HEIGHTS AND LOGARITHMIC gcd ON ALGEBRAIC CURVES

Abstract

Let F(x,y) be an irreducible polynomial over ℚ, satisfying F(0,0) = 0. Skolem proved that the integral solutions of F(x,y) = 0 with fixed gcd are bounded [13] and Walsh gave an explicit bound in terms of d = gcd (x,y) and F [16]. Assuming that (0,0) is a non-singular point of the plane curve F(x,y) = 0, we extend this result to algebraic solution, and obtain an asymptotic equality instead of inequality. We show that for any algebraic solution (α,β), the quotient h(α)/ log d is approximatively equal to degyF and the quotient h(β)/ log d to deg x F; here h(·) is the absolute logarithmic height and d is the (properly defined) "greatest common divisor" of α and β.