Sparsity of the intersection of polynomial images of an interval
Volume 165 / 2014
Acta Arithmetica 165 (2014), 243-249
MSC: Primary 11P21, 11D79.
DOI: 10.4064/aa165-3-3
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 $$|f([0,M])\cap g([0, M])|\lesssim M^{1-\varepsilon }.$$
