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Fundam. Prikl. Mat., 2003, Volume 9, Issue 3, Pages 89–102 (Mi fpm746)  

This article is cited in 3 scientific papers (total in 3 papers)

On standard bases in rings of differential polynomials

A. I. Zobnin

M. V. Lomonosov Moscow State University

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 $\beta$-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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English version:
Journal of Mathematical Sciences (New York), 2006, 135:5, 3327–3335

Bibliographic databases:

UDC: 512.628.2+512.714+512.711

Citation: A. I. Zobnin, “On standard bases in rings of differential polynomials”, Fundam. Prikl. Mat., 9:3 (2003), 89–102; J. Math. Sci., 135:5 (2006), 3327–3335

Citation in format AMSBIB
\Bibitem{Zob03}
\by A.~I.~Zobnin
\paper On standard bases in rings of differential polynomials
\jour Fundam. Prikl. Mat.
\yr 2003
\vol 9
\issue 3
\pages 89--102
\mathnet{http://mi.mathnet.ru/fpm746}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2094331}
\zmath{https://zbmath.org/?q=an:1073.16022}
\transl
\jour J. Math. Sci.
\yr 2006
\vol 135
\issue 5
\pages 3327--3335
\crossref{https://doi.org/10.1007/s10958-006-0161-3}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33744773795}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kondratieva, M, “Membership problem for differential ideals generated by a composition of polynomials”, Programming and Computer Software, 32:3 (2006), 123  crossref  mathscinet  zmath  isi
    2. A. I. Zobnin, “Differential standard bases under composition”, J. Math. Sci., 152:4 (2008), 522–539  mathnet  crossref  mathscinet  zmath
    3. A. I. Zobnin, “One-element differential standard bases with respect to inverse lexicographical orderings”, J. Math. Sci., 163:5 (2009), 523–533  mathnet  crossref  mathscinet
  • Фундаментальная и прикладная математика
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