Diophantine equations in separated variables View Full Text


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Article Info

DATE

2017-08-01

AUTHORS

Dijana Kreso, Robert F. Tichy

ABSTRACT

We study Diophantine equations of type f ( x ) = g ( y ) , where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials ( f n ) n are such that f n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if f ( x ) = g ( h ( x ) ) with g , h K [ x ] and deg g > 1 , then either deg h 2 , or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points. More... »

PAGES

47-67

References to SciGraph publications

  • 2002-04. Diophantine Equations and Bernoulli Polynomials in COMPOSITIO MATHEMATICA
  • 2008-04-24. On composite lacunary polynomials and the proof of a conjecture of Schinzel in INVENTIONES MATHEMATICAE
  • 2009-11-17. On equal values of trinomials in MONATSHEFTE FÜR MATHEMATIK
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10998-017-0195-y

    DOI

    http://dx.doi.org/10.1007/s10998-017-0195-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1090943566

    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/30636814


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