# Quasi-Equivalence of Heights and Runge’s Theorem

@article{Habegger2017QuasiEquivalenceOH, title={Quasi-Equivalence of Heights and Runge’s Theorem}, author={Philipp Habegger}, journal={arXiv: Number Theory}, year={2017}, pages={257-280} }

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 Neron 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… CONTINUE READING