Quasi-Equivalence of Heights and Runge’s Theorem

Philipp Habegger ()
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Philipp Habegger: University of Basel, Department of Mathematics and Computer Science

A chapter in Number Theory – Diophantine Problems, Uniform Distribution and Applications, 2017, pp 257-280 from Springer

Abstract: 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)∕q − h(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.

Keywords: Primary: 11G50; Secondary: 11D41; 11G30; 14H25; 14H50 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-55357-3_13

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DOI: 10.1007/978-3-319-55357-3_13

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