On the Győry-Sárközy-Stewart conjecture in function fields

summary:We consider function field analogues of the conjecture of Győry, Sárközy 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 $F,G,H \in {\mathbb C}(X)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $\deg H$ if $\deg H \ge \max \{\deg F, \deg G\} $. We also obtain a version of this result for rational functions over a finite field