Effective results for hyper- and superelliptic equations over number fields

Bérczes, Attila and Evertse, J. -H. and Győry, Kálmán (2013) Effective results for hyper- and superelliptic equations over number fields. PUBLICATIONES MATHEMATICAE DEBRECEN, 82 (3-4). pp. 727-756. ISSN 0033-3883

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Abstract

Let f be a polynomial with coefficients in the ring OS of S-integers of a given number field K, b a non-zero S-integer, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) bym = f (x) in x, y ∈ OS . In the present paper we give explicit upper bounds in terms of K, S, b, f, m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K, S, b, f such that if m > C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over Z

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