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D.4.24.1 markov4ti2
Procedure from library sing4ti2.lib (see sing4ti2_lib).
- Usage:
- markov4ti2(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 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:
- 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 Markov basis thereof
Example:
| LIB "sing4ti2.lib";
ring r=0,(x,y,z),dp;
matrix M[2][3]=0,1,2,2,1,0;
markov4ti2(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.
==> -------------------------------------------------
==> Using 64 bit integers.
==> Computing generating set (Hybrid) ...
==> Phase 1:
==> Computing generating set (Saturation) ...
==> Saturating 2 variable(s).
==>
Sat 2: Col: 1 (U) Size: 1, ToDo: 1
Sat 2: Col: \
1 (U) Size: 1, ToDo: 0
Sat 2: Col: 1 (U) Size: \
1, Time: -0.00 / -0.00 secs.
==> Saturated already on 1 variable(s).
==> Done. Size: 1, Time: -0.00 / -0.00 secs
==> Phase 2:
==> Lifting 1 variable(s).
==>
Lift 1: Col: 2 (F) Size: 1, Index: 0
Lift 1: Col: \
2 (F) Size: 1, Time: -0.00 / -0.00 secs.
==> Done. Size: 1, Time: -0.00 / -0.00 secs
==> Computing Miminal Generation Set (Fast)...
==>
Size: 1, Time: -0.00 / -0.00 secs. Done.
==> 4ti2 Total Time: -0.00 secs.
==> _[1]=-y2+xz
matrix N[1][3]=1,2,1;
markov4ti2(N,1);
==> -------------------------------------------------
==> 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.
==> -------------------------------------------------
==> Using 64 bit integers.
==> Computing generating set (Hybrid) ...
==> 4ti2 Total Time: -0.00 secs.
==> _[1]=xy2z-1
==> _[2]=xy2z-1
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