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version 1.5, 2000/01/26 01:37:33 version 1.10, 2003/04/19 10:36:30
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.4 2000/01/20 03:00:34 noro Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.9 2002/09/03 01:50:58 noro Exp $
 \BJP  \BJP
 @node $B7?(B,,, Top  @node $B7?(B,,, Top
 @chapter $B7?(B  @chapter $B7?(B
Line 83  x  afo  (2.3*x+y)^10
Line 83  x  afo  (2.3*x+y)^10
   
 \BJP  \BJP
 $BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B  $BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B
 1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B (@xref{$BJ,;6I=8=B?9`<0(B}).  1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B. (@xref{$BJ,;6I=8=B?9`<0(B}.)
 $B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B.  $B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B.
 \E  \E
 \BEG  \BEG
 Every polynomial is maintained internally in its full expanded form,  Every polynomial is maintained internally in its full expanded form,
 represented as a nested univariate polynomial, according to the current  represented as a nested univariate polynomial, according to the current
 variable ordering, arranged by the descending order of exponents.  variable ordering, arranged by the descending order of exponents.
 (@xref{Distributed polynomial}).  (@xref{Distributed polynomial}.)
 In the representation, the indeterminate (or variable), appearing in  In the representation, the indeterminate (or variable), appearing in
 the polynomial, with maximum ordering is called the @b{main variable}.  the polynomial, with maximum ordering is called the @b{main variable}.
 Moreover, we call the coefficient of the maximum degree term of  Moreover, we call the coefficient of the maximum degree term of
Line 289  afotake
Line 289  afotake
 newstruct(afo)  newstruct(afo)
 @end example  @end example
   
 \JP $B9=B$BN$K4X$7$F$O(B, $B>O$r2~$a$F2r@b$9$kM=Dj$G$"$k(B.  \BJP
 \EG For type @b{structure}, we shall describe it in a later chapter.  Asir $B$K$*$1$k9=B$BN$O(B, C $B$K$*$1$k9=B$BN$r4J0W2=$7$?$b$N$G$"$k(B.
 (Not written yet.)  $B8GDjD9G[Ns$N3F@.J,$rL>A0$G%"%/%;%9$G$-$k%*%V%8%'%/%H$G(B,
   $B9=B$BNDj5AKh$KL>A0$r$D$1$k(B.
   \E
   
   \BEG
   The type @b{structure} is a simplified version of that in C language.
   It is defined as a fixed length array and each entry of the array
   is accessed by its name. A name is associated with each structure.
   \E
   
 \JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}  \JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}
 \EG @item 9 @b{distributed polynomial}  \EG @item 9 @b{distributed polynomial}
   
Line 347  This is used for basis conversion in finite fields of 
Line 355  This is used for basis conversion in finite fields of 
 \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.  \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.
 \EG This expresses a first order formula used in quantifier elimination.  \EG This expresses a first order formula used in quantifier elimination.
   
   @item 15 @b{matrix over GF(p)}
   @*
   \JP $B>.I8?tM-8BBN>e$N9TNs(B.
   \EG A matrix over a small finite field.
   
   @item 16 @b{byte array}
   @*
   \JP $BId9f$J$7(B byte $B$NG[Ns(B
   \EG An array of unsigned bytes.
   
 \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}  \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}
 \EG @item -1 @b{VOID object}  \EG @item -1 @b{VOID object}
 @*  @*
Line 474  not guarantee the accuracy of the result,
Line 492  not guarantee the accuracy of the result,
 but it indicates the representation size of numbers with which internal  but it indicates the representation size of numbers with which internal
 operations of @b{PARI} are performed.  operations of @b{PARI} are performed.
 \E  \E
 (@ref{eval}, @xref{pari})  (@xref{eval deval}, @ref{pari}.)
   
 @item 4  @item 4
 \JP @b{$BJ#AG?t(B}  \JP @b{$BJ#AG?t(B}
Line 607  coefficients of a polynomial.
Line 625  coefficients of a polynomial.
 \E  \E
   
 @end itemize  @end itemize
   
   
   @item 8
   \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B}
   \EG @b{element of a finite field of characteristic @var{p^n}}
   
   \BJP
   $B0L?t$,(B @var{p^n} (@var{p} $B$OG$0U$NAG?t(B, @var{n} $B$O@5@0?t(B) $B$O(B,
   $BI8?t(B @var{p} $B$*$h$S(B @var{GF(p)} $B>e4{Ls$J(B @var{n} $B<!B?9`<0(B @var{m(x)}
   $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
   $B$3$NBN$N85$O(B @var{m(x)} $B$rK!$H$9$k(B @var{GF(p)} $B>e$NB?9`<0$H$7$F(B
   $BI=8=$5$l$k(B.
   \E
   \BEG
   A finite field of order @var{p^n}, where @var{p} is an arbitrary prime
   and @var{n} is a positive integer, is set by @code{setmod_ff}
   by specifying its characteristic @var{p} and an irreducible polynomial
   of degree @var{n} over @var{GF(p)}. An element of this field
   is represented by a polynomial over @var{GF(p)} modulo @var{m(x)}.
   \E
   
   @item 9
   \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)}
   \EG @b{element of a finite field of characteristic @var{p^n} (small order)}
   
   \BJP
   $B0L?t$,(B @var{p^n} $B$NM-8BBN(B (@var{p^n} $B$,(B @var{2^29} $B0J2<(B, @var{p} $B$,(B @var{2^14} $B0J>e(B
   $B$J$i(B @var{n} $B$O(B 1) $B$O(B,
   $BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}
   $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
   $B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B,
   @var{GF(p^n)} $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B
   $B$K$h$j(B, $B$3$N85$N$Y$-$H$7$FI=$5$l$k(B. $B$3$l$K$h$j(B, $B$3$NBN$N(B 0 $B$G$J$$85(B
   $B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B
   $B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B
   $BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B.
   
   \E
   \BEG
   A finite field of order @var{p^n}, where @var{p^n} must be less than
   @var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or
   equal to @var{2^14}@,
   is set by @code{setmod_ff}
   by specifying its characteristic @var{p} the extension degree
   @var{n}. If @var{p} is less than @var{2^14}, each non-zero element
   of this field
   is a power of a fixed element, which is a generator of the multiplicative
   group of the field, and it is represented by its exponent.
   Otherwise, each element is represented by the redue modulo @var{p}.
   This specification is useful for treating both cases in a single
   program.
   \E
   
 @end table  @end table
   
 \BJP  \BJP
 $BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}  $B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B.
 $B$G@_Dj$9$k(B.  $BM-8BBN$N85$I$&$7$N1i;;$G$O(B,
 $BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B  
 modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B.  
 $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B  $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
 $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.  $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.
 \E  \E
 \BEG  \BEG
 The characteristic of a large finite prime field and the defining  Finite fields other than small finite prime fields are
 polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.  set by @code{setmod_ff}.
 Elements of finite fields do not have informations about the modulus.  Elements of finite fields do not have informations about the modulus.
 Upon an arithmetic operation, the modulus set by @code{setmod_ff} is  Upon an arithmetic operation, i
 used. If one of the operands is a rational number, it is automatically  f one of the operands is a rational number, it is automatically
 converted into an element of the finite field currently set and  converted into an element of the finite field currently set and
 the operation is done in the finite field.  the operation is done in the finite field.
 \E  \E
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