
C.6.1 Toric idealsLet 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 componentwise, let denote the monomial .
The ideal
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
where The required lattice basis can be computed using the LLLalgorithm ( system, see see [Coh93]). For the computation of the saturation, there are various possibilities described in the section Algorithms.
