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A draft on sections on ox servers for RIMS kokyuroku of the workshop
Dec 2000.

%% $OpenXM: OpenXM/doc/Papers/rims2000-tmp-ja.tex,v 1.1 2001/01/18 00:32:09 takayama Exp $
\documentclass{jarticle}
%\usepackage{epsfig}
\def\epsffile#1{ \fbox{PS file {\tt #1} is included here.} }
\def\epsfxsize{ }

\def\color#1{ }  % do nothing.

\begin{document}
\section{OX $B%5!<%P$HDs6!$5$l$k?t3X4X?t(B}

OpenXM $B$N3F%5!<%P$*$h$S$=$N?t3XE*5!G=$N$&$A4v$D$+$rNc$r(B
$B$"$2$F@bL@$7$h$&(B.

\subsection{{\tt ox\_asir}}
Risa/Asir $B$O(B Free $B$GG[I[$5$l$F$$$kHFMQ?t<0=hM}%=%U%H$G$"$k(B.
$B$?$H$($P<!$N$h$&$J5!G=$r;}$D(B.
\begin{enumerate}
\item ${\bf Q}$ $B78?t$N(B $n$ $BJQ?tB?9`<0$N0x?tJ,2r(B ($B4X?t(B {\tt fctr}). \\
$BNc(B: 
\begin{verbatim}
  fctr(y^5-4*y^4+(-x^2+2*x+3)*y^3-x^2*y^2+4*x^2*y+x^4-2*x^3-3*x^2);
  [[1,1],[y^3-x^2,1],[y-x-1,1],[y+x-3,1]
\end{verbatim}
\item ${\bf Q}$ $B$NBe?t3HBgBN$K$*$1$k(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B.  
($B4X?t(B {\tt af})\\
$B<!$NNc$OB?9`<0(B $x^6-1$ $B$r(B ${\bf Q}$ $B$K(B $x^2+x+1=0$ $B$N:,$rE:2C$7$?(B
$BBN$G0x?tJ,2r$9$kNc$G$"$k(B.
\begin{verbatim}
[374] load("gr")$ load("sp")$
[375] A1=newalg(x^2+x+1);
(#0)
[476] af(x^6-1,[A1]);
[[1,1],[x+(#0+1),1],[x+(#0),1],[x-1,1],[x+(-#0),1],[x+1,1],[x+(-#0-1),1]]
\end{verbatim}
$x^2+x+1=0$ $B$N:,$r(B $\omega$ $B$H$7$h$&(B.
($\omega^3=1$, $\omega+1 = -\omega^2$  $B$G$"$k(B.)
$B>e$N7k2L$O(B, $\omega$ $B$rMQ$$$k$H(B, 
$$ x^6-1 = (x+\omega+1)(x+\omega)(x-1)(x-\omega)(x+1)(x-\omega-1)$$
$B$H0x?tJ,2r$5$l$k$3$H$r<($7$F$$$k(B.
\item $BB?9`<04D$*$h$SHyJ,:nMQAG4D(B ($B%o%$%kBe?t(B) $B$K$*$1$k%0%l%V%J4pDl7W;;(B
($B4X?t(B {\tt gr} $B$*$h$S(B {\tt dp\_gr\_weyl\_main}). 
$B1~MQ$H$7$F=`AG%$%G%"%kJ,2r$N5!G=$b$b$D(B ($B4X?t(B {\tt primadec}).
\\
$BNc$H$7$FJQ?t$N>C5n$r%0%l%V%J4pDl$N7W;;$G$*$3$J$C$F$_$h$&(B.
$I$ $B$r(B $n$ $BJQ?tB?9`<04D(B ${\bf Q}[x_1, \ldots, x_n]$ $B$N%$%G%"%k$H$9$k$H$-(B
$I \cap {\bf Q}[x_1]$ $B$N@8@.85$O(B, $I$ $B$N%0%l%V%J4pDl$r(B $B<-=q<0=g=x(B
(lexicographic order) $x_n > \cdots > x_2 > x_1$ $B$G7W;;$7$F(B,
$B%0%l%V%J4pDl$NCf$+$i(B, $x_1$ $B$N$_$NB?9`<0$r<h$j=P$7$F$/$l$P5a$^$k(B.
$B$?$H$($P(B, $I$ $B$H$7$F(B, $x_1^2+x_2^2-4$, $x_1 x_2 -1$ $B$G@8@.$5$l$k(B
$B%$%G%"%k$r9M$($k(B.
\begin{verbatim}
[375] load("gr")$
[461] gr([x1^2+x2^2-4,x1*x2-1],[x2,x1],2);
[-x1^4+4*x1^2-1,x2+x1^3-4*x1]
\end{verbatim}
$-x_1^4+4*x_1^2-1$ $B$,(B, $I \cap {\bf Q}[x_1]$ $B$N@8@.85$G$"$k(B.
$B>C5nK!$H%0%l%V%J4pDl$H$N4XO"$K$D$$$F$O(B, $B$?$H$($P(B
Cox, Little, O'Shea $B$N652J=q(B \cite{OpenXM:CLO} $B$N(B 2 $B>O(B, 3 $B>O$K(B
$BF~LgE*$J@bL@$,$"$k(B.
\end{enumerate}

