Valuations of rational solutions of linear difference equations at irreducible polynomials

Abstract

We discuss two algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, construct a finite set M of polynomials, irreducible in k [ x ] , such that if the given equation has a solution F ( x ) ∈ k ( x ) and val p ( x ) F ( x ) < 0 for an irreducible p ( x ) , then p ( x ) ∈ M . After this for each p ( x ) ∈ M the algorithms compute a lower bound for val p ( x ) F ( x ) , which is valid for any rational function solution F ( x ) of the initial equation. The algorithms are applicable to scalar linear equations of arbitrary orders as well as to linear systems of first-order equations. The algorithms are based on a combination of renewed approaches used in earlier algorithms for finding a universal denominator (Abramov and Barkatou (1998) [6]), and on a denominator bound (van Hoeij (1998) [12]). A complexity analysis of the two proposed algorithms is presented.