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  • Author or Editor: Clemens Fuchs x
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Polynomial-exponential equations involving multi-recurrences

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 46: Issue 3
DOI:
https://doi.org/10.1556/sscmath.2009.1098
Author: Clemens Fuchs

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 G n ( i ) are multi-recurrences, i.e. polynomial-exponential functions in variables n = ( n 1 ,..., n k ). Under suitable (but restrictive) conditions we prove that there are finitely many multi-recurrences H n (1) ,..., H n ( s ) such that for all solutions ( n 1 ,..., n k , 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.

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Diophantine m-tuples for linear polynomials

Periodica Mathematica Hungarica
Volume/Issue:
Volume 45: Issue 1-2
DOI:
https://doi.org/10.1023/a:1022389711839
Authors: Andrej Dujella, Clemens Fuchs and Robert Tichy

Abstract  

In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients.

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