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- Author or Editor: Clemens Fuchs x
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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.
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.