Asir $B$,MQ$$$F$$$k?t3X%"%k%4%j%:%`$K$D$$$F$O(B,
$BLnO$$K$h$k2r@b(B \cite{OpenXM:noro-book} $B$r;2>H(B.

\subsection{ {\tt ox\_sm1} }
{\tt Kan/sm1} $B$*$h$S(B {\tt Kan/k0} $B$OHyJ,:nMQAG4D(B($B%o%$%kBe?t(B) $D$ $B$K$*$1$k(B
$B%0%l%V%J4pDl7W;;$r$b$H$K$7$F(B, $D$ $B2C72$N<o!9$N9=@.$d(B, $BBe?tB?MMBN$N(B
$B%3%[%b%m%8$N7W;;$r$*$3$J$&(B.
$B8=:_(B, $B0x?tJ,2r(B, $B=`AG%$%G%"%kJ,2r(B, $b$-$B4X?t$N(B $D$ $B$G$N7W;;$J$I$O(B,
OpenXM $B%W%m%H%3%k(B (OX-RFC 100, 101) $B$rMxMQ$7$F(B, {\tt ox\_asir} $B$,(B
$BC4Ev$7$F$*$j(B, {\tt ox\_sm1} $B$O%5!<%P$G$"$k$N$_$J$i$:(B,
asir $B$N(B OpenXM $B%5!<%P5!G=$r%U%k$KMxMQ$7$F$$$k%/%i%$%"%s%H$G$b$"$k(B.
Kan $B$O$?$H$($P<!$N$h$&$J5!G=$r$b$D(B.
\begin{enumerate}
\item $D$ $B$G$N%0%l%V%J4pDl7W;;(B ($B4X?t(B {\tt gb}). \\
$B%0%l%V%J4pDl7W;;$N1~MQ$H$7$F(B, $B$"$?$($i$l$?O"N)@~7AJPHyJ,J}Dx<07O$N(B
$B2r6u4V$N<!85$N7W;;(B (holonomic rank) $B$,$"$k(B.  $B$3$l$rNc$H$7$F5s$2$F(B
$B$*$3$&(B.
\begin{verbatim}
sm1>(cohom.sm1) run ;
sm1> [ [( x*Dx + y*Dy -3 ) ( Dx-Dy )] (x,y)] rank :: 
   1
\end{verbatim}
$B$3$N=PNO$OHyJ,J}Dx<07O(B
$$ \left[
     x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}-3 \right]f 
 = \left[ \frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right]f = 0$$
$B$N2r6u4V$N<!85$O(B $1$ $B$G$"$k$3$H$r0UL#$9$k(B.
$B$3$N>l9g$O<B:]$K2r$r=q$-=P$9$3$H$,$G$-$F(B, $f=(x+y)^3$ $B$,2r$G$"$k(B.
\item $BBe?tB?MMBN(B, $B$?$H$($P(B ${\bf C}^n \setminus V(f)$ $B$N%3%[%b%m%872$N(B
$B7W;;(B ( $B4X?t(B {\tt deRham} ). \\
$B$?$H$($P(B Kan/k0 $B$G$N7W;;$O<!$N$h$&$K$J$k(B.
\begin{verbatim}
% cd OpenXM/src/k097/lib/restriction
% k0
This is kan/k0 Version 1998,12/15
sm1 version = 3.001203
Default ring is Z[x,h].
In(2)= load("demo.k");
In(3)= DeRham2WithAsir( x^3-y^2 ):
Step1: Annhilating ideal (II)[    -3*x^2*Dy-2*y*Dx , -2*x*Dx-3*y*Dy-6 ] 
Step2: (-1,1)-minimal resolution (Res0)  [ 
  [ 
    [    2*x*Dx+3*y*Dy-h^2 ] 
    [    -3*y*Dx^2+2*x*Dy*h ] 
  ]
  [ 
    [    3*y*Dx^2-2*x*Dy*h , 2*x*Dx+3*y*Dy ] 
  ]
 ]
Step3: computing the cohomology of the truncated complex.
Roots and b-function are [    [    1 ]  , [    9*s^3-18*s^2+11*s-2 ]  ] 


