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

11/25/2021
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by   Ruyong Feng, et al.
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For points (a,b) on an algebraic curve over a field K with height š”„, the asymptotic relation between š”„(a) and š”„(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 š”„ 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) š”„(b)-C ≤(P,X) š”„(a) ≤ (1+ϵ) (P,Y) š”„(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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