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#### Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials

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##### Citation

Shkaravska, O., & Van Eekelen, M. (2021). Polynomial solutions of algebraic difference
equations and homogeneous symmetric polynomials.* Journal of Symbolic Computation,* *103*, 22-45. doi:10.1016/j.jsc.2019.10.022.

Cite as: http://hdl.handle.net/21.11116/0000-0007-A4A6-4

##### Abstract

This article addresses the problem of computing an upper bound of
the degree d of a polynomial solution P(x) of an algebraic differ-
ence equation of the form Gx)(P(x −τ1), . . . , P(x −τs) + G0(x) =
0 when such P(x) with the coeﬃcients in a ﬁeld K of character-
istic zero exists and where G is a non-linear s-variable polynomial
with coeﬃcients in K[x] and G0 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 polynomi-
als fl(u0) such that if there exists a (minimal) index l0 with fl0(u0)
being a non-zero polynomial, then the degree d is one of its roots
or d ≤ l0, or d < deg(G0). Moreover, the existence of such l0 will
be proven for K being the ﬁeld of real numbers. These results are
based on the properties of the modules generated by special fami-
lies 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.