Composite values of polynomial power sums

Fuchs, Clemens; Karolus, Christina

Annales mathématiques Blaise Pascal, Tome 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 $n\in \mathbb{N}$ 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.

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

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

[2] Bilu, Yuri; Fuchs, Clemens; Luca, Florian; Pintér, Ákos 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] Bilu, Yuri; Tichy, Robert The Diophantine equation $f\left(x\right)=g\left(y\right)$, Acta Arith., Volume 95 (2000) no. 3, pp. 261-288 | DOI | Zbl

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

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

[6] Corvaja, Pietro; Zannier, Umberto 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] Eichler, Martin 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] Fuchs, Clemens On the Diophantine equation $G_n\left(x\right)=G_m\left(P\left(x\right)\right)$ for third order linear recurring sequences, Port. Math. (N.S.), Volume 61 (2004) no. 1, pp. 1-24 | MR | Zbl

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

[10] Fuchs, Clemens; Mantova, Vincenzo; Zannier, Umberto 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] Fuchs, Clemens; Pethő, Attila 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] Fuchs, Clemens; Pethő, Attila 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] Fuchs, Clemens; Pethő, Attila; Tichy, Robert On the Diophantine equation $G_n\left(x\right)=G_m\left(P\left(x\right)\right)$, Monatsh. Math., Volume 137 (2002) no. 3, pp. 173-196 | Zbl

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

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

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

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

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

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

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

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

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

[25] Zannier, Umberto 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] Zannier, Umberto 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