Univariate Polynomial Solutions of Nonlinear Polynomial Difference Equations

We study real-polynomial solutions P (x) of difference equations of the formG(P (x−τ1), . . . , P (x− τs)) +G0(x)=0, where τi are real numbers, G(x1, . . . , xs) is a real polynomial of a total degree D ≥ 2, and G0(x) is a polynomial in x. We consider the following problem: given τi, G and G0, find an upper bound on the degree d of a real-polynomial solution P (x), if exists. We reduce this problem to finding a univariate polynomial for which d is a root. We formulate a sufficient condition under which such polynomial exists. Using this condition, we can give an effective bound on d, for instance, for all difference equations G ( P (x− 1), P (x− 2), P (x− 3) ) + G0(x) = 0 with quadratic G, and all difference equations G ( P (x), P (x− τ) ) +G0(x) = 0 with G of an arbitrary degree. In the constructions we use Newton-Girard identities between elementary and power-sum symmetric polynomials.