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Univariate polynomial solutions of algebraic difference equations

Shkaravska, O., & Van Eekelen, M. (2014). Univariate polynomial solutions of algebraic difference equations. Journal of Symbolic Computation, 60, 15-28. doi:10.1016/j.jsc.2013.10.010.
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.
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