[    [    0 , 1 , 1 ]  , 
[    [    0 , [   ]  ]  , [    1 , [   ]  ]  , [    1 , [   ]  ]  ]  ] 
\end{verbatim}
Step1$B$N(B $1/(x^3-y^2)$ $B$NK~$?$9:GBg$NHyJ,J}Dx<0$N7W;;(B,
Step3$B$N0lHL2=$5$l$?(B $b$-$B4X?t$H$=$N:,$N7W;;$G$O(B {\tt ox\_asir} $B$r8F$S=P$7$F(B
$B7W;;$7$F$$$k(B.
\item $D^m/I$ $B$N(B $(u,v)$-minimal free resolution
($B4X?t(B {\tt Sminimal}).
\end{enumerate}
$D$ $B2C72$N%"%k%4%j%:%`$N%"%k%4%j%:%`$K$D$$$F$O(B,
$BK\(B \cite{OpenXM:SST} $B$,F~LgE*OCBj$+$i:G@hC<$NOCBj$^$G$r07$C$F$$$k(B.

\subsection{PHC}
PHC pack $B$O(B Jan Verschelde $B$K$h$j3+H/$5$l$?(B,
$BB?LLBN%[%b%H%T!<K!$K$h$jBe?tJ}Dx<07O$N?tCM2r$r5a$a$k(B
$B%7%9%F%`$G$"$k(B.
$B%[%b%H%T!<K!$H$$$&$N$O(B, $B$?$H$($P(B
$BBe?tJ}Dx<0(B 
$ x^n - x^3+x^2-5 = 0$
$B$r2r$/$?$a$K(B, $B$^$:%9%?!<%H%7%9%F%`(B
$ x^n - 5 = 0$ $B$r2r$-(B, $B$=$N2r$rO"B3E*$K(B 
$B$b$H$NJ}Dx<0$N2r$K1dD9$9$kJ}K!$G$"$k(B.
$BO"N)Be?tJ}Dx<07O$N>l9g$K$O(B,  $B$h$$@-<A$r$b$D%9%?!<%H%7%9%F%`$r:n$k(B
$BI,MW$,$"$j(B, $B$=$N$?$a$KB?LLBN$NJ,3d$r$b$A$$$k(B. $B$3$l$,(B
$BB?LLBN%[%b%H%T!<K!$G$"$k(B.
$BK\(B \cite{OpenXM:CLO2} $B$N(B 7.5 $B@a$K$3$NJ}K!$N4pAC$H$J$k(B
$BB?LLBN$N4v2?$H:,$N?t$N4*Dj$K4X$9$k@bL@$,$"$k(B.
$BB?LLBN%[%b%H%T!<K!$K$D$$$F$O(B, Verschelde $B$K$h$k2r@b$NB>(B,
$BO@J8(B \cite{OpenXM:HS} $B$J$I$,FI$_$d$9$$(B.

