On the Győry-Sárközy-Stewart conjecture in function fields. (English) Zbl 1499.11323

Shparlinski, Igor E.

Czech. Math. J. 68, No. 4, 1067-1077 (2018).

Summary: We consider function field analogues of the conjecture of K. Győry, A. Sárközy and C. L. Stewart [Acta Arith. 74, No. 4, 365–385 (1996; Zbl 0857.11047)] 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 \(\deg H\geq\max\{\deg F,\deg G\}\). We also obtain a version of this result for rational functions over a finite field.

MSC:

11R09	Polynomials (irreducibility, etc.)
11S05	Polynomials
12E05	Polynomials in general fields (irreducibility, etc.)

Keywords:

shifted polynomial product; number of zeros

Citations:

Zbl 0857.11047