Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f(x), g(x)\in \mathbb F_{p}[x]$ be polynomials of degrees $d$ and $e$ with $d\ge e\ge 2$. Suppose $M\in \mathbb Z$ satisfies $$p^{\frac 1E(1+\frac {\kappa }{1-\kappa })}>M>p^{\varepsilon },$$ where $E=e(e+1)/2$ and $\kappa =(\frac 1d-\frac 1{d^2})\frac {E-1}{E}+\varepsilon$. Assume $f(x)-g(y)$ is absolutely irreducible.Then $$|f([0,M])\cap g([0, M])|\lesssim M^{1-\varepsilon }.$$