Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials

This article addresses the problem of computing an upper bound of the degree d of a polynomial solution P(x) of an algebraic difference equation of the form G(x)(P(x−τ1),…,P(x−τs))+G0(x)=0 when such P(x) with the coefficients in a field K of characteristic zero exists and where G is a non-linear s-v...

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Bibliographic Details
Published in:Journal of symbolic computation 2021-03, Vol.103, p.22-45
Main Authors: Shkaravska, Olha, van Eekelen, Marko
Format: Article
Language:English
Subjects:
Algebraic difference equation
Homogeneous symmetric polynomial
Partition
Power-sum symmetric polynomial
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Summary:This article addresses the problem of computing an upper bound of the degree d of a polynomial solution P(x) of an algebraic difference equation of the form G(x)(P(x−τ1),…,P(x−τs))+G0(x)=0 when such P(x) with the coefficients in a field K of characteristic zero exists and where G is a non-linear s-variable polynomial with coefficients in K[x] and G0 is a polynomial with coefficients in K. It will be shown that if G is a quadratic polynomial with constant coefficients then one can construct a countable family of polynomials fl(u0) such that if there exists a (minimal) index l0 with fl0(u0) being a non-zero polynomial, then the degree d is one of its roots or d≤l0, or d
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2019.10.022