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C.6.1 Toric ideals

Let $A$ denote an $ m \times n $ matrix with integral coefficients. For $u
\in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., $u_i^+=0$ or $u_i^-=0$ for each component $i$) such that $u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the monomial $x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.

The ideal

\begin{displaymath}I_A:=<x^{u^+}-x^{u^-} \vert u\in\ker(A)\cap Z\!\!\! Z^n>\ \subset
K[x_1,\ldots,x_n] \end{displaymath}

is called a toric ideal.

The first problem in computing toric ideals is to find a finite generating set: Let $v_1,
\ldots, v_r$ be a lattice basis of $\ker(A)\cap
Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then

\begin{displaymath}I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty \end{displaymath}

where

\begin{displaymath}I=<x^{v_i^+}-x^{v_i^-}\vert i=1,\ldots,r> \end{displaymath}

The required lattice basis can be computed using the LLL-algorithm (see [Coh93]). For the computation of the saturation, there are various possibilities described in the section Algorithms.


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