Quasiequivalence of Heights and Runge's Theorem
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2016
 arXiv:
 arXiv:1608.04206
 Bibcode:
 2016arXiv160804206H
 Keywords:

 Mathematics  Number Theory;
 Primary: 11G50;
 secondary: 11D41;
 11G30;
 14H25;
 14H50