FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2003, VOLUME 9, NUMBER 3, PAGES 89-102

On standard bases in rings of differential polynomials

A. I. Zobnin

Abstract

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We consider Ollivier's standard bases (also known as differential Gröbner bases) in an ordinary ring of differential polynomials in one indeterminate. We establish a link between these bases and Levi's reduction process. We prove that the ideal $[xp]$ has a finite standard basis (w.r.t. the so-called $$b-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the following problem: whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering.

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Last modified: January 24, 2005.