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ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

Part of: Algebraic number theory: global fields Sequences and sets Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 November 2020

CLEMENS FUCHS Open the ORCID record for CLEMENS FUCHS [Opens in a new window]  and
SEBASTIAN HEINTZE Open the ORCID record for SEBASTIAN HEINTZE [Opens in a new window]
CLEMENS FUCHS*
Affiliation:
Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020Salzburg, Austria
SEBASTIAN HEINTZE
Affiliation:
Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020Salzburg, Austria e-mail: sebastian.heintze@sbg.ac.at
*
e-mail: clemens.fuchs@sbg.ac.at
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Abstract

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Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

Keywords

function fieldslinear recurrencesS-units

MSC classification

Primary: 11B37: Recurrences
Secondary: 11J87: Schmidt Subspace Theorem and applications 11R58: Arithmetic theory of algebraic function fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 11 - 20
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Footnotes

Supported by Austrian Science Fund (FWF): I4406.

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