, arg$(x)$}
614: \li{square/nth root of $x$}{sqrt$(x)$, sqrtn$(x,n,\&z)$}
615: \li{trig functions}{sin, cos, tan, cotan}
616: \li{inverse trig functions}{asin, acos, atan}
617: \li{hyperbolic functions}{sinh, cosh, tanh}
618: \li{inverse hyperbolic functions}{asinh, acosh, atanh}
619: \li{exponential of $x$}{exp$(x)$}
620: \li{natural log of $x$}{ln$(x)$ {\rm or} log$(x)$}
621: %
622: \li{gamma function $\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}dt$}{gamma$(x)$}
623: %\li{half-integer gamma function $\Gamma(n+1/2)$}{gammah$(n)$}
624: \li{logarithm of gamma function}{lngamma$(x)$}
625: \li{$\psi(x)=\Gamma'(x)/\Gamma(x)$}{psi$(x)$}
626: \li{incomplete gamma function ($y=\Gamma(s)$)}{incgam$(s,x,\{y\})$}
627: \li{exponential integral $\int_x^\infty e^{-t}/t\,dt$}{eint1$(x)$}
628: \li{error function $2/\sqrt\pi\int_x^\infty e^{-t^2}dt$}{erfc$(x)$}
629: \li{dilogarithm of $x$}{dilog$(x)$}
630: \li{$m$th polylogarithm of $x$}{polylog$(m,x,\fl)$}
631: \li{$U$-confluent hypergeometric function}{hyperu$(a,b,u)$}
632: \li{$J$-Bessel function $J_{n+1/2}(x)$}{besseljh$(n,x)$}
633: \li{$K$-Bessel function of index \var{nu}}{besselk$(\var{nu},x)$}
634:
635: \section{Elementary Arithmetic Functions}
636: \li{vector of binary digits of $|x|$}{binary$(x)$}
637: \li{give bit number $n$ of integer $x$}{bittest$(x,n)$}
638: \li{ceiling of $x$}{ceil$(x)$}
639: \li{floor of $x$}{floor$(x)$}
640: \li{fractional part of $x$}{frac$(x)$}
641: \li{round $x$ to nearest integer}{round$(x,\{\&e\})$}
642: \li{truncate $x$}{truncate$(x,\{\&e\})$}
643: \li{gcd of $x$ and $y$}{gcd$(x,y)$}
644: \li{LCM of $x$ and $y$}{lcm$(x,y)$}
645: \li{gcd of entries of a vector/matrix}{content$(x)$}
646: \par
647: \subsec{Primes and Factorization}
648: \li{add primes in $v$ to the prime table}{addprimes$(v)$}
649: \li{the $n$th prime}{prime$(n)$}
650: \li{vector of first $n$ primes}{primes$(n)$}
651: \li{smallest prime $\ge x$}{nextprime$(x)$}
652: \li{largest prime $\le x$}{precprime$(x)$}
653: \li{factorization of $x$}{factor$(x,\{lim\})$}
654: \li{reconstruct $x$ from its factorization}{factorback$(fa,\{nf\})$}
655: \par
656: \subsec{Divisors}
657: \li{number of distinct prime divisors}{omega$(x)$}
658: \li{number of prime divisors with mult}{bigomega$(x)$}
659: \li{number of divisors of $x$}{numdiv$(x)$}
660: \li{row vector of divisors of $x$}{divisors$(x)$}
661: \li{sum of ($k$-th powers of) divisors of $x$}{sigma$(x,\{k\})$}
662: \par
663: \subsec{Special Functions and Numbers}
664: \li{binomial coefficient $x\choose y$}{binomial$(x,y)$}
665: \li{Bernoulli number $B_n$ as real}{bernreal$(n)$}
666: \li{Bernoulli vector $B_0,B_2,\ldots,B_{2n}$}{bernvec$(n)$}