\subsection{gnuplot, M2, tigers, pari, mathematica, OMproxy}

$BD9$5$N4X78$b$"$j>\$7$/$O>R2p$G$-$J$$$N$G(B, 
$BI=Bj$N%=%U%H$r4JC1$K>R2p$7$h$&(B.
\begin{enumerate}
\item gnuplot $B$O%0%i%U$r:n@.$9$k%=%U%H(B.
\item M2 (Macaulay 2) $B$O(B D.Grayson $B$H(B M.Stillman $B$K$h$j3+H/$5$l$F$$$k(B,
$B7W;;Be?t4v2?MQ$N%=%U%H(B. 
\item tigers $B$O(B B.Hubert $B$K$h$j3+H/$5$l$?(B, affine toric variety $B$NA4$F$N(B
$B%0%l%V%J4pDl$r5a$a$k%=%U%H(B.
\item pari $B$O(B A.Cohen $B$i$K$h$j3+H/$5$l$F$$$k(B, $B@0?tO@MQ$N%=%U%H(B.
\item Mathematica $B$OM-L>$J$N$G@bL@$NI,MW$O$J$$$G$"$m$&(B.
\item OMproxy $B$O(B Java $B$G$+$+$l$?(B, OpenXM $B$N%/%i%9%i%$%V%i%j$*$h$S(B
OpenMath $B$H$N%G!<%?$NAj8_JQ49$N$?$a$N%=%U%H$G$"$k(B.
\end{enumerate}

\subsection{ 1077 functions are available on our servers and libraries}

OX server $B$NDs6!$9$k4X?t$K$I$N$h$&$J$b$N$,$"$k$+(B,
$B%j%9%H$H4JC1$J%G%b$r7G:\$7$F$*$3$&(B.
$B$3$l$i$N4X?t$N$J$+$N0lIt$K$D$$$F$O(B, $BA0@a$^$G$G4JC1$K@bL@$7$?(B.
$B$h$j>\$7$/$O3F%3%s%]!<%M%s%H%7%9%F%`$N%^%K%e%"%k$d;2>H$5$l$F$$$k(B
$BO@J8(B, $BK\$r8+$kI,MW$,$"$k(B.


\fbox{\huge {\color{green}Operations on Integers}}

\noindent
{\color{red} idiv},{\color{red} irem} (division with remainder),
{\color{red} ishift} (bit shifting),
{\color{red} iand},{\color{red} ior},{\color{red} ixor} (logical operations),
{\color{red} igcd},(GCD by various methods such as Euclid's algorithm and
the accelerated GCD algorithm),
{\color{red} fac} (factorial),
{\color{red} inv} (inverse modulo an integer),
{\color{red} random} (random number generator by the Mersenne twister algorithm).



\medbreak

\noindent
\fbox{\huge {\color{green}Ground Fields}}

\noindent
Arithmetics on various fields: the rationals, 
${\bf Q}(\alpha_1,\alpha_2,\ldots,\alpha_n)$
($\alpha_i$ is algebraic over ${\bf Q}(\alpha_1,\ldots,\alpha_{i-1})$),
$GF(p)$ ($p$ is a prime of arbitrary size), $GF(2^n)$.

\medbreak

\noindent
\fbox{\huge {\color{green}Operations on Polynomials}}

\noindent
{\color{red} sdiv }, {\color{red} srem } (division with remainder),
{\color{red} ptozp } (removal of the integer content),
{\color{red} diff } (differentiation),
{\color{red} gcd } (GCD over the rationals),
{\color{red} res } (resultant),
{\color{red} subst } (substitution),
{\color{red} umul} (fast multiplication of dense univariate polynomials 
by a hybrid method with Karatsuba and FFT+Chinese remainder),
{\color{red} urembymul\_precomp} (fast dense univariate polynomial 
division with remainder by the fast multiplication and 
the precomputed inverse of a divisor),

\noindent
\fbox{\huge {\color{green}Polynomial Factorization}}
{\color{red} fctr } (factorization over the rationals),
{\color{red} fctr\_ff } (univariate factorization over finite fields),
{\color{red} af } (univariate factorization over algebraic number fields),
{\color{red} sp} (splitting field computation).

