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C.6.1 Toric idealsLet 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.
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