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Annotation of OpenXM/doc/Papers/rims2000-tmp-ja.tex, Revision 1.1

1.1     ! takayama    1: %% $OpenXM$
        !             2: \documentclass{jarticle}
        !             3: %\usepackage{epsfig}
        !             4: \def\epsffile#1{ \fbox{PS file {\tt #1} is included here.} }
        !             5: \def\epsfxsize{ }
        !             6:
        !             7: \def\color#1{ }  % do nothing.
        !             8:
        !             9: \begin{document}
        !            10: \section{OX $B%5!<%P$HDs6!$5$l$k?t3X4X?t(B}
        !            11:
        !            12: OpenXM $B$N3F%5!<%P$*$h$S$=$N?t3XE*5!G=$N$&$A4v$D$+$rNc$r(B
        !            13: $B$"$2$F@bL@$7$h$&(B.
        !            14:
        !            15: \subsection{{\tt ox\_asir}}
        !            16: Risa/Asir $B$O(B Free $B$GG[I[$5$l$F$$$kHFMQ?t<0=hM}%=%U%H$G$"$k(B.
        !            17: $B$?$H$($P<!$N$h$&$J5!G=$r;}$D(B.
        !            18: \begin{enumerate}
        !            19: \item ${\bf Q}$ $B78?t$N(B $n$ $BJQ?tB?9`<0$N0x?tJ,2r(B ($B4X?t(B {\tt fctr}). \\
        !            20: $BNc(B:
        !            21: \begin{verbatim}
        !            22:   fctr(y^5-4*y^4+(-x^2+2*x+3)*y^3-x^2*y^2+4*x^2*y+x^4-2*x^3-3*x^2);
        !            23:   [[1,1],[y^3-x^2,1],[y-x-1,1],[y+x-3,1]
        !            24: \end{verbatim}
        !            25: \item ${\bf Q}$ $B$NBe?t3HBgBN$K$*$1$k(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B.
        !            26: ($B4X?t(B {\tt af})\\
        !            27: $B<!$NNc$OB?9`<0(B $x^6-1$ $B$r(B ${\bf Q}$ $B$K(B $x^2+x+1=0$ $B$N:,$rE:2C$7$?(B
        !            28: $BBN$G0x?tJ,2r$9$kNc$G$"$k(B.
        !            29: \begin{verbatim}
        !            30: [374] load("gr")$ load("sp")$
        !            31: [375] A1=newalg(x^2+x+1);
        !            32: (#0)
        !            33: [476] af(x^6-1,[A1]);
        !            34: [[1,1],[x+(#0+1),1],[x+(#0),1],[x-1,1],[x+(-#0),1],[x+1,1],[x+(-#0-1),1]]
        !            35: \end{verbatim}
        !            36: $x^2+x+1=0$ $B$N:,$r(B $\omega$ $B$H$7$h$&(B.
        !            37: ($\omega^3=1$, $\omega+1 = -\omega^2$  $B$G$"$k(B.)
        !            38: $B>e$N7k2L$O(B, $\omega$ $B$rMQ$$$k$H(B,
        !            39: $$ x^6-1 = (x+\omega+1)(x+\omega)(x-1)(x-\omega)(x+1)(x-\omega-1)$$
        !            40: $B$H0x?tJ,2r$5$l$k$3$H$r<($7$F$$$k(B.
        !            41: \item $BB?9`<04D$*$h$SHyJ,:nMQAG4D(B ($B%o%$%kBe?t(B) $B$K$*$1$k%0%l%V%J4pDl7W;;(B
        !            42: ($B4X?t(B {\tt gr} $B$*$h$S(B {\tt dp\_gr\_weyl\_main}).
        !            43: $B1~MQ$H$7$F=`AG%$%G%"%kJ,2r$N5!G=$b$b$D(B ($B4X?t(B {\tt primadec}).
        !            44: \\
        !            45: $BNc$H$7$FJQ?t$N>C5n$r%0%l%V%J4pDl$N7W;;$G$*$3$J$C$F$_$h$&(B.
        !            46: $I$ $B$r(B $n$ $BJQ?tB?9`<04D(B ${\bf Q}[x_1, \ldots, x_n]$ $B$N%$%G%"%k$H$9$k$H$-(B
        !            47: $I \cap {\bf Q}[x_1]$ $B$N@8@.85$O(B, $I$ $B$N%0%l%V%J4pDl$r(B $B<-=q<0=g=x(B
        !            48: (lexicographic order) $x_n > \cdots > x_2 > x_1$ $B$G7W;;$7$F(B,
        !            49: $B%0%l%V%J4pDl$NCf$+$i(B, $x_1$ $B$N$_$NB?9`<0$r<h$j=P$7$F$/$l$P5a$^$k(B.
