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1.1     ! takayama    1: /* $OpenXM$ */
        !             2:
        !             3:
        !             4: /*&C-texi
        !             5: @node TIGERS Functions,,, Top
        !             6: */
        !             7: /*&eg-texi
        !             8: @chapter TIGERS Functions
        !             9:
        !            10: This chapter describes  interface functions for
        !            11: tigers ox server @code{ox_sm1_tigers}.
        !            12: */
        !            13: /*&jp-texi
        !            14: @chapter TIGERS $BH!?t(B
        !            15:
        !            16: $B$3$N>O$G$O(B,
        !            17: tigers ox server @code{ox_sm1_tigers}
        !            18: $B$K$?$$$9$k%$%s%?%U%'!<%9H!?t$r@bL@$9$k(B.
        !            19: */
        !            20:
        !            21: /*&eg-texi
        !            22: @menu
        !            23: * tigers.tigers::
        !            24: @end menu
        !            25: @node tigers.tigers,,, TIGERS Functions
        !            26: @subsection @code{tigers.tigers}
        !            27: @findex tigers.tigers
        !            28: @table @t
        !            29: @item tigers.tigers(@var{a}|proc=@var{a})
        !            30: ::   It asks the @code{tigers} server of the descriptor number @var{p}
        !            31: to enumerate all Grobner bases associated to the toric variaty
        !            32: defined by the matrix @var{a}.
        !            33: @end table
        !            34:
        !            35: @table @var
        !            36: @item return
        !            37: List
        !            38: @item p
        !            39: Number
        !            40: @item a
        !            41: List
        !            42: @end table
        !            43:
        !            44: @itemize @bullet
        !            45: @item  It asks the @code{tigers} server of the descriptor number @var{p}
        !            46: to enumerate all Grobner bases associated to the toric variaty
        !            47: defined by the matrix @var{a}.
        !            48: @item
        !            49: The system tigers is an expert system to enumerate
        !            50: all Gr\"obner bases of affine toric ideals.
        !            51: In other words, it can be used to determine the state polytope
        !            52: of a given affine toric ideal.
        !            53: As to a theoretical background, see the book @*
        !            54: B.Sturmfels, Grobner bases and Convex Polytopes. @*
        !            55: The original tigers is written by Birk Hubert.
        !            56: The algorithm used in explained in the paper @*
        !            57: B.Huber and R.Thomas, Computing Grobner Fans of Toric Ideals.
        !            58: @end itemize
        !            59: */
        !            60:
        !            61: /*&jp-texi
        !            62: @menu
        !            63: * tigers.tigers::
        !            64: @end menu
        !            65: @node tigers.tigers,,, TIGERS Functions
        !            66: @subsection @code{tigers.tigers}
        !            67: @findex tigers.tigers
        !            68: @table @t
        !            69: @item tigers.tigers(@var{a}|proc=@var{a})
        !            70: ::   $B$3$NH!?t$O<1JL;R(B @var{p} $B$N(B tigers $B%5!<%P$K(B
        !            71: $B9TNs(B @var{a} $B$KIU?o$7$?%H!<%j%C%/%$%G%"%k$N$9$Y$F$N%0%l%V%J4pDl$r(B
        !            72: $B7W;;$7$F$/$l$k$h$&$K$?$N$`(B.
        !            73: @end table
        !            74:
        !            75: @table @var
        !            76: @item $BLa$jCM(B
        !            77: $B%j%9%H(B
        !            78: @item p
        !            79: $B?t(B
        !            80: @item a
        !            81: $B%j%9%H(B
        !            82: @end table
        !            83:
        !            84: @itemize @bullet
        !            85: @item
        !            86:  $B$3$NH!?t$O<1JL;R(B @var{p} $B$N(B tigers $B%5!<%P$K(B
        !            87: $B9TNs(B @var{a} $B$KIU?o$7$?%H!<%j%C%/%$%G%"%k$N$9$Y$F$N%0%l%V%J4pDl$r(B
        !            88: $B7W;;$7$F$/$l$k$h$&$K$?$N$`(B.
