1 research outputs found
Computing Gröbner bases of pure binomial ideals via submodules of Zn
2A binomial ideal is an ideal of the polynomial ring which is gener-
ated by binomials. In a previous paper, we gave a correspondence
between pure saturated binomial ideals of K [x1 , . . . , xn ] and sub-
modules of Zn and we showed that it is possible to construct a the-
ory 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 pa-
per, 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.nonemixedBoffi G.; Logar A.Boffi, G.; Logar, Alessandr