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D.4.24.1 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 : 1
==> multiplicity_denom : 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 : 1
==> multiplicity_denom : 6
==> hilbert_series_num : 1,-1,1
==> hilbert_series_denom : 1,6
==> class_group : 0
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See also:
ehrhartRing;
intclMonIdeal;
normalToricRing.
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