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version 1.7, 2001/03/16 05:18:04 version 1.11, 2003/04/20 08:01:24
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/appendix.texi,v 1.6 2000/03/17 08:27:28 noro Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/appendix.texi,v 1.10 2002/09/10 01:40:01 noro Exp $
 \BJP  \BJP
 @node $BIUO?(B,,, Top  @node $BIUO?(B,,, Top
 @appendix $BIUO?(B  @appendix $BIUO?(B
Line 80 
Line 80 
     <list>      <list>
 \E  \E
 @end example  @end example
 \JP (@xref{$B$5$^$6$^$J<0(B})  \JP (@xref{$B$5$^$6$^$J<0(B}.)
 \EG (@xref{various expressions})  \EG (@xref{various expressions}.)
   
 @example  @example
 \BJP  \BJP
Line 133 
Line 133 
 \E  \E
 \BEG  \BEG
 <option>:  <option>:
     Character sequence beginning with an alphabetical letter @samp{=} <expression>      Character sequence beginning with an alphabetical letter @samp{=} <expr>
 \E  \E
 @end example  @end example
   
Line 174 
Line 174 
    (X,Y,Japan etc.)     (X,Y,Japan etc.)
 \E  \E
 @end example  @end example
 \JP (@xref{$BJQ?t$*$h$SITDj85(B})  \JP (@xref{$BJQ?t$*$h$SITDj85(B}.)
 \EG (@xref{variables and indeterminates})  \EG (@xref{variables and indeterminates}.)
   
 @example  @example
 \BJP  \BJP
Line 215 
Line 215 
    (a,bCD,c1_2 etc.)     (a,bCD,c1_2 etc.)
 \E  \E
 @end example  @end example
 \JP (@xref{$BJQ?t$*$h$SITDj85(B})  \JP (@xref{$BJQ?t$*$h$SITDj85(B}.)
 \EG (@xref{variables and indeterminates})  \EG (@xref{variables and indeterminates}.)
   
 @example  @example
 \BJP  \BJP
Line 234 
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    <complex number>     <complex number>
 \E  \E
 @end example  @end example
 \JP (@xref{$B?t$N7?(B})  \JP (@xref{$B?t$N7?(B}.)
 \EG (@xref{Types of numbers})  \EG (@xref{Types of numbers}.)
   
 @example  @example
 \JP <$BM-M}?t(B>:  \JP <$BM-M}?t(B>:
Line 254 
Line 254 
 \EG <algebraic number>:  \EG <algebraic number>:
    newalg(x^2+1), alg(0)^2+1     newalg(x^2+1), alg(0)^2+1
 @end example  @end example
 \JP (@xref{$BBe?tE*?t$K4X$9$k1i;;(B})  \JP (@xref{$BBe?tE*?t$K4X$9$k1i;;(B}.)
 \EG (@xref{Algebraic numbers})  \EG (@xref{Algebraic numbers}.)
   
 @example  @example
 \JP <$BJ#AG?t(B>:  \JP <$BJ#AG?t(B>:
Line 284 
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    @samp{<<} <expr list> @samp{>>}     @samp{<<} <expr list> @samp{>>}
 \E  \E
 @end example  @end example
 \JP (@xref{$B%0%l%V%J4pDl$N7W;;(B})  \JP (@xref{$B%0%l%V%J4pDl$N7W;;(B}.)
 \EG (@xref{Groebner basis computation})  \EG (@xref{Groebner basis computation}.)
   
 @example  @example
 \BJP  \BJP
Line 321 
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     @samp{end(quit)} <terminator>      @samp{end(quit)} <terminator>
 \E  \E
 @end example  @end example
 \JP (@xref{$BJ8(B})  \JP (@xref{$BJ8(B}.)
 \EG (@xref{statements})  \EG (@xref{statements}.)
   
 @example  @example
 \JP <$B=*C<(B>:  \JP <$B=*C<(B>:
Line 386  Here, we explain some of them.
Line 386  Here, we explain some of them.
   
