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 h, the asymptotic relation between h(a) and h(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 h in K: Given P∈K[X,Y] irreducible over K and ϵ>0, there is a constant C only depending on P and ϵ such that for each (a,b)∈K2 with P(a,b)=0,(1−ϵ)deg⁡(P,Y)h(b)−C≤deg⁡(P,X)h(a)≤(1+ϵ)deg⁡(P,Y)h(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 ϵ. This result is expected to have applications in some other areas such as symbolic computation of differential and difference equations.