667: \li{$n$th Fibonacci number}{fibonacci$(n)$}
668: \li{Euler $\phi$-function}{eulerphi$(x)$}
669: \li{M\"obius $\mu$-function}{moebius$(x)$}
670: \li{Hilbert symbol of $x$ and $y$ (at $p$)}{hilbert$(x,y,\{p\})$}
671: \li{Kronecker-Legendre symbol $({x\over y})$}{kronecker$(x,y)$}
672: \par
673: \subsec{Miscellaneous}
674: \li{integer or real factorial of $x$}{$x!$ {\rm or} fact$(x)$}
675: \li{integer square root of $x$}{sqrtint$(x)$}
676: \li{solve $z\equiv x$ and $z\equiv y$}{chinese$(x,y)$}
677: \li{minimal $u,v$ so $xu+yv=\gcd(x,y)$}{bezout$(x,y)$}
678: \li{multiplicative order of $x$ (intmod)}{znorder$(x)$}
679: \li{primitive root mod prime power $q$}{znprimroot$(q)$}
680: \li{structure of $(\ZZ/n\ZZ)^*$}{znstar$(n)$}
681: \li{continued fraction of $x$}{contfrac$(x,\{b\},\{lmax\})$}
682: \li{last convergent of continued fraction $x$}{contfracpnqn$(x)$}
683: \li{best rational approximation to $x$}{bestappr$(x,k)$}
684:
685: \section{True-False Tests}
686: \li{is $x$ the disc. of a quadratic field?}{isfundamental$(x)$}
687: \li{is $x$ a prime?}{isprime$(x)$}
688: \li{is $x$ a strong pseudo-prime?}{ispseudoprime$(x)$}
689: \li{is $x$ square-free?}{issquarefree$(x)$}
690: \li{is $x$ a square?}{issquare$(x,\{\&n\})$}
691: \li{is \var{pol}\ irreducible?}{polisirreducible$(\var{pol})$}
692:
693: % This goes at the bottom of the second page (column 6)
694: \copyrightnotice
695: %
696:
697: %%%%%%%%%%% Extra Material (part II)
698: %
699: \newcolumn
700: \title{PARI-GP Reference Card (2)}
701: \centerline{(PARI-GP version \PARIversion)}
702:
703: \section{Elliptic Curves}
704: %
705: Elliptic curve initially given by $5$-tuple $E=$\kbd{[a1,a2,a3,a4,a6]}.
706: Points are \kbd{[x,y]}, the origin is \kbd{[0]}.
707: \hfill\break
708: \li{Initialize elliptic struct. $\var{ell}$, i.e create}{ellinit$(E,\fl)$}
709: \leavevmode\strut\hskip1em
710: $a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,disc,j$. This data can be
711: recovered by typing \kbd{\var{ell}.a1},$\dots$,\kbd{\var{ell}.j}.
712: If $\var{fl}$ omitted, also
713: \hfill\break
714: \beginindentedkeys
715: \li{$E$ defined over $\RR$}{}
716: \begindoubleindentedkeys
717: \key{$x$-coords. of points of order $2$}{\var{ell}.roots}
718: \key{real and complex periods}{\var{ell}.omega}
719: \key{associated quasi-periods}{\var{ell}.eta}
720: \key{volume of complex lattice}{\var{ell}.area}
721: \enddoubleindentedkeys
722: \li{$E$ defined over $\QQ_p$, $|j|_p>1$}{}
723: \begindoubleindentedkeys
724: \key{$x$-coord. of unit $2$ torsion point}{\var{ell}.roots}
725: \key{Tate's $[u^2, u, q]$}{\var{ell}.tate}
726: \key{Mestre's $w$}{\var{ell}.w}
727: \endindentedkeys