 @table @samp  @table @samp
 @item fff  @item fff
 \JP $BBgI8?tAGBN$*$h$SI8?t(B 2 $B$NM-8BBN>e$N0lJQ?tB?9`<00x?tJ,2r(B (@xref{$BM-8BBN$K4X$9$k1i;;(B})  \JP $BBgI8?tAGBN$*$h$SI8?t(B 2 $B$NM-8BBN>e$N0lJQ?tB?9`<00x?tJ,2r(B (@xref{$BM-8BBN$K4X$9$k1i;;(B}.)
 \EG Univariate factorizer over large finite fields (@xref{Finite fields})  \EG Univariate factorizer over large finite fields (@xref{Finite fields}.)
 @item gr  @item gr
 \JP $B%0%l%V%J4pDl7W;;%Q%C%1!<%8(B.  (@xref{$B%0%l%V%J4pDl$N7W;;(B})  \JP $B%0%l%V%J4pDl7W;;%Q%C%1!<%8(B.  (@xref{$B%0%l%V%J4pDl$N7W;;(B}.)
 \EG Groebner basis package.  (@xref{Groebner basis computation})  \EG Groebner basis package.  (@xref{Groebner basis computation}.)
 @item sp  @item sp
 \JP $BBe?tE*?t$N1i;;$*$h$S0x?tJ,2r(B, $B:G>.J,2rBN(B. (@xref{$BBe?tE*?t$K4X$9$k1i;;(B})  \JP $BBe?tE*?t$N1i;;$*$h$S0x?tJ,2r(B, $B:G>.J,2rBN(B. (@xref{$BBe?tE*?t$K4X$9$k1i;;(B}.)
 \EG Operations over algebraic numbers and factorization, Splitting fields. (@xref{Algebraic numbers})  \EG Operations over algebraic numbers and factorization, Splitting fields. (@xref{Algebraic numbers}.)
 @item alpi  @item alpi
 @itemx bgk  @itemx bgk
 @itemx cyclic  @itemx cyclic
Line 401  Here, we explain some of them.
Line 401  Here, we explain some of them.
 @itemx kimura  @itemx kimura
 \JP $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $B%Y%s%A%^!<%/$=$NB>$GMQ$$$i$l$kNc(B.  \JP $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $B%Y%s%A%^!<%/$=$NB>$GMQ$$$i$l$kNc(B.
 \EG Example polynomial sets for benchmarks of Groebner basis computation.  \EG Example polynomial sets for benchmarks of Groebner basis computation.
 (@xref{katsura hkatsura cyclic hcyclic})  (@xref{katsura hkatsura cyclic hcyclic}.)
 @item defs.h  @item defs.h
 \JP $B$$$/$D$+$N%^%/%mDj5A(B. (@xref{$B%W%j%W%m%;%C%5(B})  \JP $B$$$/$D$+$N%^%/%mDj5A(B. (@xref{$B%W%j%W%m%;%C%5(B}.)
 \EG Macro definitions. (@xref{preprocessor})  \EG Macro definitions. (@xref{preprocessor}.)
 @item fctrtest  @item fctrtest
 \BJP  \BJP
 $B@0?t>e$NB?9`<0$N0x?tJ,2r$N%F%9%H(B. REDUCE $B$N(B @samp{factor.tst} $B$*$h$S(B  $B@0?t>e$NB?9`<0$N0x?tJ,2r$N%F%9%H(B. REDUCE $B$N(B @samp{factor.tst} $B$*$h$S(B
Line 423  correctly.
Line 423  correctly.
 @item fctrdata  @item fctrdata
 \BJP  \BJP
 @samp{fctrtest} $B$G;H$o$l$F$$$kNc$r4^$`(B, $B0x?tJ,2r%F%9%HMQ$NNc(B.  @samp{fctrtest} $B$G;H$o$l$F$$$kNc$r4^$`(B, $B0x?tJ,2r%F%9%HMQ$NNc(B.
 @code{Alg[]} $B$K<}$a$i$l$F$$$kNc$O(B, @code{af()} (@xref{asq af af_noalg}) $BMQ$NNc$G$"$k(B.  @code{Alg[]} $B$K<}$a$i$l$F$$$kNc$O(B, @code{af()} (@ref{asq af af_noalg}) $BMQ$NNc$G$"$k(B.
 \E  \E
 \BEG  \BEG
 This contains example polynomials for factorization.  It includes  This contains example polynomials for factorization.  It includes
 polynomials used in @samp{fctrtest}.  polynomials used in @samp{fctrtest}.
 Polynomials contained in vector @code{Alg[]} is for the algebraic  Polynomials contained in vector @code{Alg[]} is for the algebraic
 factorization @code{af()} (@xref{asq af af_noalg}).  factorization @code{af()}. (@xref{asq af af_noalg}.)
 \E  \E
 @example  @example
 [45] load("sp")$  [45] load("sp")$
Line 440  factorization @code{af()} (@xref{asq af af_noalg}).
