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

Abstract

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-equivalence for 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 depending on $P$ and $\epsilon$ such that for each $(a,b)\in K^2$ with $P(a,b)=0$, $$(1-\epsilon )°(P,Y)\mathfrak{h}(b)-C\le °(P,X)\mathfrak{h}(a)\le (1+\epsilon )°(P,Y)\mathfrak{h}(b)+C.$$ 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 $\epsilon$. This result is expected to have applications in some other areas such as symbolic computation of differential and difference equations.