728: \li{change curve $E$ using $v=[u,r,s,t]$}{ellchangecurve$(ell,v)$}
729: \li{change point $z$ using $v=[u,r,s,t]$}{ellchangepoint$(z,v)$}
1.2 ! noro 730: \li{cond, min mod, Tamagawa nmbr $[N,v,c]$}{ellglobalred$(ell)$}
1.1 noro 731: \li{Kodaira type of $p$ fiber of $E$}{elllocalred$(ell,p)$}
732: \li{add points $z1+z2$}{elladd$(ell,z1,z2)$}
733: \li{subtract points $z1-z2$}{ellsub$(ell,z1,z2)$}
734: \li{compute $n\cdot z$}{ellpow$(ell,z,n)$}
735: \li{check if $z$ is on $E$}{ellisoncurve$(ell,z)$}
736: \li{order of torsion point $z$}{ellorder$(ell,z)$}
737: \li{torsion subgroup with generators}{elltors$(ell)$}
738: \li{$y$-coordinates of point(s) for $x$}{ellordinate$(ell,x)$}
739: \li{canonical bilinear form taken at $z1$, $z2$}{ellbil$(ell,z1,z2)$}
740: \li{canonical height of $z$}{ellheight$(ell,z,\fl)$}
741: \li{height regulator matrix for pts in $x$}{ellheightmatrix$(ell,x)$}
742: \li{$p$th coeff $a_p$ of $L$-function, $p$ prime}{ellap$(ell,p)$}
743: \li{$k$th coeff $a_k$ of $L$-function}{ellak$(ell,k)$}
744: \li{vector of first $n$ $a_k$'s in $L$-function}{ellan$(ell,n)$}
745: \li{$L(E,s)$, set $A\approx1$}{elllseries$(ell,s,\{A\})$}
746: \li{root number for $L(E,.)$ at $p$}{ellrootno$(ell,\{p\})$}
747: \li{modular parametrization of $E$}{elltaniyama$(ell)$}
748: \li{point $[\wp(z),\wp'(z)]$ corresp. to $z$}{ellztopoint$(ell,z)$}
749: \li{complex $z$ such that $p=[\wp(z),\wp'(z)]$}{ellpointtoz$(ell,p)$}
750:
751: \section{Elliptic \& Modular Functions}
752: %
753: \li{arithmetic-geometric mean}{agm$(x,y)$}
754: \li{elliptic $j$-function $1/q+744+\cdots$}{ellj$(x)$}
755: \li{Weierstrass $\sigma$ function}{ellsigma$(ell,z,\fl)$}
756: \li{Weierstrass $\wp$ function}{ellwp$(ell,\{z\},\fl)$}
757: \li{Weierstrass $\zeta$ function}{ellzeta$(ell,z)$}
758: \li{modified Dedekind $\eta$ func. $\prod(1-q^n)$}{eta$(x,\fl)$}
759: \li{Jacobi sine theta function}{theta$(q,z)$}
760: \li{k-th derivative at z=0 of \kbd{theta}$(q,z)$}{thetanullk$(q,k)$}
761: \li{Weber's $f$ functions}{weber$(x,\fl)$}
762: \li{Riemann's zeta $\zeta(s)=\sum n^{-s}$}{zeta$(s)$}
763: %
764: \shortcopyrightnotice
765: \newcolumn
766:
767: \section{Graphic Functions}
768: \li{crude graph of \expr\ between $a$ and $b$}{plot$(X=a,b,expr)$}
769: \subsec{High-resolution plot {\rm (immediate plot)}}
770: \li{plot \expr\ between $a$ and $b$}{ploth$(X=a,b,expr,\fl,\{n\})$}
771: \li{plot points given by lists $lx$, $ly$}{plothraw$(lx,ly,\fl)$}
772: \li{terminal dimensions}{plothsizes$()$}
773: %
774: \subsec{Rectwindow functions}
775: \li{init window $w$, with size $x$,$y$}{plotinit$(w,x,y)$}
776: \li{erase window $w$}{plotkill$(w)$}
777: \li{copy $w$ to $w2$ with offset $(dx,dy)$}{plotcopy$(w,w2,dx,dy)$}
