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C.6.1 Toric ideals

Let 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

         +    −
IA := < xu − xu |u ∈ ker(A)∩ ZZn > ⊂ K [x1,...,xn]
is called a toric 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

IA := I : (x1 ⋅...⋅xn)∞
where
I = < xv+i − xv−i |i = 1,...,r >

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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