# Quasi-equivalence of heights in algebraic function fields of one variable

For points (a,b) on an algebraic curve over a field K with height π₯, the asymptotic relation between π₯(a) and π₯(b) has been extensively studied in diophantine geometry. When K=k(t) is the field of algebraic functions in t over a field k of characteristic zero, Eremenko in 1998 proved the following quasi-equivalence for an absolute logarithmic height π₯ in K: Given Pβ K[X,Y] irreducible over K and Ο΅>0, there is a constant C only depending on P and Ο΅ such that for each (a,b)β K^2 with P(a,b)=0, (1-Ο΅) (P,Y) π₯(b)-C β€(P,X) π₯(a) β€ (1+Ο΅) (P,Y) π₯(b)+C. In this article, we shall give an explicit bound for the constant C in terms of the total degree of P, the height of P and Ο΅. This result is expected to have applications in some other areas such as symbolic computation of differential and difference equations.

READ FULL TEXT