Computing Grobner bases of pure binomial ideals via submodules of Z(n)
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[x(1,) ... ,x(n)] and sub-modules of Z(n) and we showed that it is possible to construct a theory of Grobner bases for submodules of Z(n). As a consequence, it is possible to follow alternative strategies for the computation of Grobner bases of submodules of Z(n) (and hence of binomial ideals) which avoid the use of Buchberger algorithm. In the present paper, we show that a Grobner basis of a Z-module M subset of Z(n) of rank m lies into a finite set of cones of Z(m) which cover a half-space of Z(m). 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 Grobner 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. (C) 2011 Elsevier Ltd. All rights reserved.
binomial ideal buchberger algorithm grobner basis polyhedral cone smith normal form hilbert basis