English

Abstract

We consider function field analogues of the conjecture of Győry, Sarkozy and Stewart (1996) on the greatest prime divisor of the product (ab+1)(ac+1)(bc+1) for distinct positive integers a, b and c. In particular, we show that, under some natural conditions on rational functions F,G,H ∈ ℂ(X), the number of distinct zeros and poles of the shifted products FH +1 and GH +1 grows linearly with degH if degH > max{deg F, degG}. We also obtain a version of this result for rational functions over a finite field.