# Singular

### C.6.1 Toric ideals

Let denote an matrix with integral coefficients. For , we define to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., or for each component ) such that . For component-wise, let denote the monomial .

The ideal

is called a toric ideal.

The first problem in computing toric ideals is to find a finite generating set: Let be a lattice basis of (i.e, a basis of the -module). Then

where

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

 C.6.2 Algorithms Various algorithms for computing toric ideals. C.6.3 The Buchberger algorithm for toric ideals Specializing it for toric ideals.