### 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.

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