Power values of polynomials and binomial Thue-Mahler equations

Kálmán Gyory, István Pink, Ákos Pintér

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

Let p 1,⋯,p s be distinct primes, and S the set of integers not divisible by primes different from p 1,⋯, p s. We give effectively computable upper bounds for n in the equations (1) f(x) = wy n, (5) F(x, z) = wy n and (9) ax n by n = c, where f ∈ Z[X] is a monic polynomial, F ∈ Z[X, Z] a monic binary form, the discriminants D(f), D(F) are contained in S, and x, y, z, w, a, b, c, n are unknown non-zero integers with z, w, a, b, c ∞ S, y ∉ S and n ≥ 3. It is a novelty in our paper that the upper bounds depend only on the product p 1 ⋯ p s, and, in case of (1) and (5), on deg f and deg F, respectively. The bounds are given explicitly in terms of p 1 ⋯ p s. Our results are established in more general forms, over an arbitrary algebraic number field. Equation (5) is reduced to an equation of type (9) over an appropriate extension of the ground field. The proofs involve among other things the best known estimates for linear forms in logarithms of algebraic numbers.

Original languageEnglish
Pages (from-to)341-362
Number of pages22
JournalPublicationes Mathematicae
Volume65
Issue number3-4
Publication statusPublished - Dec 1 2004

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Keywords

  • Binomial Thue-Mahler equations
  • Discriminants of polynomials and binary forms
  • S-unit equations
  • Superelliptic equations

ASJC Scopus subject areas

  • Mathematics(all)

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