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C.6.1 Toric idealsLet A denote an m × n matrix with integral coefficients. For u ∈ ZZn, we define u+,u− to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., ui+ = 0 or ui− = 0 for each component i) such that u = u+ − u−. For u ≥ 0 component-wise, let xu denote the monomial x1u1 ⋅… ⋅ xnun ∈ K[x1,…,xn].The ideal ![]() The first problem in computing toric ideals is to find a finite generating set: Let v1,…,vr be a lattice basis of ker(A) ∩ ZZn (i.e, a basis of the ZZ-module). Then ![]() ![]() The required lattice basis can be computed using the LLL-algorithm (see [Coh93]). For the computation of the saturation, there are various possibilities described in the section Algorithms.
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