A. Gheffar & S. Abramov

Valuations of rational solutions of linear difference equations at irreducible polynomials.

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 irreducible in $k[x]$ polynomials such that
if the given equation has a solution $F(x)\in k(x)$ and $\val _{p(x)}F(x)<0$ for an irreducible $p(x)$
then $p(x)\in M$. After this for each $p(x)\in M$ 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, Barkatou), and on a bound for the denominator (van Hoeij). A complexity comparison of two proposed algorithms is presented.