\medbreak

\noindent
\fbox{\huge {\color{green} Groebner basis}} 

\noindent
{\color{red} dp\_gr\_main } (Groebner basis computation of a polynomial ideal 
over the rationals by the trace lifting),
{\color{red} dp\_gr\_mod\_main } (Groebner basis over small finite fields),
{\color{red} tolex } (Modular change of ordering for a zero-dimensional ideal),
{\color{red} tolex\_gsl } (Modular rational univariate representation 
for a zero-dimensional ideal),
{\color{red} dp\_f4\_main } ($F_4$ over the rationals),
{\color{red} dp\_f4\_mod\_main } ($F_4$ over small finite fields).

\medbreak
\noindent
\fbox{\huge {\color{green} Ideal Decomposition}} 

\noindent
{\color{red} primedec} (Prime decomposition of the radical),
{\color{red} primadec} (Primary decomposition of ideals by Shimoyama/Yokoyama algorithm).

\medbreak

\noindent
\fbox{\huge {\color{green} Quantifier Elimination}} 

\noindent
{\color{red} qe} (real quantifier elimination in a linear and 
quadratic first-order formula),
{\color{red} simpl} (heuristic simplification of a first-order formula).

{\scriptsize
\begin{verbatim}
[0] MTP2 = ex([x11,x12,x13,x21,x22,x23,x31,x32,x33],
x11+x12+x13 @== a1 @&& x21+x22+x23 @== a2 @&& x31+x32+x33 @== a3 
@&& x11+x21+x31 @== b1 @&& x12+x22+x32 @== b2 @&& x13+x23+x33 @== b3
@&& 0 @<= x11 @&& 0 @<= x12 @&& 0 @<= x13 @&& 0 @<= x21
@&& 0 @<= x22 @&& 0 @<= x23 @&& 0 @<= x31 @&& 0 @<= x32 @&& 0 @<= x33)$
[1] TSOL= a1+a2+a3@=b1+b2+b3 @&& a1@>=0 @&& a2@>=0 @&& a3@>=0
@&& b1@>=0 @&& b2@>=0 @&& b3@>=0$
[2] QE_MTP2 = qe(MTP2)$
[3] qe(all([a1,a2,a3,b1,b2,b3],QE_MTP2 @equiv TSOL));
@true
\end{verbatim}}
\medbreak

\noindent
\fbox{\huge {\color{green} Visualization of curves}} 

\noindent
{\color{red} plot} (plotting of a univariate function),
{\color{red} ifplot} (plotting zeros of a bivariate polynomial),
{\color{red} conplot} (contour plotting of a bivariate polynomial function).

\medbreak

\noindent
\fbox{\huge {\color{green} Miscellaneous functions}} 

\noindent
{\color{red} det} (determinant),
{\color{red} qsort} (sorting of an array by the quick sort algorithm),
{\color{red} eval} (evaluation of a formula containing transcendental functions
such as 
{\color{red} sin}, {\color{red} cos}, {\color{red} tan}, {\color{red} exp},
{\color{red} log})
{\color{red} roots} (finding all roots of a univariate polynomial),
{\color{red} lll} (computation of an LLL-reduced basis of a lattice).

\noindent
\fbox{\huge {\color{green} $D$-modules}} ($D$ is the Weyl algebra)

\noindent
{\color{red} gb } (Gr\"obner basis),
{\color{red} syz} (syzygy),
{\color{red} annfs} (Annhilating ideal of $f^s$),
{\color{red} bfunction},
{\color{red} schreyer} (free resolution by the Schreyer method),
{\color{red} vMinRes} (V-minimal free resolution),
{\color{red} characteristic} (Characteristic variety),
{\color{red} restriction} in the derived category of $D$-modules,
{\color{red} integration} in the derived category,
{\color{red} tensor}  in the derived category,
{\color{red} dual} (Dual as a D-module),
{\color{red} slope}.

