Effective bounds for the zeros of linear recurrences in function fields

Clemens Fuchs; Attila Pethő

doi:10.5802/jtnb.518

Clemens Fuchs ¹ ; Attila Pethő ²

¹ Institut für Mathematik Technische Universität Graz Steyrergasse 30 8010 Graz, Austria Current Address: Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, Postbus 9512, 2300 RA Leiden, The Netherlands
² Institute of Informatics University of Debrecen, Debrecen Pf. 12 4010 Debrecen, Hungary

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 749-766.

Résumé
Abstract

Dans cet article, on utilise la généralisation de l’inégalité de Mason (due à Brownawell et Masser [8]) afin d’exhiber des bornes supérieures effectives pour les zéros d’une suite linéaire récurrente définie sur un corps de fonctions à une variable.

De plus, on étudie de problèmes similairs dans ce contexte, comme l’équation $G_{n} (x) = G_{m} (P (x)), (m, n) \in ℕ^{2}$ , où $(G_{n} (x))$ est une suite récurrente de polynômes et $P (x)$ un polynôme fixé. Ce problème a été étudié auparavant dans [14,15,16,17,32].

In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.

Moreover, we study similar problems in this context as the equation $G_{n} (x) = G_{m} (P (x)), (m, n) \in ℕ^{2}$ , where $(G_{n} (x))$ is a linear recurring sequence of polynomials and $P (x)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

MR Zbl

DOI : 10.5802/jtnb.518

Affiliations des auteurs :

Clemens Fuchs ¹ ; Attila Pethő ²

¹ Institut für Mathematik Technische Universität Graz Steyrergasse 30 8010 Graz, Austria Current Address: Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, Postbus 9512, 2300 RA Leiden, The Netherlands
² Institute of Informatics University of Debrecen, Debrecen Pf. 12 4010 Debrecen, Hungary

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Clemens Fuchs; Attila Pethő. Effective bounds for the zeros of linear recurrences in function fields. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 749-766. doi : 10.5802/jtnb.518. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.518/

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