C.6.1 Toric ideals
Let denote an matrix with integral coefficients. For , we define to be the uniquely determined vectors with nonnegative coefficients and disjoint support (i.e., or for each component ) such that . For component-wise, let denote the monomial .
is called a toric ideal.
The first problem in computing toric ideals is to find a finite
generating set: Let
be a lattice basis of
(i.e, a basis of the -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.