Computing Gröbner bases of pure binomial ideals via submodules of Zn

A binomial ideal is an ideal of the polynomial ring which is generated by binomials. In a previous paper, we gave a correspondence between pure saturated binomial ideals of K[x1,…,xn] and submodules of Zn and we showed that it is possible to construct a theory of Gröbner bases for submodules of Zn. As a consequence, it is possible to follow alternative strategies for the computation of Gröbner bases of submodules of Zn (and hence of binomial ideals) which avoid the use of Buchberger algorithm. In the present paper, we show that a Gröbner basis of a Z-module M⊆Zn of rank m lies into a finite set of cones of Zm which cover a half-space of Zm. More precisely, in each of these cones C, we can find a suitable subset Y(C) which has the structure of a finite abelian group and such that a Gröbner basis of the module M (and hence of the pure saturated binomial ideal represented by M) is described using the elements of the groups Y(C) together with the generators of the cones.