        !            50: $B$?$H$($P(B, $I$ $B$H$7$F(B, $x_1^2+x_2^2-4$, $x_1 x_2 -1$ $B$G@8@.$5$l$k(B
        !            51: $B%$%G%"%k$r9M$($k(B.
        !            52: \begin{verbatim}
        !            53: [375] load("gr")$
        !            54: [461] gr([x1^2+x2^2-4,x1*x2-1],[x2,x1],2);
        !            55: [-x1^4+4*x1^2-1,x2+x1^3-4*x1]
        !            56: \end{verbatim}
        !            57: $-x_1^4+4*x_1^2-1$ $B$,(B, $I \cap {\bf Q}[x_1]$ $B$N@8@.85$G$"$k(B.
        !            58: $B>C5nK!$H%0%l%V%J4pDl$H$N4XO"$K$D$$$F$O(B, $B$?$H$($P(B
        !            59: Cox, Little, O'Shea $B$N652J=q(B \cite{OpenXM:CLO} $B$N(B 2 $B>O(B, 3 $B>O$K(B
        !            60: $BF~LgE*$J@bL@$,$"$k(B.
        !            61: \end{enumerate}
        !            62:
        !            63: Asir $B$,MQ$$$F$$$k?t3X%"%k%4%j%:%`$K$D$$$F$O(B,
        !            64: $BLnO$$K$h$k2r@b(B \cite{OpenXM:noro-book} $B$r;2>H(B.
        !            65:
        !            66: \subsection{ {\tt ox\_sm1} }
        !            67: {\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
        !            68: $B%0%l%V%J4pDl7W;;$r$b$H$K$7$F(B, $D$ $B2C72$N<o!9$N9=@.$d(B, $BBe?tB?MMBN$N(B
        !            69: $B%3%[%b%m%8$N7W;;$r$*$3$J$&(B.
        !            70: $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,
        !            71: OpenXM $B%W%m%H%3%k(B (OX-RFC 100, 101) $B$rMxMQ$7$F(B, {\tt ox\_asir} $B$,(B
        !            72: $BC4Ev$7$F$*$j(B, {\tt ox\_sm1} $B$O%5!<%P$G$"$k$N$_$J$i$:(B,
        !            73: asir $B$N(B OpenXM $B%5!<%P5!G=$r%U%k$KMxMQ$7$F$$$k%/%i%$%"%s%H$G$b$"$k(B.
        !            74: Kan $B$O$?$H$($P<!$N$h$&$J5!G=$r$b$D(B.
        !            75: \begin{enumerate}
        !            76: \item $D$ $B$G$N%0%l%V%J4pDl7W;;(B ($B4X?t(B {\tt gb}). \\
        !            77: $B%0%l%V%J4pDl7W;;$N1~MQ$H$7$F(B, $B$"$?$($i$l$?O"N)@~7AJPHyJ,J}Dx<07O$N(B
        !            78: $B2r6u4V$N<!85$N7W;;(B (holonomic rank) $B$,$"$k(B.  $B$3$l$rNc$H$7$F5s$2$F(B
        !            79: $B$*$3$&(B.
        !            80: \begin{verbatim}
        !            81: sm1>(cohom.sm1) run ;
        !            82: sm1> [ [( x*Dx + y*Dy -3 ) ( Dx-Dy )] (x,y)] rank ::
        !            83:    1
        !            84: \end{verbatim}
        !            85: $B$3$N=PNO$OHyJ,J}Dx<07O(B
        !            86: $$ \left[
        !            87:      x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}-3 \right]f
        !            88:  = \left[ \frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right]f = 0$$
        !            89: $B$N2r6u4V$N<!85$O(B $1$ $B$G$"$k$3$H$r0UL#$9$k(B.
        !            90: $B$3$N>l9g$O<B:]$K2r$r=q$-=P$9$3$H$,$G$-$F(B, $f=(x+y)^3$ $B$,2r$G$"$k(B.
        !            91: \item $BBe?tB?MMBN(B, $B$?$H$($P(B ${\bf C}^n \setminus V(f)$ $B$N%3%[%b%m%872$N(B
        !            92: $B7W;;(B ( $B4X?t(B {\tt deRham} ). \\
        !            93: $B$?$H$($P(B Kan/k0 $B$G$N7W;;$O<!$N$h$&$K$J$k(B.
        !            94: \begin{verbatim}
        !            95: % cd OpenXM/src/k097/lib/restriction
        !            96: % k0
        !            97: This is kan/k0 Version 1998,12/15
        !            98: sm1 version = 3.001203
        !            99: Default ring is Z[x,h].
