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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.
