> | ### C.6.1 Toric ideals Let 190#190 denote an 68#68 matrix with integral coefficients. For 737#737, we define 738#738 to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., 739#739 or 740#740 for each component 57#57) such that 741#741. For 742#742 component-wise, let 743#743 denote the monomial 744#744. The ideal 745#745 is called a toric ideal. The first problem in computing toric ideals is to find a finite generating set: Let 585#585 be a lattice basis of 746#746 (i.e, a basis of the 747#747-module). Then 748#748 where 749#749 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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