        !           100: In(2)= load("demo.k");
        !           101: In(3)= DeRham2WithAsir( x^3-y^2 ):
        !           102: Step1: Annhilating ideal (II)[    -3*x^2*Dy-2*y*Dx , -2*x*Dx-3*y*Dy-6 ]
        !           103: Step2: (-1,1)-minimal resolution (Res0)  [
        !           104:   [
        !           105:     [    2*x*Dx+3*y*Dy-h^2 ]
        !           106:     [    -3*y*Dx^2+2*x*Dy*h ]
        !           107:   ]
        !           108:   [
        !           109:     [    3*y*Dx^2-2*x*Dy*h , 2*x*Dx+3*y*Dy ]
        !           110:   ]
        !           111:  ]
        !           112: Step3: computing the cohomology of the truncated complex.
        !           113: Roots and b-function are [    [    1 ]  , [    9*s^3-18*s^2+11*s-2 ]  ]
        !           114:
        !           115:
        !           116: [    [    0 , 1 , 1 ]  ,
        !           117: [    [    0 , [   ]  ]  , [    1 , [   ]  ]  , [    1 , [   ]  ]  ]  ]
        !           118: \end{verbatim}
        !           119: Step1$B$N(B $1/(x^3-y^2)$ $B$NK~$?$9:GBg$NHyJ,J}Dx<0$N7W;;(B,
        !           120: Step3$B$N0lHL2=$5$l$?(B $b$-$B4X?t$H$=$N:,$N7W;;$G$O(B {\tt ox\_asir} $B$r8F$S=P$7$F(B
        !           121: $B7W;;$7$F$$$k(B.
        !           122: \item $D^m/I$ $B$N(B $(u,v)$-minimal free resolution
        !           123: ($B4X?t(B {\tt Sminimal}).
        !           124: \end{enumerate}
        !           125: $D$ $B2C72$N%"%k%4%j%:%`$N%"%k%4%j%:%`$K$D$$$F$O(B,
        !           126: $BK\(B \cite{OpenXM:SST} $B$,F~LgE*OCBj$+$i:G@hC<$NOCBj$^$G$r07$C$F$$$k(B.
        !           127:
        !           128: \subsection{PHC}
        !           129: PHC pack $B$O(B Jan Verschelde $B$K$h$j3+H/$5$l$?(B,
        !           130: $BB?LLBN%[%b%H%T!<K!$K$h$jBe?tJ}Dx<07O$N?tCM2r$r5a$a$k(B
        !           131: $B%7%9%F%`$G$"$k(B.
        !           132: $B%[%b%H%T!<K!$H$$$&$N$O(B, $B$?$H$($P(B
        !           133: $BBe?tJ}Dx<0(B
        !           134: $ x^n - x^3+x^2-5 = 0$
        !           135: $B$r2r$/$?$a$K(B, $B$^$:%9%?!<%H%7%9%F%`(B
        !           136: $ x^n - 5 = 0$ $B$r2r$-(B, $B$=$N2r$rO"B3E*$K(B
        !           137: $B$b$H$NJ}Dx<0$N2r$K1dD9$9$kJ}K!$G$"$k(B.
        !           138: $BO"N)Be?tJ}Dx<07O$N>l9g$K$O(B,  $B$h$$@-<A$r$b$D%9%?!<%H%7%9%F%`$r:n$k(B
        !           139: $BI,MW$,$"$j(B, $B$=$N$?$a$KB?LLBN$NJ,3d$r$b$A$$$k(B. $B$3$l$,(B
        !           140: $BB?LLBN%[%b%H%T!<K!$G$"$k(B.
        !           141: $BK\(B \cite{OpenXM:CLO2} $B$N(B 7.5 $B@a$K$3$NJ}K!$N4pAC$H$J$k(B
        !           142: $BB?LLBN$N4v2?$H:,$N?t$N4*Dj$K4X$9$k@bL@$,$"$k(B.
        !           143: $BB?LLBN%[%b%H%T!<K!$K$D$$$F$O(B, Verschelde $B$K$h$k2r@b$NB>(B,
        !           144: $BO@J8(B \cite{OpenXM:HS} $B$J$I$,FI$_$d$9$$(B.
        !           145:
        !           146: \subsection{gnuplot, M2, tigers, pari, mathematica, OMproxy}
        !           147:
        !           148: $BD9$5$N4X78$b$"$j>\$7$/$O>R2p$G$-$J$$$N$G(B,
        !           149: $BI=Bj$N%=%U%H$r4JC1$K>R2p$7$h$&(B.
        !           150: \begin{enumerate}
        !           151: \item gnuplot $B$O%0%i%U$r:n@.$9$k%=%U%H(B.
        !           152: \item M2 (Macaulay 2) $B$O(B D.Grayson $B$H(B M.Stillman $B$K$h$j3+H/$5$l$F$$$k(B,
        !           153: $B7W;;Be?t4v2?MQ$N%=%U%H(B.
