affineToricRing -- computes the toric ring associated to a monomial map

Synopsis

Usage:

affineToricRing L

affineToricRing M
Inputs:
- L, a list, of lists; each list corresponds to a vector in $\mathbb Z^n$
- M, a matrix, with each column corresponding to a vector in $\mathbb Z^n$
Outputs:
- a ring,

Description

Given a list $\{v_1,...,v_d\}$ of vectors in $\mathbb Z^n$ this function computes the toric ring $R/I$ where $R$ is the polynomial ring $\mathbb{Q}[x_1,\dots,x_d]$ with $x_i$ having degree $v_i$ and $I$ is the associated toric ideal. In particular $I$ is the kernel of the map $R \to \mathbb{Q}[y_1,\dots,y_n]$ defined by $x_i \mapsto \mathbb y^{v_i}$.

i1 : L = {{2,0},{1,1},{0,2}};

i2 : X = affineToricRing L; -- The singular quadric in A^3

i3 : I = ideal X

            2
o3 = ideal(x  - x x )
            1    0 2

o3 : Ideal of QQ[x ..x ]
                  0   2

i4 : hilbertSeries I

                   2 2
              1 - T T
                   0 1
o4 = --------------------------
           2                 2
     (1 - T )(1 - T T )(1 - T )
           1       0 1       0

o4 : Expression of class Divide

Ways to use affineToricRing :

affineToricRing(List)
affineToricRing(Matrix)

For the programmer

The object affineToricRing is a method function.