> | ### C.6.1 Toric ideals Let 191#191 denote an 68#68 matrix with integral coefficients. For 739#739, we define 740#740 to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., 741#741 or 742#742 for each component 57#57) such that 743#743. For 744#744 component-wise, let 745#745 denote the monomial 746#746. The ideal 747#747 is called a toric ideal. The first problem in computing toric ideals is to find a finite generating set: Let 587#587 be a lattice basis of 748#748 (i.e, a basis of the 749#749-module). Then 750#750 where 751#751 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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