Quasi-equivalence of Heights and Runge's Theorem

TL;DR

This paper provides an explicit bound on the difference of normalized heights of algebraic solutions to bivariate polynomial equations and applies it to refine Runge's theorem for integral solutions.

Contribution

It derives a fully explicit bound for height differences in algebraic solutions and uses it to simplify conditions in Runge's theorem.

Findings

Explicit bound for |h(x)/q - h(y)/p| in terms of polynomial P

Bound grows as the square root of maximum height of solutions

Application to simplified criteria for integral solutions in polynomial equations

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 N\'eron $h(x)/q$ 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.