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For points $(a,b)$ on an algebraic curve over a field $K$ with height$\mathfrak{h}$, the asymptotic relation between $\mathfrak{h}(a)$ and$\mathfrak{h}(b)$ has been extensively studied in diophantine geometry. When$K=\overline{k(t)}$ is the field of algebraic functions in $t$ over a field $k$of characteristic zero, Eremenko in 1998 proved the following quasi-equivalencefor an absolute logarithmic height $\mathfrak{h}$ in $K$: Given $P\in K[X,Y]$irreducible over $K$ and $\epsilon>0$, there is a constant $C$ only dependingon $P$ and $\epsilon$ such that for each $(a,b)\in K^2$ with $P(a,b)=0$, $$  (1-\epsilon) \deg(P,Y) \mathfrak{h}(b)-C \leq \deg(P,X) \mathfrak{h}(a) \leq(1+\epsilon) \deg(P,Y) \mathfrak{h}(b)+C. $$ In this article, we shall give anexplicit bound for the constant $C$ in terms of the total degree of $P$, theheight of $P$ and $\epsilon$. This result is expected to have applications insome other areas such as symbolic computation of differential and differenceequations.

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