Quasi-Equivalence of Heights and Runge’s Theorem

  title={Quasi-Equivalence of Heights and Runge’s Theorem},
  author={Philipp Habegger},
  journal={arXiv: Number Theory},
  • Philipp Habegger
  • Published 2017
  • Mathematics
  • arXiv: Number Theory
  • 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