778: \li{scale coordinates in $w$}{plotscale$(w,x_1,x_2,y_1,y_2)$}
779: \li{\kbd{ploth} in $w$}{plotrecth$(w,X=a,b,expr,\fl,\{n\})$}
780: \li{\kbd{plothraw} in $w$}{plotrecthraw$(w,data,\fl)$}
781: \li{draw window $w_1$ at $(x_1,y_1)$, \dots} {plotdraw$([[w_1,x_1,y_1],\dots])$}
782: %
783: \subsec{Low-level Rectwindow Functions}
784: %\li{}{plotlinetype$(w,)$}
785: %\li{}{plotpointtype$(w,)$}
786: %\li{}{plotterm$(w,)$}
787: \li{set current drawing color in $w$ to $c$}{plotcolor$(w,c)$}
788: \li{current position of cursor in $w$}{plotcursor$(w)$}
789: %
790: \li{write $s$ at cursor's position}{plotstring$(w,s)$}
791: \li{move cursor to $(x,y)$}{plotmove$(w,x,y)$}
792: \li{move cursor to $(x+dx,y+dy)$}{plotrmove$(w,dx,dy)$}
793: \li{draw a box to $(x_2,y_2)$}{plotbox$(w,x_2,y_2)$}
794: \li{draw a box to $(x+dx,y+dy)$}{plotrbox$(w,dx,dy)$}
795: \li{draw polygon}{plotlines$(w,lx,ly,\fl)$}
796: \li{draw points}{plotpoints$(w,lx,ly)$}
797: \li{draw line to $(x+dx,y+dy)$}{plotrline$(w,dx,dy)$}
798: \li{draw point $(x+dx,y+dy)$}{plotrpoint$(w,dx,dy)$}
799: %
800: \subsec{Postscript Functions}
801: \li{as {\tt ploth}}{psploth$(X=a,b,expr,\fl,\{n\})$}
802: \li{as {\tt plothraw}}{psplothraw$(lx,ly,\fl)$}
803: \li{as {\tt plotdraw}}{psdraw$([[w_1,x_1,y_1],\dots])$}
804: \newcolumn
805:
806: \section{Binary Quadratic Forms}
807: %
808: \li{create $ax^2+bxy+cy^2$ (distance $d$) }{Qfb$(a,b,c,\{d\})$}
809: \li{reduce $x$ ($s =\sqrt{D}$, $l=\lfloor s\rfloor$)}
810: {qfbred$(x,\fl,\{D\},\{l\},\{s\})$}
811: \li{composition of forms}{$x*y$ {\rm or }qfbnucomp$(x,y,l)$}
812: \li{$n$-th power of form}{$x$\pow$n$ {\rm or }qfbnupow$(x,n)$}
813: \li{composition without reduction}{qfbcompraw$(x,y)$}
814: \li{$n$-th power without reduction}{qfbpowraw$(x,n)$}
815: \li{prime form of disc. $x$ above prime $p$}{qfbprimeform$(x,p)$}
816: \li{class number of disc. $x$}{qfbclassno$(x)$}
817: \li{Hurwitz class number of disc. $x$}{qfbhclassno$(x)$}
818:
819: \section{Quadratic Fields}
820: %
821: \li{quadratic number $\omega=\sqrt x$ or $(1+\sqrt x)/2$}{quadgen$(x)$}
822: \li{minimal polynomial of $\omega$}{quadpoly$(x)$}
823: \li{discriminant of $\QQ(\sqrt{D})$}{quaddisc$(x)$}
824: \li{regulator of real quadratic field}{quadregulator$(x)$}
825: \li{fundamental unit in real $\QQ(x)$}{quadunit$(x)$}
826: \li{class group of $\QQ(\sqrt{D})$}{quadclassunit$(D,\fl,\{t\})$}
827: \li{Hilbert class field of $\QQ(\sqrt{D})$}{quadhilbert$(D,\fl)$}
828: \li{ray class field modulo $f$ of $\QQ(\sqrt{D})$}{quadray$(D,f,\fl)$}
829:
830: \section{General Number Fields: Initializations}
831: A number field $K$ is given by a monic irreducible $f\in\ZZ[X]$.\hfill\break
