Singular

D.4.18.1 intclToricRing

Usage:

intclToricRing(ideal I);
intclToricRing(ideal I, intvec grading);

Return:

The toric ring S is the subalgebra of the basering generated by the leading monomials of the elements of I (considered as a list of polynomials). The function computes the integral
closure T of S in the basering and returns an ideal listing the algebra generators of T over the coefficient field.
The function returns the input ideal I if one of the options supp, triang, volume, or hseries has been activated.
However, in this case some numerical invariants are computed, and some other data may be contained in files that you can read into Singular (see showNuminvs, exportNuminvs).

Note:

A mathematical remark: the toric ring depends on the list of monomials given, and not only on the ideal they generate!

LIB "normaliz.lib";
ring R=37,(x,y,t),dp;
ideal I=x3,x2y,y3;
intclToricRing(I);
==> _[1]=x
==> _[2]=y
showNuminvs();
==> hilbert_basis_elements : 2
==> number_extreme_rays : 2
==> rank : 2
==> index : 3
==> number_support_hyperplanes : 2
==> homogeneous : 1
==> height_1_elements : 2
==> homogeneous_weights : 1,1,0
==> multiplicity : 1
//now the same example with another grading
intvec grading = 2,3,1;
intclToricRing(I,grading);
==> _[1]=x
==> _[2]=y
showNuminvs();
==> hilbert_basis_elements : 2
==> number_extreme_rays : 2
==> rank : 2
==> index : 3
==> number_support_hyperplanes : 2
==> homogeneous : 1
==> height_1_elements : 2
==> homogeneous_weights : 1,1,0
==> multiplicity : 1