Explicit bounds for the solutions of superelliptic equations over number fields

2023/10/15 by Attila Bérczes, Bérczes, Attila, Yann Bugeaud +9 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2310.09704

openalex publication_date 2023/10/15 · openalex created_date 2023/10/18 · openalex updated_date 2026/07/28

Abstract

Let f be a polynomial with coefficients in the ring O_S of S-integers of a number field K, b a non-zero S-integer, and m an integer ≥ 2. We consider the equation ( ⋆ ): f(x) = b y^m in x,y ∈ O_S. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of K, S, f, m and the S-norm of b for the heights of the solutions x of the equation ( ⋆). Further, we give an explicit bound C in terms of K, S, f and the S-norm of b such that if m > C the equation (⋆) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of Bérczes, Evertse, and Győry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the S-norm of b instead of its height.

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