\medbreak
\noindent
\fbox{\huge {\color{green} Cohomology groups}} 

\noindent
{\color{red} deRham} (The de Rham cohomology groups of
${\bf C}^n \setminus V(f)$,
{\color{red} ext} (Ext modules for a holonomic $D$-module $M$
and the ring of formal power series).

\medbreak
\noindent
\fbox{\huge 
 {\color{green} Differential equations}}

\noindent
Helping to derive and prove {\color{red} combinatorial} and
{\color{red} special function identities},
{\color{red} gkz} (GKZ hypergeometric differential equations),
{\color{red} appell} (Appell's hypergeometric differential equations),
{\color{red} indicial} (indicial equations),
{\color{red} rank} (Holonomic rank),
{\color{red} rrank} (Holonomic rank of regular holonomic systems),
{\color{red} dsolv} (series solutions of holonomic systems).

\medbreak
\noindent
\fbox{\huge 
 {\color{green} OpenMATH support}}

\noindent
{\color{red} om\_xml} (CMO to OpenMATH XML),
{\color{red} om\_xml\_to\_cmo} (OpenMATH XML to CMO).

\medbreak
\noindent
\fbox{\huge 
 {\color{green} Homotopy Method}}

\noindent
{\color{red} phc} (Solving systems of algebraic equations by 
numerical and polyhedral homotopy methods).

\medbreak
\noindent
\fbox{\huge 
 {\color{green} Toric ideal}}

\noindent
{\color{red} tigers} (Enumerate all Gr\"obner basis of a toric ideal.
Finding test sets for integer program),
{\color{red} arithDeg} (Arithmetic degree of a monomial ideal),
{\color{red} stdPair} (Standard pair decomposition of a monomial ideal).

\medbreak
\noindent
\fbox{\huge {\color{green} Communications}}

\noindent
{\color{red} ox\_launch} (starting a server), 
{\color{red} ox\_launch\_nox}, 
{\color{red} ox\_shutdown},
{\color{red} ox\_launch\_generic}, 
{\color{red} generate\_port}, 
{\color{red} try\_bind\_listen}, 
{\color{red} try\_connect}, 
{\color{red} try\_accept}, 
{\color{red} register\_server}, 
{\color{red} ox\_rpc}, 
{\color{red} ox\_cmo\_rpc}, 
{\color{red} ox\_execute\_string}, 
{\color{red} ox\_reset} (reset the server),
{\color{red} ox\_intr},
{\color{red} register\_handler}, 
{\color{red} ox\_push\_cmo}, 
{\color{red} ox\_push\_local},
{\color{red} ox\_pop\_cmo}, 
{\color{red} ox\_pop\_local}, 
{\color{red} ox\_push\_cmd}, 
{\color{red} ox\_sync},
{\color{red} ox\_get}, 
{\color{red} ox\_pops}, 
{\color{red} ox\_select}, 
{\color{red} ox\_flush},
{\color{red} ox\_get\_serverinfo}

\medbreak
\noindent
In addition to these functions, {\color{green} Mathematica functions} 
can be called as server functions.
\medbreak
\noindent
\fbox{\huge {\color{green} Examples}}
{\footnotesize
\begin{verbatim}
[345] sm1_deRham([x^3-y^2*z^2,[x,y,z]]);
[1,1,0,0]
/* dim H^i = 1 (i=0,1), =0 (i=2,3) */
\end{verbatim}} 

\noindent
{\footnotesize \begin{verbatim}
[287] phc(katsura(7)); B=map(first,Phc)$
[291] gnuplot_plotDots(B,0)$
\end{verbatim} }

\epsfxsize=3cm
\begin{center}
\epsffile{../calc2000/katsura7.ps}
%\epsffile{katsura7.ps}
\end{center}
%%The first components of the solutions to the system of algebraic equations Katsura 7.

