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

Let (G n (x)) n=0 be a d-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let m2 be a given integer. We ask for n such that the equation G n (x)=gh is satisfied for a polynomial g[x] with degg=m and some polynomial h[x] 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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