Univariate polynomial solutions of algebraic difference equations

Contrary to linear difference equations, there is no general theory of difference equations of the form G(P (x − τ1), . . . , P (x − τs)) + G0(x)=0, with τi ∈ K, G(x1, . . . , xs) ∈ K[x1, . . . , xs] of total degree D ≥ 2 and G0(x) ∈ K[x], where K is a field of characteristic zero. This article is concerned with the following problem: given τi, G and G0, find an upper bound on the degree d of a polynomial solution P (x), if it exists. In the presented approach the problem is reduced to constructing a univariate polynomial for which d is a root. The authors formulate a sufficient condition under which such a polynomial exists. Using this condition, they give an effective bound on d, for instance, for all difference equations of the form G ( P (x − a), P (x − a − 1), P (x − a − 2) ) + G0(x) = 0 with quadratic G, and all difference equations of the form G ( P (x), P (x− τ) ) +G0(x) = 0 with G having an arbitrary degree.