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Quasi-Equivalence of Heights and Runge’s Theorem

  • Philipp Habegger
Chapter

Abstract

Let P be a polynomial that depends on two variables X and Y and has algebraic coefficients. If x and y are algebraic numbers with P(x, y) = 0, then by work of Néron h(x)∕q is asymptotically equal to h(y)∕p where p and q are the partial degrees of P in X and Y, respectively. In this paper we compute a completely explicit bound for | h(x)∕qh(y)∕p | in terms of P which grows asymptotically as max{h(x), h(y)}1∕2. We apply this bound to obtain a simple version of Runge’s Theorem on the integral solutions of certain polynomial equations.

Mathematics Subject Classification.

Primary: 11G50 Secondary: 11D41 11G30 14H25 14H50 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland

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