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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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