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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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