Universal Groebner Bases in Weyl Algebras

TL;DR

The paper introduces a topological framework for orderings in Weyl algebras, proving the existence of universal Groebner bases for all left ideals using compactness and division theorems.

Contribution

It establishes the compactness of the space of normal orderings in Weyl algebras and proves that every left ideal has a universal Groebner basis.

Findings

The space of normal orderings is compact.

Every left ideal in Weyl algebra admits a universal Groebner basis.

The proof relies on topological and division properties of Weyl algebras.

Abstract

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.