832: \li{init number field structure \var{nf}}{nfinit$(f,\fl)$}
833: \subsec{nf members:}
834: \beginindentedkeys
835: \key{polynomial defining \var{nf}, $f(\theta)=0$}{\var{nf}.pol}
836: \key{number of [real,complex] places}{\var{nf}.sign}
837: \key{discriminant of \var{nf}}{\var{nf}.disc}
838: \key{$T_2$ matrix}{\var{nf}.t2}
839: \key{vector of roots of $f$}{\var{nf}.roots}
840: \key{integral basis of $\ZZ_K$ as powers of $\theta$}{\var{nf}.zk}
841: \key{different}{\var{nf}.diff}
842: \key{codifferent}{\var{nf}.codiff}
843: \endindentedkeys
844: \li{recompute \var{nf}\ using current precision}{nfnewprec$(nf)$}
845: \li{init relative \var{rnf}\ given by $g=0$ over $K$}{rnfinit$(\var{nf},g)$}
846: %
847: \li{init big number field structure \var{bnf}}{bnfinit$(f,\fl)$}
848: \subsec{bnf members: {\rm same as \var{nf}, plus}}
849: \beginindentedkeys
850: \key{underlying \var{nf}}{\var{bnf}.nf}
851: \key{classgroup}{\var{bnf}.clgp}
852: \key{regulator}{\var{bnf}.reg}
853: \key{fundamental units}{\var{bnf}.fu}
854: \key{torsion units}{\var{bnf}.tu}
855: \key{$[tu,fu]$, $[fu,tu]$}{\var{bnf}.tufu{\rm/}futu}
856: \endindentedkeys
857: \li{compute a \var{bnf}\ from small \var{bnf}}{bnfmake$(\var{sbnf})$}
858: %
859: \li{add $S$-class group and units, yield \var{bnfs}}{bnfsunit$(\var{nf},S)$}
860: \li{init class field structure \var{bnr}}{bnrinit$(\var{bnf},m,\fl)$}
861: %
862: \subsec{bnr members: {\rm same as \var{bnf}, plus}}
863: \beginindentedkeys
864: \key{underlying \var{bnf}}{\var{bnr}.bnf}
865: \key{structure of $(\ZZ_K/m)^*$}{\var{bnr}.zkst}
866: \endindentedkeys
867:
868: \section{Simple Arithmetic Invariants (nf)}
869: Elements are rational numbers, polynomials, polmods, or column vectors (on
870: integral basis \kbd{\var{nf}.zk}).\hfill\break
871: \li{integral basis of field def. by $f=0$}{nfbasis$(f)$}
872: \li{field discriminant of field $f=0$}{nfdisc$(f)$}
873: \li{reverse polmod $a=A(X)\mod T(X)$}{modreverse$(a)$}
874: \li{Galois group of field $f=0$, $\deg f\le 11$}{polgalois$(f)$}
875: \li{smallest poly defining $f=0$}{polredabs$(f,\fl)$}
876: \li{small polys defining subfields of $f=0$}{polred$(f,\fl,\{p\})$}
877: \li{small polys defining suborders of $f=0$}{polredord$(f)$}
878: \li{poly of degree $\le k$ with root $x\in\CC$}{algdep$(x,k)$}
879: \li{small linear rel.\ on coords of vector $x$}{lindep$(x)$}
880: \li{are fields $f=0$ and $g=0$ isomorphic?}{nfisisom$(f,g)$}
881: \li{is field $f=0$ a subfield of $g=0$?}{nfisincl$(f,g)$}
882: \li{compositum of $f=0$, $g=0$}{polcompositum$(f,g,\fl)$}
883: %
884: \li{basic element operations (prefix \kbd{nfelt}):}{}
885: \leavevmode\strut\hskip1em
886: $($\kbd{nfelt}$)$\kbd{mul}, \kbd{pow}, \kbd{div}, \kbd{diveuc},