Line 440  factorization @code{af()} (@xref{asq af af_noalg}).
 x^9-15*x^6-87*x^3-125  x^9-15*x^6-87*x^3-125
 0msec  0msec
 [177] af(Alg[5],[newalg(Alg[5])]);  [177] af(Alg[5],[newalg(Alg[5])]);
 [[1,1],[75*x^2+(10*#0^7-175*#0^4-470*#0)*x+(3*#0^8-45*#0^5-261*#0^2),1],  [[1,1],[75*x^2+(10*#0^7-175*#0^4-470*#0)*x
 [75*x^2+(-10*#0^7+175*#0^4+395*#0)*x+(3*#0^8-45*#0^5-261*#0^2),1],  +(3*#0^8-45*#0^5-261*#0^2),1],
 [25*x^2+(25*#0)*x+(#0^8-15*#0^5-87*#0^2),1],[x^2+(#0)*x+(#0^2),1],  [75*x^2+(-10*#0^7+175*#0^4+395*#0)*x
 [x+(-#0),1]]  +(3*#0^8-45*#0^5-261*#0^2),1],
   [25*x^2+(25*#0)*x+(#0^8-15*#0^5-87*#0^2),1],
   [x^2+(#0)*x+(#0^2),1],[x+(-#0),1]]
 3.600sec + gc : 1.040sec  3.600sec + gc : 1.040sec
 @end example  @end example
 @item ifplot  @item ifplot
 \BJP  \BJP
 $BIA2h(B (@xref{ifplot conplot plot plotover}) $B$N$?$a$NNc(B. @code{IS[]} $B$K$OM-L>$J(B  $BIA2h(B (@ref{ifplot conplot plot polarplot plotover}) $B$N$?$a$NNc(B. @code{IS[]} $B$K$OM-L>$J(B
 $B6J@~$NNc(B, $BJQ?t(B @code{H, D, C, S} $B$K$O%H%i%s%W$N%O!<%H(B, $B%@%$%d(B, $B%/%i%V(B,  $B6J@~$NNc(B, $BJQ?t(B @code{H, D, C, S} $B$K$O%H%i%s%W$N%O!<%H(B, $B%@%$%d(B, $B%/%i%V(B,
 $B%9%Z!<%I(B ($B$i$7$-(B) $B6J@~$NNc$,F~$C$F$$$k(B.  $B%9%Z!<%I(B ($B$i$7$-(B) $B6J@~$NNc$,F~$C$F$$$k(B.
 \E  \E
 \BEG  \BEG
 Examples for plotting (@xref{ifplot conplot plot plotover}).  Examples for plotting. (@xref{ifplot conplot plot polarplot plotover}.)
 Vector @code{IS[]} contains several famous algebraic curves.  Vector @code{IS[]} contains several famous algebraic curves.
 Variables @code{H, D, C, S} contains something like the suits  Variables @code{H, D, C, S} contains something like the suits
 (Heart, Diamond, Club, and Spade) of cards.  (Heart, Diamond, Club, and Spade) of cards.
Line 480  is defined.  Its returns a rather complex result.
Line 482  is defined.  Its returns a rather complex result.
 [84] load("ratint")$  [84] load("ratint")$
 [102] ratint(x^6/(x^5+x+1),x);  [102] ratint(x^6/(x^5+x+1),x);
 [1/2*x^2,  [1/2*x^2,
 [[(#2)*log(-140*x+(-2737*#2^2+552*#2-131)),161*t#2^3-23*t#2^2+15*t#2-1],  [[(#2)*log(-140*x+(-2737*#2^2+552*#2-131)),
   161*t#2^3-23*t#2^2+15*t#2-1],
 [(#1)*log(-5*x+(-21*#1-4)),21*t#1^2+3*t#1+1]]]  [(#1)*log(-5*x+(-21*#1-4)),21*t#1^2+3*t#1+1]]]
 @end example  @end example
 \BJP  \BJP
Line 513  result and then summing them up all.
Line 516  result and then summing them up all.
 \E  \E
 @item primdec  @item primdec
 \BJP  \BJP
 $BB?9`<0%$%G%"%k$N=`AG%$%G%"%kJ,2r$H$=$N:,4p$NAG%$%G%"%kJ,2r(B  $BM-M}?tBN>e$NB?9`<0%$%G%"%k$N=`AG%$%G%"%kJ,2r$H$=$N:,4p$NAG%$%G%"%kJ,2r(B
 (@pxref{primadec primedec}).  (@pxref{primadec primedec}).
 \E  \E
 \BEG  \BEG
 Primary ideal decomposition of polynomial ideals and prime compotision  Primary ideal decomposition of polynomial ideals and prime compotision
 of radicals (@pxref{primadec primedec}).  of radicals over the rationals (@pxref{primadec primedec}).
 \E  \E
   @item primdec_mod
   \BJP
   $BM-8BBN>e$NB?9`<0%$%G%"%k$N:,4p$NAG%$%G%"%kJ,2r(B
   (@pxref{primedec_mod}).
   \E
   \BEG
   Prime decomposition of radicals of polynomial ideals
   over finite fields (@pxref{primedec_mod}).
   \E
 @end table  @end table
   
