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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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amss.ac.cn
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University of the Chinese Academy of Sciences
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For points (a,b) on an algebraic curve over a field K with height π”₯, the asymptotic relation between π”₯(a) and π”₯(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 π”₯ in K: Given P∈ K[X,Y] irreducible over K and Ο΅>0, there is a constant C only depending on P and Ο΅ such that for each (a,b)∈ K^2 with P(a,b)=0, (1-Ο΅) (P,Y) π”₯(b)-C ≀(P,X) π”₯(a) ≀ (1+Ο΅) (P,Y) π”₯(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 Ο΅. This result is expected to have applications in some other areas such as symbolic computation of differential and difference equations.

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