normalToricRing(Ideal,Thing) -- normalization of a toric ring given by a binomial ideal

Synopsis

Function: normalToricRing
Usage:

normalToricRing(I,t)
Inputs:
- an ideal, a binomial ideal defining the toric ring
- a thing, letter naming variables in new polynomial ring
Optional inputs:
- allComputations (missing documentation) => ..., default value false,
- grading (missing documentation) => ..., default value {},
Outputs:
- a monomial subalgebra, the normalization of the toric ring

Description

The ideal I is generated by binomials of type X^a-X^b (multiindex notation) in the surrounding polynomial ring K[X]=K[X_1,...,X_n]. The binomials represent a congruence on the monoid \ZZ^n with residue monoid M. Let N be the image of M in gp(M)/torsion. Then N is universal in the sense that every homomorphism from M to an affine monoid factors through N. If I is a prime ideal, then K[N] \cong K[X]/I. In general, K[N]\cong K[X]/P where P is the unique minimal prime ideal of I generated by binomials of type X^a-X^b.

The function computes the normalization of K[N] and returns it as a monomial subalgebra in a newly created polynomial ring of the same Krull dimension, whose variables are t_1,\ldots,t_{n-r}, where r is the rank of the matrix with rows a-b. (In general there is no canonical choice for such an embedding.)

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

i2 : I=ideal(x*z-y^2, x*w-y*z);

o2 : Ideal of R

i3 : normalToricRing(I,t)

o3 = MonomialSubalgebra{cache => CacheTable{...1...}      }
                                        3     2   2     3
                        generators => {t , t t , t t , t }
                                        2   1 2   1 2   1
                                ZZ
                        ring => --[t ..t ]
                                37  1   2

                           ZZ
o3 : MonomialSubalgebra of --[t ..t ]
                           37  1   2

Ways to use this method: