Details:
Title  Univariate polynomial solutions of algebraic difference equations  Author(s)  M. van Eekelen, O. Shkaravska  Type  Article in Journal  Abstract  Abstract Contrary to linear difference equations, there is no general theory of difference equations of the form G ( P ( x − τ_1 ) , … , P ( x − τ_s ) ) + G_0 ( x ) = 0 , with τ_i ∈ K , G_( x 1 , … , x s ) ∈ K [ x 1 , … , x s ] of total degree D ⩾ 2 and G 0 ( x ) ∈ K [ x ] , where K is a field of characteristic zero. This article concerns the following problem: given τ_i , G and G_0 , 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 ) ) + G_0 ( x ) = 0 with quadratic G, and all difference equations of the form G ( P ( x ) , P ( x − τ ) ) + G_0 ( x ) = 0 with G having an arbitrary degree.  Keywords  Difference equation, Elementary symmetric polynomials, Powersum symmetric polynomials, Newton–Girard formulæ, System of linear equations  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717113001296 
Language  English  Journal  Journal of Symbolic Computation  Volume  60  Number  0  Pages  15  28  Year  2014  Edition  0  Translation 
No  Refereed 
No 