 \BJP  \BJP
Line 594  available for UNIX commands, including @samp{asir}.
Line 606  available for UNIX commands, including @samp{asir}.
 [[1,1],[x-1,1],[x^4+x^3+x^2+x+1,1]]  [[1,1],[x-1,1],[x^4+x^3+x^2+x+1,1]]
 [1] !!                              /* !!+Return                      */  [1] !!                              /* !!+Return                      */
 \BJP  \BJP
 fctr(x^5-1);                        /* $BD>A0$NF~NO$,8=$l$k$FJT=8$G$-$k(B */  fctr(x^5-1);                        /* $BD>A0$NF~NO$,8=$l$FJT=8$G$-$k(B */
 ...                                 /* $BJT=8(B+Return                    */  ...                                 /* $BJT=8(B+Return                  */
 \E  \E
 \BEG  \BEG
 fctr(x^5-1);                        /* The last input appears.        */  fctr(x^5-1);                        /* The last input appears.        */
Line 679  distribution (@code{http://www.math.kobe-u.ac.jp/OpenX
Line 691  distribution (@code{http://www.math.kobe-u.ac.jp/OpenX
 It is possible to link an @b{Asir} library to use the functionalities of  It is possible to link an @b{Asir} library to use the functionalities of
 @b{Asir} from other programs.  @b{Asir} from other programs.
 The necessary libraries are included in the @b{OpenXM} distribution  The necessary libraries are included in the @b{OpenXM} distribution
   @ifhtml
   (<A HREF="http://www.math.kobe-u.ac.jp/OpenXM">OpenXM </A>)
   @end ifhtml
 (@code{http://www.math.kobe-u.ac.jp/OpenXM}).  (@code{http://www.math.kobe-u.ac.jp/OpenXM}).
 At present only the @b{OpenXM} interfaces are available. Here we assume  At present only the @b{OpenXM} interfaces are available. Here we assume
 that @b{OpenXM} is already installed. In the following  that @b{OpenXM} is already installed. In the following
Line 1205  J. Symb. Comp. 20 (1995), 364-397.
Line 1220  J. Symb. Comp. 20 (1995), 364-397.
 Traverso, C., "Groebner trace algorithms", Proc. ISSAC '88(LNCS 358), 125-138.  Traverso, C., "Groebner trace algorithms", Proc. ISSAC '88(LNCS 358), 125-138.
 @item [Weber]  @item [Weber]
 Weber, K., "The accelerated Integer GCD Algorithm", ACM TOMS, 21, 1(1995), 111-122.  Weber, K., "The accelerated Integer GCD Algorithm", ACM TOMS, 21, 1(1995), 111-122.
   @item [Yokoyama]
   Yokoyama, K., "Prime decomposition of polynomial ideals over finite fields",
   Proc. ICMS, (2002), 217-227.
 @end table  @end table
   
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