        !           154: \item tigers $B$O(B B.Hubert $B$K$h$j3+H/$5$l$?(B, affine toric variety $B$NA4$F$N(B
        !           155: $B%0%l%V%J4pDl$r5a$a$k%=%U%H(B.
        !           156: \item pari $B$O(B A.Cohen $B$i$K$h$j3+H/$5$l$F$$$k(B, $B@0?tO@MQ$N%=%U%H(B.
        !           157: \item Mathematica $B$OM-L>$J$N$G@bL@$NI,MW$O$J$$$G$"$m$&(B.
        !           158: \item OMproxy $B$O(B Java $B$G$+$+$l$?(B, OpenXM $B$N%/%i%9%i%$%V%i%j$*$h$S(B
        !           159: OpenMath $B$H$N%G!<%?$NAj8_JQ49$N$?$a$N%=%U%H$G$"$k(B.
        !           160: \end{enumerate}
        !           161:
        !           162: \subsection{ 1077 functions are available on our servers and libraries}
        !           163:
        !           164: OX server $B$NDs6!$9$k4X?t$K$I$N$h$&$J$b$N$,$"$k$+(B,
        !           165: $B%j%9%H$H4JC1$J%G%b$r7G:\$7$F$*$3$&(B.
        !           166: $B$3$l$i$N4X?t$N$J$+$N0lIt$K$D$$$F$O(B, $BA0@a$^$G$G4JC1$K@bL@$7$?(B.
        !           167: $B$h$j>\$7$/$O3F%3%s%]!<%M%s%H%7%9%F%`$N%^%K%e%"%k$d;2>H$5$l$F$$$k(B
        !           168: $BO@J8(B, $BK\$r8+$kI,MW$,$"$k(B.
        !           169:
        !           170:
        !           171: \fbox{\huge {\color{green}Operations on Integers}}
        !           172:
        !           173: \noindent
        !           174: {\color{red} idiv},{\color{red} irem} (division with remainder),
        !           175: {\color{red} ishift} (bit shifting),
        !           176: {\color{red} iand},{\color{red} ior},{\color{red} ixor} (logical operations),
        !           177: {\color{red} igcd},(GCD by various methods such as Euclid's algorithm and
        !           178: the accelerated GCD algorithm),
        !           179: {\color{red} fac} (factorial),
        !           180: {\color{red} inv} (inverse modulo an integer),
        !           181: {\color{red} random} (random number generator by the Mersenne twister algorithm).
        !           182:
        !           183:
        !           184:
        !           185: \medbreak
        !           186:
        !           187: \noindent
        !           188: \fbox{\huge {\color{green}Ground Fields}}
        !           189:
        !           190: \noindent
        !           191: Arithmetics on various fields: the rationals,
        !           192: ${\bf Q}(\alpha_1,\alpha_2,\ldots,\alpha_n)$
        !           193: ($\alpha_i$ is algebraic over ${\bf Q}(\alpha_1,\ldots,\alpha_{i-1})$),
        !           194: $GF(p)$ ($p$ is a prime of arbitrary size), $GF(2^n)$.
        !           195:
        !           196: \medbreak
        !           197:
        !           198: \noindent
        !           199: \fbox{\huge {\color{green}Operations on Polynomials}}
        !           200:
        !           201: \noindent
        !           202: {\color{red} sdiv }, {\color{red} srem } (division with remainder),
        !           203: {\color{red} ptozp } (removal of the integer content),
        !           204: {\color{red} diff } (differentiation),
        !           205: {\color{red} gcd } (GCD over the rationals),
        !           206: {\color{red} res } (resultant),
        !           207: {\color{red} subst } (substitution),
        !           208: {\color{red} umul} (fast multiplication of dense univariate polynomials
        !           209: by a hybrid method with Karatsuba and FFT+Chinese remainder),
        !           210: {\color{red} urembymul\_precomp} (fast dense univariate polynomial
        !           211: division with remainder by the fast multiplication and
        !           212: the precomputed inverse of a divisor),
        !           213:
        !           214: \noindent
        !           215: \fbox{\huge {\color{green}Polynomial Factorization}}
        !           216: {\color{red} fctr } (factorization over the rationals),
        !           217: {\color{red} fctr\_ff } (univariate factorization over finite fields),
        !           218: {\color{red} af } (univariate factorization over algebraic number fields),
        !           219: {\color{red} sp} (splitting field computation).
