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Zbl 1080.11028
Zannier, Umberto
On the integer solutions of exponential equations in function fields. (English)
[J] Ann. Inst. Fourier 54, No. 4, 849-874 (2004). ISSN 0373-0956; ISSN 1777-5310/e

Author's summary: 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 values of certain pairs of recurrence sequences (Cor. 2). In particular we substantially sharpen (Cor. 3) recent bounds for the number of integer solutions $(m,n)$ of $G_m(P(X))=c_{m,n}G_n(X)$, where $G_n$ is a recurrence of polynomials, $P$ is a polynomial and $c_{m,n}$ is a variable constant. Finally, we estimate the number of solutions to an $S$-unit type equation in two variables (Cor. 4), improving on known bounds.
[Maurice Mignotte (Strasbourg)]

MSC 2000:: *11D45 Counting solutions of Diophantine equations
11D61 Exponential diophantine equations
11D99 Diophantine equations

Keywords: Diophantine equations; function fields; number of integer solutions; exponential equations in several variables; recurrence of polynomials; $S$-unit type equation

Cited in: Zbl 1185.11022

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