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D.4.22.2 hilbert4ti2

Procedure from library sing4ti2.lib (see sing4ti2_lib).

Usage:
hilbert4ti2(A[,i]);
A=intmat
i=int

Assume:
- A is a matrix with integer entries which describes the lattice
as ker(A), if second argument is not present,
as the left image Im(A) = {zA : z \in ZZ^k}, if second argument is a positive integer
- number of variables of basering equals number of columns of A
(for ker(A)) resp. of rows of A (for Im(A))

Create:
temporary files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar
in the current directory (I/O files for communication with 4ti2)

Note:
input rules for 4ti2 also apply to input to this procedure
hence ker(A)={x|Ax=0} and Im(A)={xA}

Return:
toric ideal specified by Hilbert basis thereof

Example:
 
LIB "sing4ti2.lib";
ring r=0,(x1,x2,x3,x4,x5,x6,x7,x8,x9),dp;
matrix M[7][9]=1,1,1,-1,-1,-1,0,0,0,1,1,1,0,0,0,-1,-1,-1,0,1,1,-1,0,0,-1,0,0,1,0,1,0,-1,0,0,-1,0,1,1,0,0,0,-1,0,0,-1,0,1,1,0,-1,0,0,0,-1,1,1,0,0,-1,0,-1,0,0;
hilbert4ti2(M);
==> -------------------------------------------------
==> 4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team.
==> 4ti2 comes with ABSOLUTELY NO WARRANTY.
==> This is free software, and you are welcome
==> to redistribute it under certain conditions.
==> For details, see the file COPYING.
==> -------------------------------------------------
==> 
==> Linear integer system to solve:
==> 
==> + + +  +  +  +  +  +  + 
==> 0 0 0  0  0  0  0  0  0 
==> 
==> 1 1 1 -1 -1 -1  0  0  0 = 0
==> 1 1 1  0  0  0 -1 -1 -1 = 0
==> 0 1 1 -1  0  0 -1  0  0 = 0
==> 1 0 1  0 -1  0  0 -1  0 = 0
==> 1 1 0  0  0 -1  0  0 -1 = 0
==> 0 1 1  0 -1  0  0  0 -1 = 0
==> 1 1 0  0 -1  0 -1  0  0 = 0
==> 
==> Linear integer system of homogeneous equalities to solve:
==> 
==> + + +  +  +  +  +  +  + 
==> 0 0 0  0  0  0  0  0  0 
==> 
==> 1 1 1 -1 -1 -1  0  0  0 = 0
==> 1 1 1  0  0  0 -1 -1 -1 = 0
==> 0 1 1 -1  0  0 -1  0  0 = 0
==> 1 0 1  0 -1  0  0 -1  0 = 0
==> 1 1 0  0  0 -1  0  0 -1 = 0
==> 0 1 1  0 -1  0  0  0 -1 = 0
==> 1 1 0  0 -1  0 -1  0  0 = 0
==> 
==> 
==> Appending variable 1... Solutions: 5, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 2... Solutions: 4, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 3... Solutions: 3, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 4... Solutions: 4, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 5... Solutions: 4, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 6... Solutions: 7, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 7... Solutions: 6, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 8... Solutions: 5, Step: 0.00 secs, Time: 0.00 secs
==> Appending variable 9... Solutions: 5, Step: 0.00 secs, Time: 0.00 secs
==> 
==> Final basis has 1 inhomogeneous, 5 homogeneous and 0 free elements.
==> 
==> 4ti2 Total Time: 0.00 secs
==> _[1]=x1^2*x3*x5*x6^2*x7*x8^2-1
==> _[2]=x1*x3^2*x4^2*x5*x8^2*x9-1
==> _[3]=x2^2*x3*x4^2*x5*x7*x9^2-1
==> _[4]=x1*x2^2*x5*x6^2*x7^2*x9-1
==> _[5]=x1*x2*x3*x4*x5*x6*x7*x8*x9-1


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