D.4.11.1 toric_ideal

Procedure from library toric.lib (see toric_lib).

Usage:

toric_ideal(A,alg); A intmat, alg string
toric_ideal(A,alg,prsv); A intmat, alg string, prsv intvec

Return:

ideal: standard basis of the toric ideal of A

Note:

These procedures return the standard basis of the toric ideal of A with respect to the term ordering in the current basering. Not all term orderings are supported: The usual global term orderings may be used, but no block orderings combining them.
One may call the procedure with several different algorithms:
- the algorithm of Conti/Traverso using elimination (ect),
- the algorithm of Pottier (pt),
- an algorithm of Bigatti/La Scala/Robbiano (blr),
- the algorithm of Hosten/Sturmfels (hs),
- the algorithm of DiBiase/Urbanke (du).
The argument ‘alg’ should be the abbreviation for an algorithm as above: ect, pt, blr, hs or du.

If ‘alg’ is chosen to be ‘blr’ or ‘hs’, the algorithm needs a vector with positive coefficients in the row space of A.
If no row of A contains only positive entries, one has to use the second version of toric_ideal which takes such a vector as its third argument.
For the mathematical background, see

Toric ideals and integer programming.

Example:

LIB "toric.lib";
ring r=0,(x,y,z),dp;
// call with two arguments
intmat A[2][3]=1,1,0,0,1,1;
A;
→ 1,1,0,
→ 0,1,1 
ideal I=toric_ideal(A,"du");
I;
→ I[1]=xz-y
I=toric_ideal(A,"blr");
→ ERROR: The chosen algorithm needs a positive vector in the row space of t\
   he matrix.
I;
→ I[1]=0
// call with three arguments
intvec prsv=1,2,1;
I=toric_ideal(A,"blr",prsv);
I;
→ I[1]=xz-y

See also: Toric ideals; intprog_lib; toric_lib; toric_std.