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)=0G(P(x−τ1),…,P(x−τs))+G0(x)=0, with τi∈Kτi∈K, G(x1,…,xs)∈K[x1,…,xs]G(x1,…,xs)∈K[x1,…,xs] of total degree D⩾2D⩾2 and G0(x)∈K[x]G0(x)∈K[x], where KK is a field of characteristic zero. This article concerns the following problem: given τiτi, G   and G0G0, find an upper bound on the degree d   of a polynomial solution P(x)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)=0G(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)=0G(P(x),P(x−τ))+G0(x)=0 with G having an arbitrary degree.