# toricIdeal -- returns a toric ideal

## Synopsis

• Usage:
toricIdeal(A,R)
• Inputs:
• A, , a d x n integer matrix
• R, a ring, a polynomial ring in n variables
• Outputs:
• an ideal, the toric ideal associated to A in R

## Description

Given a $d\times n$ Matrix A and a polynomial ring in $n$ variables $R$, this method returns the toric ideal associated to $A$ in $R$. To do this, toricIdeal saturates the lattice basis ideal of the kernel of $A$ with respect to the product of the variables of $R$.

 i1 : A=matrix{{1,1,1,1,1,1},{1,2,1,2,3,0},{0,2,2,0,1,1}} o1 = | 1 1 1 1 1 1 | | 1 2 1 2 3 0 | | 0 2 2 0 1 1 | 3 6 o1 : Matrix ZZ <--- ZZ i2 : R=QQ[a..f] o2 = R o2 : PolynomialRing i3 : toricIdeal(A,R) 3 3 2 2 2 2 o3 = ideal (c*d - e*f, a*b - e*f, c e - b f, a*c e - b d*f, a c*e - b*d f, ------------------------------------------------------------------------ 3 3 3 2 2 2 2 2 3 2 2 3 2 2 2 2 a e - d f, b*d - a e , b d - a*c*e , b d - c e , a*c - b f , a c - ------------------------------------------------------------------------ 2 3 2 2 b*d*f , a c - d f ) o3 : Ideal of R

 i4 : A=matrix{{1,1,1,1,1},{0,0,1,1,0},{0,1,1,0,-2}} o4 = | 1 1 1 1 1 | | 0 0 1 1 0 | | 0 1 1 0 -2 | 3 5 o4 : Matrix ZZ <--- ZZ i5 : R=toGradedRing(A,QQ[a..e]) o5 = R o5 : PolynomialRing i6 : toricIdeal(A,R) 2 2 2 3 2 o6 = ideal (a*c - b*d, a*d - c e, a d - b*c*e, a - b e) o6 : Ideal of R

## Ways to use toricIdeal :

• "toricIdeal(Matrix,Ring)"

## For the programmer

The object toricIdeal is .