Composite values of polynomial power sums

Clemens Fuchs; Christina Karolus

doi:10.5802/ambp.380

Clemens Fuchs ¹ ; Christina Karolus ¹

¹ University of Salzburg Hellbrunnerstr. 34/I A-5020 Salzburg AUSTRIA

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

Zbl

DOI : 10.5802/ambp.380

Classification : 11B37, 12Y05, 11R58
Mots clés : Decomposable polynomials, linear recurrence sequences, Brownawell–Masser inequality

Affiliations des auteurs :

Clemens Fuchs ¹ ; Christina Karolus ¹

¹ University of Salzburg Hellbrunnerstr. 34/I A-5020 Salzburg AUSTRIA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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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/

Bibliographie
Cité par

[1] Roberto M. Avanzi; Umberto Zannier The equation $f (X) = f (Y)$ in rational functions $X = X (t)$ , $Y = Y (t)$ , Compos. Math., Volume 139 (2003) no. 3, pp. 263-295 | DOI | MR | Zbl

[2] Yuri Bilu; Clemens Fuchs; Florian Luca; Ákos Pintér Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations, Publ. Math., Volume 82 (2013) no. 1, pp. 219-254 | MR | Zbl

[3] Yuri Bilu; Robert Tichy The Diophantine equation $f (x) = g (y)$ , Acta Arith., Volume 95 (2000) no. 3, pp. 261-288 | DOI | Zbl

[4] Arnaud Bodin Decomposition of polynomials and approximate roots, Proc. Am. Math. Soc., Volume 138 (2010) no. 6, pp. 1989-1994 | DOI | MR | Zbl

[5] W. Dale Brownawell; David W. Masser Vanishing sums in function fields, Math. Proc. Camb. Philos. Soc., Volume 11 (1986) no. 3, pp. 427-434 | DOI | MR | Zbl

[6] Pietro Corvaja; Umberto Zannier Finiteness of integral values for the ratio of two linear recurrences, Invent. Math., Volume 149 (2002) no. 2, pp. 431-451 | DOI | MR | Zbl

[7] Martin Eichler Einführung in die Theorie der algebraischen Zahlen und Funktionen, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Birkhäuser, 1963 no. 27 | Zbl

[8] Clemens Fuchs On the Diophantine equation $G_{n} (x) = G_{m} (P (x))$ for third order linear recurring sequences, Port. Math. (N.S.), Volume 61 (2004) no. 1, pp. 1-24 | MR | Zbl

[9] Clemens Fuchs; Christina Karolus; Dijana Kreso Decomposable polynomials in second order linear recurrence sequences, Manuscr. Math., Volume 159 (2019) no. 3-4, pp. 321-346 | DOI | MR | Zbl

[10] Clemens Fuchs; Vincenzo Mantova; Umberto Zannier On fewnomials, integral points, and a toric version of Bertini’s theorem, J. Am. Math. Soc., Volume 31 (2018) no. 1, pp. 107-134 | DOI | MR | Zbl

[11] Clemens Fuchs; Attila Pethő Effective bounds for the zeros of linear recurrences in function fields, J. Théor. Nombres Bordeaux, Volume 17 (2005) no. 3, pp. 749-766 | DOI | Numdam | MR | Zbl

[12] Clemens Fuchs; Attila Pethő Composite rational functions having a bounded number of zeros and poles, Proc. Am. Math. Soc., Volume 139 (2011) no. 1, pp. 31-38 | DOI | MR | Zbl

[13] Clemens Fuchs; Attila Pethő; Robert Tichy On the Diophantine equation $G_{n} (x) = G_{m} (P (x))$ , Monatsh. Math., Volume 137 (2002) no. 3, pp. 173-196 | Zbl

[14] Clemens Fuchs; Attila Pethő; Robert Tichy On the Diophantine equation $G_{n} (x) = G_{m} (P (x))$ : higher-order recurrences, Trans. Am. Math. Soc., Volume 355 (2003) no. 11, pp. 4657-4681 | DOI | Zbl

[15] Clemens Fuchs; Attila Pethő; Robert Tichy On the Diophantine equation $G_{n} (x) = G_{m} (y)$ with $Q (x, y) = 0$ , Diophantine Approximation. Festschrift for Wolfgang Schmidt (Developments in Mathematics), Volume 16, Springer, 2008, pp. 199-209 | DOI | Zbl

[16] Clemens Fuchs; Umberto Zannier Composite rational functions expressible with few terms, J. Eur. Math. Soc., Volume 14 (2012) no. 1, pp. 175-208 | DOI | MR | Zbl

[17] James Rickards When is a polynomial a composition of other polynomials?, Am. Math. Mon., Volume 118 (2011) no. 4, pp. 358-363 | DOI | MR | Zbl

[18] Helmut Salzmann; Theo Grundhöfer; Hermann Hähl; Rainer Löwen The classical fields. Structural features of the real and rational numbers, Encyclopedia of Mathematics and Its Applications, 112, Cambridge University Press, 2007 | Zbl

[19] Andrzej Schinzel Polynomials with special regard to reducibility, Encyclopedia of Mathematics and Its Applications, 77, Cambridge University Press, 2000 (With an appendix by Umberto Zannier) | MR | Zbl

[20] Hans P. Schlickewei; Wolfgang M. Schmidt The intersection of recurrence sequences, Acta Arith., Volume 72 (1995) no. 1, pp. 1-44 | DOI | MR | Zbl

[21] Henning Stichtenoth Function Fields and Codes, Universitext, Springer, 1993 | Zbl

[22] Robert J. Walker Algebraic curves, Dover Publications, 1962 | MR | Zbl

[23] Umberto Zannier On the integer solutions of exponential equations in function fields, Ann. Inst. Fourier, Volume 54 (2004) no. 4, pp. 849-874 | DOI | Numdam | MR | Zbl

[24] Umberto Zannier On the number of terms of a composite polynomial, Acta Arith., Volume 127 (2007) no. 2, pp. 157-167 | DOI | MR | Zbl

[25] Umberto Zannier On composite lacunary polynomials and the proof of a conjecture of Schinzel, Invent. Math., Volume 174 (2008) no. 1, pp. 127-138 | DOI | MR | Zbl

[26] Umberto Zannier Addendum to the paper: “On the number of terms of a composite polynomial”, Acta Arith., Volume 140 (2009) no. 1, pp. 93-99 | DOI | Zbl

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