Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials. (March 2021) - Shkaravska, Olha; van Eekelen, Marko
Abstract: This article addresses the problem of computing an upper bound of the degree d of a polynomial solution P ( x ) of an algebraic difference equation of the form G ( x ) ( P ( x − τ 1 ), …, P ( x − τ s ) ) + G 0 ( x ) = 0 when such P ( x ) with the coefficients in a field K of characteristic zero exists and where G is a non-linear s -variable polynomial with coefficients in K [ x ] and G 0 is a polynomial with coefficients in K . It will be shown that if G is a quadratic polynomial with constant coefficients then one can construct a countable family of polynomials f l ( u 0 ) such that if there exists a (minimal) index l 0 with f l 0 ( u 0 ) being a non-zero polynomial, then the degree d is one of its roots or d ≤ l 0, or d < deg ⁡ ( G 0 ) . Moreover, the existence of such l 0 will be proven for K being the field of real numbers. These results are based on the properties of the modules generated by special families of homogeneous symmetric polynomials. A sufficient condition for the existence of a similar bound of the degree of a polynomial solution for an algebraic difference equation with G of arbitrary total degree and with variable coefficients will be proven as well.