887: \kbd{mod}, \kbd{divrem}, \kbd{val}
888: \hfill\break
889: %
890: \li{express $x$ on integer basis}{nfalgtobasis$(\var{nf},x)$}
891: \li{express element\ $x$ as a polmod}{nfbasistoalg$(\var{nf},x)$}
892: \li{quadratic Hilbert symbol (at $p$)}{nfhilbert$(\var{nf},a,b,\{p\})$}
893: \li{roots of $g$ belonging to {\tt nf}}{nfroots$(\var{nf},g)$}
894: \li{factor $g$ in {\tt nf}}{nffactor$(\var{nf},g)$}
895: \li{factor $g$ mod prime $pr$ in {\tt nf}}{nffactormod$(\var{nf},g,pr)$}
896: \li{number of roots of $1$ in {\tt nf}}{nfrootsof1$(nf)$}
897: \li{conjugates of a root $\theta$ of {\tt nf}}{nfgaloisconj$(\var{nf},\fl)$}
898: \li{apply Galois automorphism $s$ to $x$}{nfgaloisapply$(\var{nf},s,x)$}
899: \li{subfields (of degree $d$) of {\tt nf}}{nfsubfields$(\var{nf},\{d\})$}
900: %
901: \subsec{Dedekind Zeta Function $\zeta_K$}
902: \li{$\zeta_K$ as Dirichlet series, $N(I)<b$}{dirzetak$(\var{nf},b)$}
903: \li{init \var{nfz}\ for field $f=0$}{zetakinit$(f)$}
904: \li{compute $\zeta_K(s)$}{zetak$(\var{nfz},s,\fl)$}
905: \li{Artin root number of $K$}{bnrrootnumber$(\var{bnr},\var{chi},\fl)$}
906:
907: \section{Class Groups \& Units (\var{bnf}, bnr)}
908: \leavevmode
909: $a1,\{a2\},\{a3\}$ usually $bnr,subgp$ or $\var{bnf},module,\{subgp\}$
910: \hfill\break
911: %
912: \li{remove GRH assumption from \var{bnf}}{bnfcertify$(\var{bnf})$}
913: \li{expo.~of ideal $x$ on class gp}{bnfisprincipal$(\var{bnf},x,\fl)$}
914: \li{expo.~of ideal $x$ on ray class gp}{bnrisprincipal$(\var{bnr},x,\fl)$}
915: \li{expo.~of $x$ on fund.~units}{bnfisunit$(\var{bnf},x)$}
916: \li{as above for $S$-units}{bnfissunit$(\var{bnfs},x)$}
917: \li{fundamental units of \var{bnf}}{bnfunit$(\var{bnf})$}
918: \li{signs of real embeddings of \kbd{\var{bnf}.fu}}{bnfsignunit$(\var{bnf})$}
919: %
920: \subsec{Class Field Theory}
921: \li{ray class group structure for mod.~$m$}{bnrclass$(\var{bnf},m,\fl)$}
922: \li{ray class number for mod.~$m$}{bnrclassno$(\var{bnf},m)$}
923: \li{discriminant of class field ext}{bnrdisc$(a1,\{a2\},\{a3\})$}
924: \li{ray class numbers, $l$ list of mods}{bnrclassnolist$(\var{bnf},l)$}
925: \li{discriminants of class fields}{bnrdisclist$(\var{bnf},l,\{arch\},\fl)$}
926: \li{decode output from \kbd{bnrdisclist}}{bnfdecodemodule$(\var{nf},fa)$}
927: \li{is modulus the conductor?}{bnrisconductor$(a1,\{a2\},\{a3\})$}
928: \li{conductor of character $chi$}{bnrconductorofchar$(\var{bnr},chi)$}
929: \li{conductor of extension}{bnrconductor$(a1,\{a2\},\{a3\},\fl)$}
930: \li{conductor of extension def.\ by $g$}{rnfconductor$(\var{bnf},g)$}
931: \li{Artin group of ext.\ def'd by $g$}{rnfnormgroup$(\var{bnr},g)$}
