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Summary: Valuations of rational solutions of linear difference
equations at irreducible polynomials
A. Gheffara
, S. Abramovb
a
XLIM, Universit´e de Limoges, CNRS, 123, Av. A. Thomas, 87060, Limoges Cedex,
France
b
Computing Centre of the Russian Academy of Sciences, ul. Vavilova, 40, Moscow
119991, GSP-1, Russia
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 valp(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 valp(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
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