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Diophantine equations with linear recurrences An overview of some recent progress

Umberto Zannier — 2005

Journal de Théorie des Nombres de Bordeaux

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. " d -th root problem"), which in short asks whether the integrality of the values of the quotient (resp. d -th root) of two (resp. one) linear recurrences implies that this quotient (resp. d -th root) is itself a recurrence. We shall also relate such...

On the integer solutions of exponential equations in function fields

Umberto Zannier — 2004

Annales de l’institut Fourier

This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal...

An a b c d theorem over function fields and applications

Pietro CorvajaUmberto Zannier — 2011

Bulletin de la Société Mathématique de France

We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u , v , in term of the degree of u , v , which is sharp whenever u , v have few distinct zeros and poles compared to their degree. This sharpens the “ a b c d -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves,...

Local-global divisibility of rational points in some commutative algebraic groups

Roberto DvornicichUmberto Zannier — 2001

Bulletin de la Société Mathématique de France

Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension with respect to r . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime.

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