Approximation of values of algebraic elements over the ring of power sums

Clemens Fuchs; Sebastian Heintze

doi:10.5802/jtnb.1247

Clemens Fuchs ¹; Sebastian Heintze ²

¹ University of Salzburg, Department of Mathematics, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
² Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II, A-8010 Graz, Austria

Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 63-86.

Abstract
Résumé

Let $ℚ ℰ_{ℤ}$ be the set of power sums whose characteristic roots belong to $ℤ$ and whose coefficients belong to $ℚ$ , i.e. $G : ℕ \to ℚ$ satisfies

G (n) = G_{n} = b_{1} c_{1}^{n} + \dots + b_{h} c_{h}^{n}

with $c_{1}, \dots, c_{h} \in ℤ$ and $b_{1}, \dots, b_{h} \in ℚ$ . Furthermore, let $f \in ℚ [x, y]$ be absolutely irreducible and $α : ℕ \to \bar{ℚ}$ be a solution $y$ of $f (G_{n}, y) = 0$ , i.e. $f (G_{n}, α (n)) = 0$ identically in $n$ . Then we will prove under suitable assumptions a lower bound, valid for all but finitely many positive integers $n$ , for the approximation error if $α (n)$ is approximated by rational numbers with bounded denominator. After that we will also consider the case that $α$ is a solution of

f (G_{n}^{(0)}, \dots, G_{n}^{(d)}, y) = 0,

i.e. defined by using more than one power sum and a polynomial $f$ satisfying some suitable conditions. This extends results of Bugeaud, Corvaja, Luca, Scremin and Zannier.

Soit $ℚ ℰ_{ℤ}$ l’ensemble des sommes de puissances dont les racines caractéristiques sont dans $ℤ$ et dont les coefficients sont dans $ℚ$ , i.e. les éléments $G : ℕ \to ℚ$ de $ℚ ℰ_{ℤ}$ sont de la forme

G (n) = G_{n} = b_{1} c_{1}^{n} + \dots + b_{h} c_{h}^{n}

avec $c_{1}, \dots, c_{h} \in ℤ$ et $b_{1}, \dots, b_{h} \in ℚ$ . De plus, soit $f \in ℚ [x, y]$ un polynôme absolument irréductible et soit $α : ℕ \to \bar{ℚ}$ une solution $y$ de $f (G_{n}, y) = 0$ , i.e. la fonction $f (G_{n}, α (n))$ est identiquement nulle en $n$ . Si $α (n)$ est approximé par des nombres rationnels à dénominateur borné, nous établissons, sous conditions appropriées, une borne inférieure pour l’erreur d’approximation qui est valable pour tous les $n$ sauf un nombre fini. Nous considérons ensuite le cas où $α$ est une solution de l’équation

f (G_{n}^{(0)}, \dots, G_{n}^{(d)}, y) = 0,

i.e. $α$ est défini à l’aide de plus d’une somme de puissances et d’un polynôme $f$ satisfaisant à des conditions appropriées. Ce résultat est une extension des résultats de Bugeaud, Corvaja, Luca, Scremin et Zannier.

Received: 2021-10-19
Revised: 2022-06-24
Accepted: 2022-07-02
Published online: 2023-05-04

DOI: 10.5802/jtnb.1247

Classification: 11B37, 11J68, 11J87
Keywords: Power sum, Diophantine approximation, Subspace theorem

Author's affiliations:

Clemens Fuchs ¹; Sebastian Heintze ²

¹ University of Salzburg, Department of Mathematics, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
² Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II, A-8010 Graz, Austria

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Clemens Fuchs; Sebastian Heintze. Approximation of values of algebraic elements over the ring of power sums. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 63-86. doi : 10.5802/jtnb.1247. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1247/

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