normalToricRing(List,allComputations=>...) -- normalization of a toric ring

Synopsis

Usage:

normalToricRing L
Inputs:
- a list, the generators of the toric ring
Outputs:
- an instance of the type MonomialSubalgebra, the normalization of the toric ring

Description

The toric ring S is the monomial subalgebra of the basering generated by the monomials in the list L. The function computes the normalization T of S, which is the integral closure in its field of fractions. If the option allComputations is set to true, all data that has been computed by Normaliz is stored in a RationalCone in the CacheTable of the monomial subalgebra returned.

i1 : R=ZZ/37[x,y,t];

i2 : L={x^3, x^2*y, y^3, x*y^2};

i3 : T=normalToricRing(allComputations=> true, L)

     ZZ  3     2   2    3
o3 = --[y , x*y , x y, x ]
     37

o3 : monomial subalgebra of R

i4 : T.cache#"cone"

o4 = RationalCone{"cgr" => | 1 1 0 3 |                                    }
                  "equ" => | 0 0 1 |
                  "gen" => | 0 3 0 |
                           | 1 2 0 |
                           | 2 1 0 |
                           | 3 0 0 |
                  "inv" => HashTable{"" => (1, 1)                        }
                                     "class group" => (0, 3)
                                     "degree 1 elements" => 4
                                     "dim max subspace" => 0
                                     "embedding dim" => 3
                                     "external index" => 3
                                     "graded" => true
                                     "grading denom" => 3
                                     "grading" => (1, 1, 0)
                                     "hilbert basis elements" => 4
                                     "hilbert quasipolynomial denom" => 1
                                     "hilbert series denom" => (1, 1)
                                     "hilbert series num" => (1, 2)
                                     "inhomogeneous" => false
                                     "integrally closed" => true
                                     "internal index" => 1
                                     "multiplicity denom" => 1
                                     "multiplicity" => 3
                                     "number extreme rays" => 2
                                     "number support hyperplanes" => 2
                                     "rank" => 2
                                     "size triangulation" => 3
                                     "sum dets" => 3
                  "sup" => | 0 1 0 |
                           | 1 0 0 |

o4 : RationalCone

Further information

Default value: false
Method: normalToricRing(List) -- normalization of a toric ring
Option key: allComputations

Functions with optional argument named allComputations :

diagInvariants(Matrix,Matrix,Ring,allComputations=>...) -- see diagInvariants -- ring of invariants of a diagonalizable group action
ehrhartRing(List,allComputations=>...) -- Ehrhart ring
ehrhartRing(List,RingElement,allComputations=>...) -- Ehrhart ring
finiteDiagInvariants(Matrix,Ring,allComputations=>...) -- see finiteDiagInvariants -- ring of invariants of a finite group action
intclMonIdeal(Ideal,allComputations=>...) -- normalization of Rees algebra
intclMonIdeal(Ideal,RingElement,allComputations=>...) -- normalization of Rees algebra
intclToricRing(List,allComputations=>...) -- integral closure of a toric ring
intclToricRing(MonomialSubalgebra,allComputations=>...) -- integral closure of a toric ring
intersectionValRingIdeals(Matrix,Ring,allComputations=>...) -- see intersectionValRingIdeals -- intersection of valuation ideals
intersectionValRings(Matrix,Ring,allComputations=>...) -- see intersectionValRings -- intersection of ring of valuations
normaliz(List,allComputations=>...) -- see normaliz(List) -- calls Normaliz with several input matrices
normaliz(Matrix,String,allComputations=>...) -- see normaliz(Matrix,String) -- calls Normaliz
normaliz(Matrix,ZZ,allComputations=>...) (missing documentation)
normalToricRing(Ideal,Thing,allComputations=>...) -- see normalToricRing(Ideal,Thing) -- normalization of a toric ring given by a binomial ideal
normalToricRing(List,allComputations=>...) -- normalization of a toric ring
normalToricRing(MonomialSubalgebra,allComputations=>...) -- normalization of a toric ring
torusInvariants(Matrix,Ring,allComputations=>...) -- see torusInvariants -- ring of invariants of torus action