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A finiteness theorem for a class of exponential congruences

Authors:	Marian Vâ jâ itu Alexandru Zaharescu

Institution:	Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, 70700 Bucharest, Romania ; Massachusetts Institute of Technology, Department of Mathematics, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139

Abstract:	For given elements $\alpha _1,\ldots,\alpha _k$ and $\beta$ belonging to the ring of integers $\mathcal{A}$ of a number field we consider the set of all tuples $(a_1,\ldots,a_k)$ in $\mathbb{N}^k$ for which $\sum _{i=1}^{k}\alpha _i\beta^{a_i}$ divides $\sum _{i=1}^{k}\alpha _i z^{a_i}$ for any $z\in\mathcal{A},$ and prove under some reasonable assumptions that the set of solutions is finite.

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