Effective results for Diophantine equations over finitely generated domains

Attila Bérczes, Jan Hendrik Evertse, Kálmán Gyory

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over double-struck Z. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations f(x) = δym in x,y ∈ A, where f ∈ A[X], δ ∈ A \ {0} and M ∈ double-struck Z≥2. Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F, f, M. These results imply that the solutions of these equations can be determined in principle. Further, we consider the Schinzel-Tijdeman equation f(x) = δyM where x,y ∈ A and M ∈ double-struck Z≥2 are the unknowns and give an effective upper bound for M. Our results extend earlier work of Gyo{combining double acute accent}ry, Brindza and Végso{combining double acute accent}, where the equations mentioned above were considered only for a restricted class of finitely generated domains.

Original languageEnglish
Pages (from-to)71-100
Number of pages30
JournalActa Arithmetica
Volume163
Issue number1
DOIs
Publication statusPublished - Jan 1 2014

Keywords

  • Diophantine equations over finitely generated domains
  • Effective results
  • Hyperelliptic equations
  • Schinzel-Tijdeman equation
  • Superelliptic equations
  • Thue equations

ASJC Scopus subject areas

  • Algebra and Number Theory

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