Diophantine equations with linear recurrences An overview of some recent progress

Zannier, Umberto

doi:10.5802/jtnb.499

Cited by 9 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Schmidt [36,37] to obtain several new and surprising results. (For these results we refer to two recent surveys in [45,11].) For example in [12] we characterized the solutions (n, y) ∈ N × Z of polynomial-exponential equation of the form (2)…”

Section: Introductionmentioning

confidence: 99%

Polynomial-exponential equations involving multi-recurrences

Fuchs

2009

Studia Scientiarum Mathematicarum Hungarica

View full text Add to dashboard Cite

In this paper we consider polynomial-exponential Diophantine equations of the form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$G_n^{(0)} y^d + G_n^{(1)} y^{d - 1} + \cdots + G_n^{(d - 1)} y + G_n^{(d)} = 0$$ \end{document} where Gn( i ) are multi-recurrences, i.e. polynomial-exponential functions in variables n = ( n1 ,..., nk ). Under suitable (but restrictive) conditions we prove that there are finitely many multi-recurrences Hn(1) ,..., Hn( s ) such that for all solutions ( n1 ,..., nk , y ) ∈ ℕ k × ℤ we either have \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_n^{(i)} = 0 or y = H_n^{(j)}$$ \end{document} for certain 1 ≦ i,j ≦ s , respectively. This generalizes earlier results of this type on such equations. The proof uses a recent result by Corvaja and Zannier.

show abstract

Section: Introductionmentioning

confidence: 99%

Polynomial-exponential equations involving multi-recurrences

Fuchs

2009

Studia Scientiarum Mathematicarum Hungarica

View full text Add to dashboard Cite

show abstract

“…A special case of Pisot's conjecture states that given integers a, b > 1, if a n − 1 divides b n − 1 for all large positive integer n, then b is a power of a, see [19] for a solution of the original problem, and see [31] for an overview of the existing improvements. The following result can be viewed as an analogue in the complex and non-Archimedean situation.…”

Section: 3mentioning

confidence: 99%

GCD Bounds for Analytic Functions

Pasten

Wang

2016

Int Math Res Notices

View full text Add to dashboard Cite

We consider the problem giving upper bounds for the counting function of common zeros (i.e. the gcd) of two entire analytic functions in various settings. Applying techniques from analytic number theory and Diophantine approximation in the context of entire and meromorphic functions, we establish bounds of this type covering several aspects that were not well-suited for the previously used methods in the context of this problem.

show abstract

“…We will use Theorem 1.3 and Theorem 1.5 to solve the following quotient problem, which can be considered an analogue of the"Hadamard quotient theorem" for recurrence sequences proved by van der Poorten (see [17] and [20], and see [3] and [25] for an overview of the existing improvements). To state the result we make the following definitions: Let G ∈ C[x1, .…”

Section: Introductionmentioning

confidence: 99%

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

Levin

Wang

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author, and consider moving targets versions of our results.

show abstract

Diophantine equations with linear recurrences An overview of some recent progress

Cited by 9 publications

References 13 publications

Polynomial-exponential equations involving multi-recurrences

Polynomial-exponential equations involving multi-recurrences

GCD Bounds for Analytic Functions

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

Contact Info

Product

Resources

About