        !           220:
        !           221: \medbreak
        !           222:
        !           223: \noindent
        !           224: \fbox{\huge {\color{green} Groebner basis}}
        !           225:
        !           226: \noindent
        !           227: {\color{red} dp\_gr\_main } (Groebner basis computation of a polynomial ideal
        !           228: over the rationals by the trace lifting),
        !           229: {\color{red} dp\_gr\_mod\_main } (Groebner basis over small finite fields),
        !           230: {\color{red} tolex } (Modular change of ordering for a zero-dimensional ideal),
        !           231: {\color{red} tolex\_gsl } (Modular rational univariate representation
        !           232: for a zero-dimensional ideal),
        !           233: {\color{red} dp\_f4\_main } ($F_4$ over the rationals),
        !           234: {\color{red} dp\_f4\_mod\_main } ($F_4$ over small finite fields).
        !           235:
        !           236: \medbreak
        !           237: \noindent
        !           238: \fbox{\huge {\color{green} Ideal Decomposition}}
        !           239:
        !           240: \noindent
        !           241: {\color{red} primedec} (Prime decomposition of the radical),
        !           242: {\color{red} primadec} (Primary decomposition of ideals by Shimoyama/Yokoyama algorithm).
        !           243:
        !           244: \medbreak
        !           245:
        !           246: \noindent
        !           247: \fbox{\huge {\color{green} Quantifier Elimination}}
        !           248:
        !           249: \noindent
        !           250: {\color{red} qe} (real quantifier elimination in a linear and
        !           251: quadratic first-order formula),
        !           252: {\color{red} simpl} (heuristic simplification of a first-order formula).
        !           253:
        !           254: {\scriptsize
        !           255: \begin{verbatim}
        !           256: [0] MTP2 = ex([x11,x12,x13,x21,x22,x23,x31,x32,x33],
        !           257: x11+x12+x13 @== a1 @&& x21+x22+x23 @== a2 @&& x31+x32+x33 @== a3
        !           258: @&& x11+x21+x31 @== b1 @&& x12+x22+x32 @== b2 @&& x13+x23+x33 @== b3
        !           259: @&& 0 @<= x11 @&& 0 @<= x12 @&& 0 @<= x13 @&& 0 @<= x21
        !           260: @&& 0 @<= x22 @&& 0 @<= x23 @&& 0 @<= x31 @&& 0 @<= x32 @&& 0 @<= x33)$
        !           261: [1] TSOL= a1+a2+a3@=b1+b2+b3 @&& a1@>=0 @&& a2@>=0 @&& a3@>=0
        !           262: @&& b1@>=0 @&& b2@>=0 @&& b3@>=0$
        !           263: [2] QE_MTP2 = qe(MTP2)$
        !           264: [3] qe(all([a1,a2,a3,b1,b2,b3],QE_MTP2 @equiv TSOL));
        !           265: @true
        !           266: \end{verbatim}}
        !           267: \medbreak
        !           268:
        !           269: \noindent
        !           270: \fbox{\huge {\color{green} Visualization of curves}}
        !           271:
        !           272: \noindent
        !           273: {\color{red} plot} (plotting of a univariate function),
        !           274: {\color{red} ifplot} (plotting zeros of a bivariate polynomial),
        !           275: {\color{red} conplot} (contour plotting of a bivariate polynomial function).
        !           276:
        !           277: \medbreak
        !           278:
        !           279: \noindent
        !           280: \fbox{\huge {\color{green} Miscellaneous functions}}
        !           281:
        !           282: \noindent
        !           283: {\color{red} det} (determinant),
        !           284: {\color{red} qsort} (sorting of an array by the quick sort algorithm),
        !           285: {\color{red} eval} (evaluation of a formula containing transcendental functions
        !           286: such as
        !           287: {\color{red} sin}, {\color{red} cos}, {\color{red} tan}, {\color{red} exp},
        !           288: {\color{red} log})
        !           289: {\color{red} roots} (finding all roots of a univariate polynomial),
        !           290: {\color{red} lll} (computation of an LLL-reduced basis of a lattice).
        !           291:
        !           292: \noindent
        !           293: \fbox{\huge {\color{green} $D$-modules}} ($D$ is the Weyl algebra)
        !           294:
        !           295: \noindent
        !           296: {\color{red} gb } (Gr\"obner basis),
        !           297: {\color{red} syz} (syzygy),
        !           298: {\color{red} annfs} (Annhilating ideal of $f^s$),
        !           299: {\color{red} bfunction},
        !           300: {\color{red} schreyer} (free resolution by the Schreyer method),
        !           301: {\color{red} vMinRes} (V-minimal free resolution),
        !           302: {\color{red} characteristic} (Characteristic variety),
        !           303: {\color{red} restriction} in the derived category of $D$-modules,
        !           304: {\color{red} integration} in the derived category,
        !           305: {\color{red} tensor}  in the derived category,
        !           306: {\color{red} dual} (Dual as a D-module),
        !           307: {\color{red} slope}.
