Effective results for hyper- and superelliptic equations over number fields

Attila Bérczes, Jan Hendrik Evertse, K. Györy

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 (*) f(x) = by m 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.

Original languageEnglish
Pages (from-to)727-756
Number of pages30
JournalPublicationes Mathematicae
Volume82
Issue number3-4
DOIs
Publication statusPublished - 2013

Fingerprint

Number field
Integer
Multiple Zeros
Explicit Bounds
Diophantine equation
Integral domain
Roots of Unity
Finitely Generated
Upper bound
Ring
Polynomial
Coefficient

Keywords

  • Baker's method
  • Hyperelliptic equations
  • Schinzel-tijdeman theorem
  • Superelliptic equations

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Effective results for hyper- and superelliptic equations over number fields. / Bérczes, Attila; Evertse, Jan Hendrik; Györy, K.

In: Publicationes Mathematicae, Vol. 82, No. 3-4, 2013, p. 727-756.

Research output: Contribution to journalArticle

Bérczes, Attila ; Evertse, Jan Hendrik ; Györy, K. / Effective results for hyper- and superelliptic equations over number fields. In: Publicationes Mathematicae. 2013 ; Vol. 82, No. 3-4. pp. 727-756.
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