arXiv subjects

Roberto Boldini

Publications and source records attributed to Roberto Boldini.

3 recordsLinked to original sources

arXiv2011

A topological approach to leading monomial ideals

We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to deduce some finiteness results about leading monomial ideals of any fixed ideal, namely: (1) the number of minimal leading monomial ideals with respect to total orderings is finite; (2) the number of leading monomial ideals with respect to degree orderings is finite; (3) the number of leading monomial ideals with respect to admissible orderings is finite under some multiplicativity assumptions on the considered algebra. Finally we are able to infer the existence of universal Groebner bases from the topological properties of degree and admissible orderings in a class of algebras that includes at least the algebras of solvable type. These existence results turn out to be independent from the finiteness results mentioned above, in contrast to the typical situation that occurs with "classical" more combinatorial proofs.

math.RA↗

arXiv2011

Critical cones of characteristic varieties

We show that certain characteristic varieties of a finitely generated module over a given Weyl algebra arising from weighted degree filtrations are equal to the critical cone of some other characteristic varieties. This behaviour of the characteristic varieties permits us to introduce a new invariant of the module. As a second consequence we are able to provide an easy and non-homological proof that the characteristic varieties of a module arising from weights in the natural polynomial region of the Weyl algebra all have the same Krull and Gelfand-Kirillov dimension, equal to the Gelfand-Kirillov dimension of the module. Third we give an upper bound for the number of distinct characteristic varieties of a cyclic module in terms of degrees of elements in universal Groebner bases and the above results allow us to conjecture a further upper bound.

math.RA↗

arXiv2011

Universal Groebner Bases in Weyl Algebras

A topological space TO(S) of total orderings on any given set S is introduced and it is shown that TO(S) is compact if S is countable. The set NO(N) of all normal orderings of the nth Weyl algebra W is a closed subspace of TO(N), where N is the set of all normal monomials of W. Hence NO(N) is compact and, as a consequence of this fact and by a division theorem valid in W, we give a proof that each left ideal of W admits a universal Groebner basis.

math.RA↗