932: \li{subgroups of {\tt bnr}, index $<=b$}{subgrouplist$(\var{bnr},b,\fl)$}
933: \li{rel.\ eq.\ for class field def'd by $sub$}{rnfkummer$(\var{bnr},sub,\{d\})$}
934: \li{same, using Stark units (real field)}{bnrstark$(\var{bnr},sub,\fl)$}
935:
936: \newcolumn
937: \title{PARI-GP Reference Card (2)}
938: \centerline{(PARI-GP version \PARIversion)}
939:
940: \section{Ideals}
941: Ideals are elements, primes, or matrix of generators in HNF.\hfill\break
942: \li{is $id$ an ideal in {\tt nf}?}{nfisideal$(\var{nf},id)$}
943: \li{is $x$ principal in {\tt bnf}?}{bnfisprincipal$(\var{bnf},x)$}
944: \li{principal ideal generated by $x$}{idealprincipal$(\var{nf},x)$}
945: \li{principal idele generated by $x$}{ideleprincipal$(\var{nf},x)$}
946: \li{give $[a,b]$, s.t.~ $a\ZZ_K+b\ZZ_K = x$}{idealtwoelt$(\var{nf},x,\{a\})$}
947: \li{put ideal $a$ ($a\ZZ_K+b\ZZ_K$) in HNF form}{idealhnf$(\var{nf},a,\{b\})$}
948: \li{norm of ideal $x$}{idealnorm$(\var{nf},x)$}
949: \li{minimum of ideal $x$ (direction $v$)}{idealmin$(\var{nf},x,v)$}
950: \li{LLL-reduce the ideal $x$ (direction $v$)}{idealred$(\var{nf},x,\{v\})$}
951: %
952: \subsec{Ideal Operations}
953: \li{add ideals $x$ and $y$}{idealadd$(\var{nf},x,y)$}
954: \li{multiply ideals $x$ and $y$}{idealmul$(\var{nf},x,y,\fl)$}
955: \li{intersection of ideals $x$ and $y$}{idealintersect$(\var{nf},x,y,\fl)$}
956: \li{$n$-th power of ideal $x$}{idealpow$(\var{nf},x,n,\fl)$}
957: \li{inverse of ideal $x$}{idealinv$(\var{nf},x)$}
958: \li{divide ideal $x$ by $y$}{idealdiv$(\var{nf},x,y,\fl)$}
959: \li{Find $[a,b]\in x\times y$, $a+b=1$}{idealaddtoone$(\var{nf},x,\{y\})$}
960: %
961: \subsec{Primes and Multiplicative Structure}
962: \li{factor ideal $x$ in {\tt nf}}{idealfactor$(\var{nf},x)$}
963: \li{recover $x$ from its factorization in {\tt nf}}{factorback$(x,nf)$}
964: \li{decomposition of prime $p$ in {\tt nf}}{idealprimedec$(\var{nf},p)$}
965: \li{valuation of $x$ at prime ideal $pr$}{idealval$(\var{nf},x,pr)$}
966: \li{weak approximation theorem in {\tt nf}}{idealchinese$(\var{nf},x,y)$}
967: \li{give $bid=$structure of $(\ZZ_K/id)^*$}{idealstar$(\var{nf},id,\fl)$}
968: \li{discrete log of $x$ in $(\ZZ_K/bid)^*$}{ideallog$(\var{nf},x,bid)$}
969: \li{\kbd{idealstar} of all ideals of norm $\le b$}{ideallist$(\var{nf},b,\fl)$}
970: \li{add archimedean places}{ideallistarch$(\var{nf},b,\{ar\},\fl)$}
971: \li{init \kbd{prmod} structure}{nfmodprinit$(\var{nf},pr)$}
972: \li{kernel of matrix $M$ in $(\ZZ_K/pr)^*$}{nfkermodpr$(\var{nf},M,prmod)$}
973: \li{solve $M x = B$ in $(\ZZ_K/pr)^*$}{nfsolvemodpr$(\var{nf},M,B,prmod)$}
974:
975: \section{Relative Number Fields (rnf)}
976: Extension $L/K$ is defined by $g\in K[x]$. We have $order\subset L$.
