A proof of Pisot's th root conjecture.
On periodic mod p sequences and G-functions (On a conjecture of Ruzsa).
Polynomial squares of the form
Proof of the existence of certain triples of polynomials
Some remarks on the S-unit equation in function fields
Acknowledgement of priority
A note on recurrent mod p sequences
On the linear independence of roots of unity over finite extensions of Q
Remarks on a question of Skolem about the integer solutions of x₁x₂ - x₃x₄ = 1
On Davenport's bound for the degree of f³ - g² and Riemann's Existence Theorem
Diophantine equations with linear recurrences An overview of some recent progress
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. "-th root problem"), which in short asks whether the integrality of the values of the quotient (resp. -th root) of two (resp. one) linear recurrences implies that this quotient (resp. -th root) is itself a recurrence. We shall also relate such...
Parametrizing SL₂(ℤ) and a question of Skolem
On the number of terms of a composite polynomial
Addendum to the paper: "On the number of terms of a composite polynomial" (Acta Arith. 127 (2007), 157-167)
On the integer solutions of exponential equations in function fields
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...
A Note on Thue's Equation Over Function Fields.
On the greatest prime factor of Markov pairs
An theorem over function fields and applications
We provide a lower bound for the number of distinct zeros of a sum for two rational functions , in term of the degree of , which is sharp whenever have few distinct zeros and poles compared to their degree. This sharpens the “-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 contains only finitely many rational or elliptic curves,...
On polynomials taking small values at integral arguments
Local-global divisibility of rational points in some commutative algebraic groups
Let be a commutative algebraic group defined over a number field . We consider the following question:A complete answer for the case of the multiplicative group is classical. We study other instances and in particular obtain an affirmative answer when is a prime and is either an elliptic curve or a torus of small dimension with respect to . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when is a prime.
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