Quasi-equivalence of Heights and Runge's Theorem

2016/08/15 by P. Habegger, Habegger, P.
Mathematics · #11G30 #14H25 #14H50 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11G50 #math.NT #msc:11D41 #msc:11G30 #msc:11G50 #msc:14H25 #msc:14H50 #secondary: 11D41

paper · pdf · doi:10.48550/arxiv.1608.04206

arxiv created 2016/08/15 · arxiv updated 2016/08/16

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.

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