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- W4225568017 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 $Pin K[X,Y]$ 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 $P(a,b)=0$, $$ (1-epsilon) deg(P,Y) mathfrak{h}(b)-C leq deg(P,X) mathfrak{h}(a) leq (1+epsilon) deg(P,Y) mathfrak{h}(b)+C. $$ In this article, we shall give 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." @default.
- W4225568017 created "2022-05-05" @default.
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- W4225568017 date "2021-11-25" @default.
- W4225568017 modified "2023-09-26" @default.
- W4225568017 title "Quasi-equivalence of heights in algebraic function fields of one variable" @default.
- W4225568017 hasPublicationYear "2021" @default.
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