        !           308:
        !           309: \medbreak
        !           310: \noindent
        !           311: \fbox{\huge {\color{green} Cohomology groups}}
        !           312:
        !           313: \noindent
        !           314: {\color{red} deRham} (The de Rham cohomology groups of
        !           315: ${\bf C}^n \setminus V(f)$,
        !           316: {\color{red} ext} (Ext modules for a holonomic $D$-module $M$
        !           317: and the ring of formal power series).
        !           318:
        !           319: \medbreak
        !           320: \noindent
        !           321: \fbox{\huge
        !           322:  {\color{green} Differential equations}}
        !           323:
        !           324: \noindent
        !           325: Helping to derive and prove {\color{red} combinatorial} and
        !           326: {\color{red} special function identities},
        !           327: {\color{red} gkz} (GKZ hypergeometric differential equations),
        !           328: {\color{red} appell} (Appell's hypergeometric differential equations),
        !           329: {\color{red} indicial} (indicial equations),
        !           330: {\color{red} rank} (Holonomic rank),
        !           331: {\color{red} rrank} (Holonomic rank of regular holonomic systems),
        !           332: {\color{red} dsolv} (series solutions of holonomic systems).
        !           333:
        !           334: \medbreak
        !           335: \noindent
        !           336: \fbox{\huge
        !           337:  {\color{green} OpenMATH support}}
        !           338:
        !           339: \noindent
        !           340: {\color{red} om\_xml} (CMO to OpenMATH XML),
        !           341: {\color{red} om\_xml\_to\_cmo} (OpenMATH XML to CMO).
        !           342:
        !           343: \medbreak
        !           344: \noindent
        !           345: \fbox{\huge
        !           346:  {\color{green} Homotopy Method}}
        !           347:
        !           348: \noindent
        !           349: {\color{red} phc} (Solving systems of algebraic equations by
        !           350: numerical and polyhedral homotopy methods).
        !           351:
        !           352: \medbreak
        !           353: \noindent
        !           354: \fbox{\huge
        !           355:  {\color{green} Toric ideal}}
        !           356:
        !           357: \noindent
        !           358: {\color{red} tigers} (Enumerate all Gr\"obner basis of a toric ideal.
        !           359: Finding test sets for integer program),
        !           360: {\color{red} arithDeg} (Arithmetic degree of a monomial ideal),
        !           361: {\color{red} stdPair} (Standard pair decomposition of a monomial ideal).
        !           362:
        !           363: \medbreak
        !           364: \noindent
        !           365: \fbox{\huge {\color{green} Communications}}
        !           366:
        !           367: \noindent
        !           368: {\color{red} ox\_launch} (starting a server),
        !           369: {\color{red} ox\_launch\_nox},
        !           370: {\color{red} ox\_shutdown},
        !           371: {\color{red} ox\_launch\_generic},
        !           372: {\color{red} generate\_port},
        !           373: {\color{red} try\_bind\_listen},
        !           374: {\color{red} try\_connect},
        !           375: {\color{red} try\_accept},
        !           376: {\color{red} register\_server},
        !           377: {\color{red} ox\_rpc},
        !           378: {\color{red} ox\_cmo\_rpc},
        !           379: {\color{red} ox\_execute\_string},
        !           380: {\color{red} ox\_reset} (reset the server),
        !           381: {\color{red} ox\_intr},
        !           382: {\color{red} register\_handler},
        !           383: {\color{red} ox\_push\_cmo},
        !           384: {\color{red} ox\_push\_local},
        !           385: {\color{red} ox\_pop\_cmo},
        !           386: {\color{red} ox\_pop\_local},
        !           387: {\color{red} ox\_push\_cmd},
        !           388: {\color{red} ox\_sync},
        !           389: {\color{red} ox\_get},
        !           390: {\color{red} ox\_pops},
        !           391: {\color{red} ox\_select},
        !           392: {\color{red} ox\_flush},
        !           393: {\color{red} ox\_get\_serverinfo}
        !           394:
        !           395: \medbreak
        !           396: \noindent
        !           397: In addition to these functions, {\color{green} Mathematica functions}
        !           398: can be called as server functions.
        !           399: \medbreak
        !           400: \noindent
        !           401: \fbox{\huge {\color{green} Examples}}
        !           402: {\footnotesize
        !           403: \begin{verbatim}
        !           404: [345] sm1_deRham([x^3-y^2*z^2,[x,y,z]]);
        !           405: [1,1,0,0]
        !           406: /* dim H^i = 1 (i=0,1), =0 (i=2,3) */
        !           407: \end{verbatim}}
        !           408:
        !           409: \noindent
        !           410: {\footnotesize \begin{verbatim}
        !           411: [287] phc(katsura(7)); B=map(first,Phc)$
        !           412: [291] gnuplot_plotDots(B,0)$
        !           413: \end{verbatim} }
        !           414:
        !           415: \epsfxsize=3cm
        !           416: \begin{center}
        !           417: \epsffile{../calc2000/katsura7.ps}
        !           418: %\epsffile{katsura7.ps}
        !           419: \end{center}
        !           420: %%The first components of the solutions to the system of algebraic equations Katsura 7.
