Composite values of polynomial power sums

Clemens Fuchs; Christina Karolus

doi:10.5802/ambp.380

Clemens Fuchs ; Christina Karolus

Annales Mathématiques Blaise Pascal, Tome 26 (2019) no. 1, pp. 1-24.

Résumé

Let ${(G_{n} (x))}_{n = 0}^{\infty}$ be a $d$ -th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m \geq 2$ be a given integer. We ask for $n \in ℕ$ such that the equation $G_{n} (x) = g \circ h$ is satisfied for a polynomial $g \in ℂ [x]$ with $deg g = m$ and some polynomial $h \in ℂ [x]$ with $deg h > 1$ . 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.

Publié le : 2020-01-21

DOI : https://doi.org/10.5802/ambp.380
Classification : 11B37, 12Y05, 11R58
Mots clés : Decomposable polynomials, linear recurrence sequences, Brownawell–Masser inequality

@article{AMBP_2019__26_1_1_0,
     author = {Clemens Fuchs and Christina Karolus},
     title = {Composite values of polynomial power sums},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {1--24},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {26},
     number = {1},
     year = {2019},
     doi = {10.5802/ambp.380},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.380/}
}

Clemens Fuchs; Christina Karolus. Composite values of polynomial power sums. Annales Mathématiques Blaise Pascal, Tome 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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