Quasi-equivalence of heights in algebraic function fields of one variable
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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)∈ K^2 with P(a,b)=0, (1-ϵ) (P,Y) 𝔥(b)-C ≤(P,X) 𝔥(a) ≤ (1+ϵ) (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.
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