        !           421:
        !           422: \medbreak
        !           423: \noindent
        !           424: \fbox{ {\color{green} Authors}}
        !           425: Castro-Jim\'enez, Dolzmann, Hubert, Murao, Noro, Oaku, Okutani,
        !           426: Shimoyama, Sturm, Takayama, Tamura, Verschelde, Yokoyama.
        !           427:
        !           428:
        !           429:
        !           430: \section{OX $B%5!<%P$rAH$_9g$o$;$FMxMQ$7$?Nc(B}
        !           431:
        !           432: $B6qBNE*$J?t3XE*LdBj$r$b$H$K?t3X%=%U%H$N3+H/$r$9$k$H$$$&$N$O(B
        !           433: $B0l$D$N7rA4$J3+H/<jK!$G$"$m$&(B.
        !           434:
        !           435: OX $B%5!<%P$N3+H/$G$b(B, $B$=$N$h$&$J3+H/<jK!$r$H$j$?$$$H;W$C$F$*$j(B,
        !           436: $B$$$m$$$m$J?t3XE*LdBj$r$5$,$7$F$$$k(B.
        !           437:
        !           438: $B$=$NCf$N0l$D$NLdBj$O(B, $BB?JQ?tD64v2?4X?t$K4X$7$F2?$G$bEz$($i$l$k%7%9%F%`$N(B
        !           439: $B3+H/$G$"$k(B.
        !           440: $B$3$3$G$O(B, $BB?JQ?tD64v2?4X?t$H$7$F(B, $B$$$o$f$k(B GKZ hypergeometric system
        !           441: $B$N2r$r9M$($F$$$k(B.
        !           442: GKZ hypergeometric system $B$O(B $n$ $B<!856u4V$NE@=89g$KIU?o$7$F$-$^$k(B
        !           443: $BO"N)@~7AJPHyJ,J}Dx<07O$G$"$k(B.
        !           444: $n$ $B<!856u4V$NE@=89g$r9M$($k$?$a(B, $BB?LLBN$N4v2?$r07$&I,MW$,$"$k$7(B,
        !           445: $BO"N)@~7AJPHyJ,J}Dx<07O$r9M$($k$?$a(B, $D$ $B2C72$r07$&I,MW$b$"$k(B.
        !           446: GKZ system $B$O(B affine toric ideal $B$r4^$`$?$a(B, affine toric ideal $B$N(B
        !           447: $B%7%9%F%`$bI,MW$G$"$k(B.
        !           448: $B4X?t$N?tCM7W;;$N$?$a$K$O(B, $B?tCM@QJ,E*$J<jK!$b=EMW$G$"$k$,(B, $B$3$l$O$^$@(B
        !           449: $BA4$/<j$r$D$1$F$$$J$$(B.
        !           450: GKZ hypergeometric system $B$K$D$$$F$O(B, $BK\(B \cite{OpenXM:SST} $B$r;2>H(B.
        !           451:
        !           452: $B8=:_$N(B {\tt OpenXM/src/asir-contrib} $B$N%Q%C%1!<%8$O(B,
        !           453: GKZ hypergeometric system $B$K4X$7$F2?$G$b$3$?$($i$l$k%7%9%F%`$rL\I8$K(B,
        !           454: 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.
        !           455:
        !           456: $BNc$r0l$D$"$2$h$&(B.
        !           457: {\tt dsolv\_starting\_term} $B$O@5B'%[%m%N%_%C%/7O$r(B cone $B>e$G<}B+$9$k(B
        !           458: $BB?JQ?t$N5i?t$G2r$/$?$a$N(B
        !           459: $B4X?t$N0l$D$G$"$j(B, $B5i?tE83+$N<gIt$r$b$H$a$k(B.
        !           460: \begin{enumerate}
        !           461: \item $B9TNs(B($BE@G[CV(B) $\pmatrix{ 1&1&1&1&1 \cr
        !           462:                               1&1&0&-1&0 \cr
        !           463:                               0&1&1&-1&0 \cr}$ $B$KIU?o$9$k(B
        !           464: GKZ hypergeometric system $B$r4X?t(B {\tt sm1\_gkz} $B$G5a$a$k(B.
