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−τ_s))+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 f_l(u₀) such that if there exists a (minimal) index l₀ with f_l0(u₀) being a non-zero polynomial, then the degree d is one of its roots or d≤l₀, or d<deg⁡(G₀). Moreover, the existence of such l₀ 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.