        !            89: @item
        !            90: Tigers $B$O(B $B%"%U%#%s%H!<%j%C%/%$%G%"%k$N(B reduced $B%0%l%V%J4pDl$r(B
        !            91: $B$9$Y$F?t$($"$2$k$?$a$N@lMQ$N%W%m%0%i%`$G$"$k(B.
        !            92: $B$3$N%W%m%0%i%`$O(B, $B%"%U%#%s%H!<%j%C%/%$%G%"%k$N(B state polytope
        !            93: $B$r$-$a$k$?$a$K;H$($k(B.
        !            94: $BM}O@E*$J%P%C%/%0%i%&%s%I$K$D$$$F$O(B,
        !            95: $BK\(B @*
        !            96: B.Sturmfels, Grobner bases and Convex Polytopes @*
        !            97: $B$r8+$h(B.
        !            98: Tigers $B$O(B Birk Hubert $B$,:n<T$G$"$k(B.
        !            99: $B$3$N%W%m%0%i%`$NMxMQ$7$F$$$k%"%k%4%j%:%`$O(B @*
        !           100: B.Huber and R.Thomas, Computing Grobner Fans of Toric Ideals @*
        !           101: $B$K@bL@$5$l$F$$$k(B.
        !           102: @end itemize
        !           103: */
        !           104:
        !           105: /*&C-texi
        !           106:
        !           107: @example
        !           108: [395] A=[[1,1,1,1],[0,1,2,3]]$
        !           109: [306] S=tigers.tigers(A)$
        !           110: [307] length(S);
        !           111: 8
        !           112: [308] S[0];
        !           113: [[[1,0,1,0],[0,2,0,0]],[[1,0,0,1],[0,1,1,0]],[[0,1,0,1],[0,0,2,0]]]
        !           114: [309] S[1];
        !           115: [[[1,0,0,1],[0,1,1,0]],[[0,2,0,0],[1,0,1,0]],[[0,1,0,1],[0,0,2,0]]]
        !           116:
        !           117: @end example
        !           118:
        !           119: */
        !           120:
        !           121: /*&eg-texi
        !           122: In this example, all reduced Grobner bases for the toric ideal
        !           123: associated to the matrix @var{A} are stored in @var{S}.
        !           124: There are eight distinct Grobner bases of @var{A}.
        !           125: [[i_1, i_2, ...],[j_1, j_2, ...]] is a set of exponents of
        !           126: two monomials and stands for a binomial.
        !           127: For example,
        !           128: the S[0] consists of @*
        !           129:   x1 x3 - x2^2,   x1 x4 - x2 x3, x2 x4 - x3^2.  @*
        !           130: <x1 x3, x1 x4, x2 x4> is the initial ideal of S[0].
        !           131:
        !           132: */
        !           133: /*&jp-texi
        !           134: $B$3$NNc$G$O(B, @var{A} $B$KIU?o$7$?%"%U%#%s%H!<%j%C%/%$%G%"%k$N(B
        !           135: $B$9$Y$F$N%0%l%V%J4pDl$,(B @var{S} $B$K3JG<$5$l$k(B.
        !           136: $B$3$NNc$G$O(B, 8 $B8D$N%0%l%V%J4pDl$,$"$k(B.
        !           137: [[i_1, i_2, ...],[j_1, j_2, ...]] $B$OFs$D$N%b%N%_%"%k$N(B
        !           138: exponent $B$r$J$i$Y$?$b$N$G$"$j(B, 2 $B9`<0$r$"$i$o$9(B.
        !           139: $B$?$H$($P(B,
        !           140: S[0] $B$O<!$NB?9`<0$N=89g(B @*
        !           141:   x1 x3 - x2^2,   x1 x4 - x2 x3, x2 x4 - x3^2  @*
        !           142: $B$G$"$j(B,
        !           143: <x1 x3, x1 x4, x2 x4> $B$,$=$N(B initial ideal $B$G$"$k(B.
        !           144:
        !           145: */
        !           146: