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1.3     ! ohara       1: /* $OpenXM: OpenXM/src/asir-contrib/packages/doc/tigers.oxweave,v 1.2 2004/03/05 15:30:50 ohara Exp $ */
1.1       takayama    2:
                      3:
1.3     ! ohara       4: /*&C
1.1       takayama    5: @node TIGERS Functions,,, Top
                      6: */
1.2       ohara       7: /*&en
1.1       takayama    8: @chapter TIGERS Functions
                      9:
                     10: This chapter describes  interface functions for
                     11: tigers ox server @code{ox_sm1_tigers}.
                     12: */
1.2       ohara      13: /*&ja
1.1       takayama   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:
1.2       ohara      21: /*&en
1.1       takayama   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:
1.2       ohara      61: /*&ja
1.1       takayama   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:
1.3     ! ohara     105: /*&C
1.1       takayama  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:
1.2       ohara     121: /*&en
1.1       takayama  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: */
1.2       ohara     133: /*&ja
1.1       takayama  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: