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C.6.1 Toric idealsLet 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.
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