        !           465: \begin{verbatim}
        !           466: [1076]   F = sm1_gkz( [ [[1,1,1,1,1],[1,1,0,-1,0],[0,1,1,-1,0]], [1,0,0]]);
        !           467: [[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,-x4*dx4+x2*dx2+x1*dx1,
        !           468:   -x4*dx4+x3*dx3+x2*dx2,
        !           469:   -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4],[x1,x2,x3,x4,x5]]
        !           470: \end{verbatim}
        !           471: \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
        !           472: $B5i?t2r$N<gIt$r5a$a$k(B.
        !           473: \begin{verbatim}
        !           474: [1077]  A= dsolv_starting_term(F[0],F[1],[1,1,1,1,0])$
        !           475: Computing the initial ideal.
        !           476: Done.
        !           477: Computing a primary ideal decomposition.
        !           478: Primary ideal decomposition of the initial Frobenius ideal
        !           479: to the direction [1,1,1,1,0] is
        !           480: [[[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,
        !           481:    x5^2-2*x5-8*x4^2+1,x5^3-3*x5^2+3*x5-1],
        !           482:  [x5-1,x4,x3,x2,x1]]]
        !           483:
        !           484: ----------- root is [ 0 0 0 0 1 ]
        !           485: ----------- dual system is
        !           486: [x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2
        !           487:  +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2,
        !           488:  x4-2*x3+3*x2-2*x1,x5-x3+x2-x1,1]
        !           489: \end{verbatim}
        !           490: \item $BE83+$N<gIt(B 4 $BDL$j$"$j(B, $B$=$l$>$l$r0x?tJ,2r$9$k$H(B,
        !           491: $B<!$N$h$&$K$J$k(B.
        !           492: $B$?$H$($P(B, 3 $BHVL\$N(B
        !           493: {\tt [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]], }
        !           494: $B$O(B,
        !           495: $$ x_5 (-\log x_1 + \log x_2 - \log x_3 + \log x_5) =
        !           496:    x+5 \log \frac{x_2 x_5}{x_1 x_3} $$
        !           497: $B$+$i;O$^$k5i?t2r$,B8:_$9$k$3$H$r0UL#$9$k(B.
        !           498: \begin{verbatim}
        !           499: [1078] A[0];
        !           500: [[ 0 0 0 0 1 ]]
        !           501: [1079] map(fctr,A[1][0]);
        !           502: [[[1/8,1],[x5,1],[log(x2)+log(x4)-2*log(x5),1],
        !           503:           [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5),1]],
        !           504:  [[1,1],[x5,1],[-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]],
        !           505:  [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]],
        !           506:  [[1,1],[x5,1]]]
        !           507: \end{verbatim}
        !           508: \end{enumerate}
        !           509: {\tt dsolv\_starting\_term} $B$G$O(B,  kan/sm1 $B$,(B D-$B2C72$N7W;;(B,
        !           510: asir $B$,=`AG%$%G%"%kJ,2r$N7W;;(B, $B2r$N7W;;$rC4Ev$7$F$$$k(B.
        !           511: $B2r$NE83+$NJ}8~$,2?DL$jB8:_$9$k$+$rD4$Y$k$K$O(B,
        !           512: tigers $B$r1gMQ$9$k$H$h$$(B.
        !           513:
        !           514:
        !           515: \begin{thebibliography}{99}
        !           516: \bibitem{OpenXM:CLO}
        !           517: Cox, D., Little, J., O'Shea;
        !           518:
        !           519: \bibitem{OpenXM:CLO2}
        !           520: Cox, D., Little, J., O'Shea,
        !           521: {\it Using Algebraic Geometry},
        !           522: Springer, 1998.
        !           523:
        !           524: \bibitem{OpenXM:HS}
        !           525: Huber, B., Sturmfels, B.,
        !           526: A Polyhedral Method for Solving Sparse Polynomial Systems,
        !           527: Mathematics of Computation,
        !           528: {\bf 64} (1995), 1541--1555.
        !           529:
        !           530: \bibitem{OpenXM:noro-book}
        !           531:  $BLnO$(B:  $B7W;;Be?tF~Lg(B, Rokko Lectures in Mathematics, 9,
        !           532: 2000.  ISBN 4-907719-09-4. \\
        !           533: {\tt ftp://ftp.math.kobe-u.ac.jp/pub/OpenXM/Head/openxm-head.tar.gz}
        !           534: $B$N%G%#%l%/%H%j(B {\tt OpenXM/doc/compalg} $B$K$3$NK\$N(B TeX $B%=!<%9$,$"$k(B.
        !           535:
        !           536: \bibitem{OpenXM:SST}
        !           537: Saito, M., Sturmfels, B., Takayama, N.,
        !           538: {\it Gr\"obner deformations of hypergeometric differential
        !           539: equations}. Algorithms and Computation in Mathematics,
        !           540: 6. Springer-Verlag, Berlin, 2000.
        !           541:
        !           542: \end{thebibliography}
        !           543:
        !           544:
        !           545: \end{document}