2021/11/25 by Ruyong Feng, Feng, Ruyong, Feng Shuang +3
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2111.13025
For points (a,b) on an algebraic curve over a field K with height\n mathfrakh, the asymptotic relation between mathfrakh(a) and\n mathfrakh(b) has been extensively studied in diophantine geometry. When\nK=\k(t) is the field of algebraic functions in t over a field k\nof characteristic zero, Eremenko in 1998 proved the following quasi-equivalence\nfor an absolute logarithmic height mathfrakh in K: Given P\∈ K[X,Y]\nirreducible over K and \ε>0, there is a constant C only depending\non P and \ε such that for each (a,b)\∈ K2 with P(a,b)=0, \n (1-
epsilon)
deg(P,Y)
mathfrakh(b)-C
leq
deg(P,X)
mathfrakh(a)
leq\n(1+
epsilon)
deg(P,Y)
mathfrakh(b)+C. In this article, we shall give an\nexplicit bound for the constant C in terms of the total degree of P, the\nheight of P and \ε. This result is expected to have applications in\nsome other areas such as symbolic computation of differential and difference\nequations.\n