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
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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Journal of Mathematical Sciences (New York), 2006, 135:5, 3327–3335
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
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\paper On standard bases in rings of differential polynomials
\jour Fundam. Prikl. Mat.
\jour J. Math. Sci.
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This publication is cited in the following articles:
Kondratieva, M, “Membership problem for differential ideals generated by a composition of polynomials”, Programming and Computer Software, 32:3 (2006), 123
A. I. Zobnin, “Differential standard bases under composition”, J. Math. Sci., 152:4 (2008), 522–539
A. I. Zobnin, “One-element differential standard bases with respect to inverse lexicographical orderings”, J. Math. Sci., 163:5 (2009), 523–533
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