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On standard bases in rings of differential polynomials

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Abstract

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 [x p] has a finite standard basis (w.r.t. the so-called β-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the question of 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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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Russia

    A. I. Zobnin

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  1. A. I. Zobnin
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 89–102, 2003.

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Zobnin, A.I. On standard bases in rings of differential polynomials. J Math Sci 135, 3327–3335 (2006). https://doi.org/10.1007/s10958-006-0161-3

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  • DOI: https://doi.org/10.1007/s10958-006-0161-3

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