977: \hfill\break
978: %
979: \li{absolute equation of $L$}{rnfequation$(\var{nf},g,\fl)$}
980: %
981: \subsec{Lifts and Push-downs}
982: \li{absolute $\rightarrow$ relative repres.\ for $x$}
983: {rnfeltabstorel$(\var{rnf},x)$}
984: \li{relative $\rightarrow$ absolute repres.\ for $x$}
985: {rnfeltreltoabs$(\var{rnf},x)$}
986: \li{lift $x$ to the relative field}{rnfeltup$(\var{rnf},x)$}
987: \li{push $x$ down to the base field}{rnfeltdown$(\var{rnf},x)$}
988: \leavevmode idem for $x$ ideal:
989: \kbd{$($rnfideal$)$reltoabs}, \kbd{abstorel}, \kbd{up}, \kbd{down}\hfill\break
990: %
991: \li{relative {\tt nfalgtobasis}}{rnfalgtobasis$(\var{rnf},x)$}
992: \li{relative {\tt nfbasistoalg}}{rnfbasistoalg$(\var{rnf},x)$}
993: \li{relative {\tt idealhnf}}{rnfidealhnf$(\var{rnf},x)$}
994: \li{relative {\tt idealmul}}{rnfidealmul$(\var{rnf},x,y)$}
995: \li{relative {\tt idealtwoelt}}{rnfidealtwoelt$(\var{rnf},x)$}
996: %
997: \subsec{Projective $\ZZ_K$-modules, maximal order}
998: \li{relative {\tt polred}}{rnfpolred$(\var{nf},g)$}
999: \li{relative {\tt polredabs}}{rnfpolredabs$(\var{nf},g)$}
1000: \li{characteristic poly.\ of $a$ mod $g$}{rnfcharpoly$(\var{nf},g,a,\{v\})$}
1001: \li{relative Dedekind criterion, prime $pr$}{rnfdedekind$(\var{nf},g,pr)$}
1002: \li{discriminant of relative extension}{rnfdisc$(\var{nf},g)$}
1003: \li{pseudo-basis of $\ZZ_L$}{rnfpseudobasis$(\var{nf},g)$}
1004: \li{relative HNF basis of $order$}{rnfhnfbasis$(\var{bnf},order)$}
1005: \li{reduced basis for $order$}{rnflllgram$(\var{nf},g,order)$}
1006: \li{determinant of pseudo-matrix $A$}{rnfdet$(\var{nf},A)$}
1007: \li{Steinitz class of $order$}{rnfsteinitz$(\var{nf},order)$}
1008: \li{is \var{order} a free $\ZZ_K$-module?}{rnfisfree$(\var{bnf},\var{order})$}
1009: \li{true basis of \var{order}, if it is free}{rnfbasis$(\var{bnf},\var{order})$}
1010: %
1011: \subsec{Norms}
1012: \li{absolute norm of ideal $x$}{rnfidealnormabs$(\var{rnf},x)$}
1013: \li{relative norm of ideal $x$}{rnfidealnormrel$(\var{rnf},x)$}
1014: \li{solutions of $N_{K/\QQ}(y)=x\in \ZZ$}{bnfisintnorm$(\var{bnf},x)$}
1015: \li{is $x\in\QQ$ a norm from $K$?}{bnfisnorm$(\var{bnf},x,\fl)$}
1.2 ! noro 1016: \li{initialize $T$ for norm eq.~solver}{rnfisnorminit$(K,pol,\fl)$}
! 1017: \li{is $a\in K$ a norm from $L$?}{rnfisnorm$(T,a,\fl)$}
1.1 noro 1018: \vfill
1019: \copyrightnotice
1020: \bye
1021: % Local variables:
1022: % compile-command: "tex PARIRefCard"
1023: % End:
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