Singular Manual: intclToricRing

| #### D.4.25.1 intclToricRing Procedure from library `normaliz.lib` (see normaliz_lib). 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 an option blocking the computation of Hilbert bases 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! Example:
    LIB "normaliz.lib";
    ring R=37,(x,y,t),dp;
    ideal I=x3,x2y,y3;
    intclToricRing(I);
    ==> _[1]=y
    ==> _[2]=x
    showNuminvs();
    ==> hilbert_basis_elements : 2
    ==> number_extreme_rays : 2
    ==> dim_max_subspace : 0
    ==> embedding_dim : 3
    ==> rank : 2
    ==> external_index : 1
    ==> internal_index : 3
    ==> number_support_hyperplanes : 2
    ==> size_triangulation : 1
    ==> sum_dets : 1
    ==> integrally_closed : 0
    ==> inhomogeneous : 0
    ==> graded : 1
    ==> degree_1_elements : 2
    ==> grading : 1,1,0
    ==> grading_denom : 1
    ==> multiplicity_num : 1
    ==> multiplicity_denom : 1
    ==> hilbert_series_cyclo_num : 1
    ==> hilbert_series_cyclo_denom : 1,1
    ==> hilbert_series_num : 1
    ==> hilbert_series_denom : 1,1
    ==> class_group : 0
    //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
    ==> dim_max_subspace : 0
    ==> embedding_dim : 3
    ==> rank : 2
    ==> external_index : 1
    ==> internal_index : 3
    ==> number_support_hyperplanes : 2
    ==> size_triangulation : 1
    ==> sum_dets : 1
    ==> integrally_closed : 0
    ==> inhomogeneous : 0
    ==> graded : 1
    ==> degree_1_elements : 0
    ==> grading : 2,3,1
    ==> grading_denom : 1
    ==> multiplicity_num : 1
    ==> multiplicity_denom : 6
    ==> hilbert_series_cyclo_num : 1
    ==> hilbert_series_cyclo_denom : 1,1,2,3
    ==> hilbert_series_num : 1,-1,1
    ==> hilbert_series_denom : 1,6
    ==> class_group : 0
See also: ehrhartRing; intclMonIdeal; normalToricRing.

User manual for Singular version 4.4.1, 2025, generated by texi2html.