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Olha Shkaravska, and Marko van Eekelen. Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials. Technical report: ICIS-R18001, December, Radboud University Nijmegen, 2018.

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-tau_1),..., P(x-tau_s))+G_0(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 G_0 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 f_l(u_0) such that if there exists a (minimal) index l_0 with f_{l_0}(u_0) being a non-zero polynomial, then the degree d is one of its roots or d <= l_0, or d < deg(G_{0}). Moreover, the existence of such l_0 will be proven for K being the field of real numbers. These results are based on the properties of the modules generated by special families of homogeneous symmetric polynomials.
A sufficient condition for the existence of a similar bound of the degree of a polynomial solution for an algebraic difference equation with G of arbitrary total degree and with variable coefficients will be proven as well.