Dyson’s lemma with moving parts

In its original form [D], Dyson’s Lemma bounds the order of vanishing of a polynomial in two variables at a finite set of points and was used in studying rational approximations to a fixed algebraic irrational number. Bombieri [B] revived interest in Dyson’s Lemma and subsequently Esnault and Viehweg [EV] and Vojta [V1] vastly extended the scope of the result, to polynomials in several variables and to a section of a line bundle on a product of two algebraic curves respectively. The theorems of Vojta and of Esnault and Viehweg are highly refined geometric results which are used to prove the Mordell Conjecture and Roth’s Theorem respectively. The Main Theorems of both [EV] and, up to a factor of two, [V1] were recovered in [N] as special cases of a more general result. Before [N] had been written, Vojta [V3] formulated a result which would vastly improve the main result of [N]. The goal of this note is to prove Vojta’s statement on a product of an arbitrary number of projective lines and to give a counterexample on a product of three or more curves of genus ≥ 1. In order to state our main result, we need to fix some notation. Let