Composite values of polynomial power sums
Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 1-24.

Let ${\left({G}_{n}\left(x\right)\right)}_{n=0}^{\infty }$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\ge 2$ be a given integer. We ask for $n\in ℕ$ such that the equation ${G}_{n}\left(x\right)=g\circ h$ is satisfied for a polynomial $g\in ℂ\left[x\right]$ with $degg=m$ and some polynomial $h\in ℂ\left[x\right]$ with $degh>1$. We prove that for all but finitely many $n$ these decompositions can be described in “finite terms” coming from a generic decomposition parameterized by an algebraic variety. All data in this description will be shown to be effectively computable.

Published online:
DOI: 10.5802/ambp.380
Classification: 11B37,  12Y05,  11R58
Keywords: Decomposable polynomials, linear recurrence sequences, Brownawell–Masser inequality
Clemens Fuchs 1; Christina Karolus 1

1 University of Salzburg Hellbrunnerstr. 34/I A-5020 Salzburg AUSTRIA
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Clemens Fuchs; Christina Karolus. Composite values of polynomial power sums. Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 1-24. doi : 10.5802/ambp.380. https://ambp.centre-mersenne.org/articles/10.5802/ambp.380/

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