\medbreak
\noindent
\fbox{ {\color{green} Authors}}
Castro-Jim\'enez, Dolzmann, Hubert, Murao, Noro, Oaku, Okutani, 
Shimoyama, Sturm, Takayama, Tamura, Verschelde, Yokoyama.



\section{OX $B%5!<%P$rAH$_9g$o$;$FMxMQ$7$?Nc(B}

$B6qBNE*$J?t3XE*LdBj$r$b$H$K?t3X%=%U%H$N3+H/$r$9$k$H$$$&$N$O(B
$B0l$D$N7rA4$J3+H/<jK!$G$"$m$&(B.

OX $B%5!<%P$N3+H/$G$b(B, $B$=$N$h$&$J3+H/<jK!$r$H$j$?$$$H;W$C$F$*$j(B,
$B$$$m$$$m$J?t3XE*LdBj$r$5$,$7$F$$$k(B.

$B$=$NCf$N0l$D$NLdBj$O(B, $BB?JQ?tD64v2?4X?t$K4X$7$F2?$G$bEz$($i$l$k%7%9%F%`$N(B
$B3+H/$G$"$k(B.
$B$3$3$G$O(B, $BB?JQ?tD64v2?4X?t$H$7$F(B, $B$$$o$f$k(B GKZ hypergeometric system
$B$N2r$r9M$($F$$$k(B.
GKZ hypergeometric system $B$O(B $n$ $B<!856u4V$NE@=89g$KIU?o$7$F$-$^$k(B
$BO"N)@~7AJPHyJ,J}Dx<07O$G$"$k(B.
$n$ $B<!856u4V$NE@=89g$r9M$($k$?$a(B, $BB?LLBN$N4v2?$r07$&I,MW$,$"$k$7(B,
$BO"N)@~7AJPHyJ,J}Dx<07O$r9M$($k$?$a(B, $D$ $B2C72$r07$&I,MW$b$"$k(B.
GKZ system $B$O(B affine toric ideal $B$r4^$`$?$a(B, affine toric ideal $B$N(B
$B%7%9%F%`$bI,MW$G$"$k(B.
$B4X?t$N?tCM7W;;$N$?$a$K$O(B, $B?tCM@QJ,E*$J<jK!$b=EMW$G$"$k$,(B, $B$3$l$O$^$@(B
$BA4$/<j$r$D$1$F$$$J$$(B.
GKZ hypergeometric system $B$K$D$$$F$O(B, $BK\(B \cite{OpenXM:SST} $B$r;2>H(B.

$B8=:_$N(B {\tt OpenXM/src/asir-contrib} $B$N%Q%C%1!<%8$O(B,
GKZ hypergeometric system $B$K4X$7$F2?$G$b$3$?$($i$l$k%7%9%F%`$rL\I8$K(B,
asir $B$r%/%i%$%"%s%H$H$7$F(B, ox servers $B$r$$$m$$$m$/$C$D$1$?%7%9%F%`$G$"$k(B.
 
$BNc$r0l$D$"$2$h$&(B. 
{\tt dsolv\_starting\_term} $B$O@5B'%[%m%N%_%C%/7O$r(B cone $B>e$G<}B+$9$k(B
$BB?JQ?t$N5i?t$G2r$/$?$a$N(B
$B4X?t$N0l$D$G$"$j(B, $B5i?tE83+$N<gIt$r$b$H$a$k(B.
\begin{enumerate}
\item $B9TNs(B($BE@G[CV(B) $\pmatrix{ 1&1&1&1&1 \cr
                              1&1&0&-1&0 \cr
                              0&1&1&-1&0 \cr}$ $B$KIU?o$9$k(B
GKZ hypergeometric system $B$r4X?t(B {\tt sm1\_gkz} $B$G5a$a$k(B.
\begin{verbatim}
[1076]   F = sm1_gkz( [ [[1,1,1,1,1],[1,1,0,-1,0],[0,1,1,-1,0]], [1,0,0]]);
[[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,-x4*dx4+x2*dx2+x1*dx1,
  -x4*dx4+x3*dx3+x2*dx2,
  -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4],[x1,x2,x3,x4,x5]]
\end{verbatim}
\item $B$3$N%7%9%F%`(B {\tt F} $B$N(B $BJ}8~(B $(1,1,1,1,0)$ $B$K$*$1$k(B
$B5i?t2r$N<gIt$r5a$a$k(B.
\begin{verbatim}
[1077]  A= dsolv_starting_term(F[0],F[1],[1,1,1,1,0])$
Computing the initial ideal.
Done.
Computing a primary ideal decomposition.
Primary ideal decomposition of the initial Frobenius ideal 
to the direction [1,1,1,1,0] is 
[[[x5+2*x4+x3-1,x5+3*x4-x2-1,x5+2*x4+x1-1,3*x5^2+(8*x4-6)*x5-8*x4+3,
   x5^2-2*x5-8*x4^2+1,x5^3-3*x5^2+3*x5-1],
 [x5-1,x4,x3,x2,x1]]]
 
