# 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, , with each column corresponding to a vector in $\mathbb Z^n$
• Outputs:

## 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 .