Binary polynomial power sums vanishing at roots of unity

On the Skolem problem and some related questions for parametric families of linear recurrence sequences

“…Since a, b, f, g are multiplicatively independent, the finiteness of the set of such α P C follows directly from [10,30] (which in turn generalise [9,Theorem 2]). See also [8] for some effective results for rational functions over Q and G m " X m ´1. To obtain the divisibility (2.4) we only need to bound the multiplicity of such zeros α P C, and this follows directly from [31, Lemma 2.9].…”

“…Let pF n q 8 n"0 be an infinite sequence of rational functions F n P CpXq, n ě 0, and let K ě 1. Assume there exist infinitely many α P C such that each F n pαq is well defined and pF n pαqq 8 n"0 is a linear recurrence of order at most K. Then pF n q 8…”

“…Let pF n q 8 n"0 be an infinite sequence of polynomials F n P CrXs, n ě 0. Assume that pF n pαqq 8 n"0 is a linear recurrence sequence for uncountably many α P C. Then pF n q 8…”

“…Remark 2.17. It is natural to ask whether it is possible to merge Theorem 2.15 and Corollary 2.16 where simply the infinitude of α P C such that pF n pαqq 8 n"0 is a linear recurrence sequence (without any restriction on the order) implies that pF n q 8…”

“…Motivation and background. We recall that a linear recurrence sequence pu n q 8 n"1 of order k over a field K is a sequence which satisfies a relation of the form (1.1) u n`k " A k´1 u n`k´1 `. .…”