----------- root is [ 0 0 0 0 1 ]
----------- dual system is 
[x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2
 +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2,
 x4-2*x3+3*x2-2*x1,x5-x3+x2-x1,1]
\end{verbatim}
\item $BE83+$N<gIt(B 4 $BDL$j$"$j(B, $B$=$l$>$l$r0x?tJ,2r$9$k$H(B,
$B<!$N$h$&$K$J$k(B.
$B$?$H$($P(B, 3 $BHVL\$N(B
{\tt [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]], }
$B$O(B,
$$ x_5 (-\log x_1 + \log x_2 - \log x_3 + \log x_5) =
   x+5 \log \frac{x_2 x_5}{x_1 x_3} $$
$B$+$i;O$^$k5i?t2r$,B8:_$9$k$3$H$r0UL#$9$k(B.
\begin{verbatim}  
[1078] A[0];
[[ 0 0 0 0 1 ]]
[1079] map(fctr,A[1][0]);
[[[1/8,1],[x5,1],[log(x2)+log(x4)-2*log(x5),1],
          [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5),1]],
 [[1,1],[x5,1],[-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]],
 [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]],
 [[1,1],[x5,1]]]
\end{verbatim}
\end{enumerate}
{\tt dsolv\_starting\_term} $B$G$O(B,  kan/sm1 $B$,(B D-$B2C72$N7W;;(B,
asir $B$,=`AG%$%G%"%kJ,2r$N7W;;(B, $B2r$N7W;;$rC4Ev$7$F$$$k(B.
$B2r$NE83+$NJ}8~$,2?DL$jB8:_$9$k$+$rD4$Y$k$K$O(B,
tigers $B$r1gMQ$9$k$H$h$$(B.


\begin{thebibliography}{99}
\bibitem{OpenXM:CLO}
Cox, D., Little, J., O'Shea;

\bibitem{OpenXM:CLO2}
Cox, D., Little, J., O'Shea,
{\it Using Algebraic Geometry},
Springer, 1998.

\bibitem{OpenXM:HS}
Huber, B., Sturmfels, B.,
A Polyhedral Method for Solving Sparse Polynomial Systems,
Mathematics of Computation,
{\bf 64} (1995), 1541--1555.

\bibitem{OpenXM:noro-book}
 $BLnO$(B:  $B7W;;Be?tF~Lg(B, Rokko Lectures in Mathematics, 9, 
2000.  ISBN 4-907719-09-4. \\
{\tt ftp://ftp.math.kobe-u.ac.jp/pub/OpenXM/Head/openxm-head.tar.gz}
$B$N%G%#%l%/%H%j(B {\tt OpenXM/doc/compalg} $B$K$3$NK\$N(B TeX $B%=!<%9$,$"$k(B.

\bibitem{OpenXM:SST}
Saito, M., Sturmfels, B., Takayama, N.,
{\it Gr\"obner deformations of hypergeometric differential
equations}. Algorithms and Computation in Mathematics,
6. Springer-Verlag, Berlin, 2000.

\end{thebibliography}


\end{document}