# Publishing house / Journals and Serials / Acta Arithmetica / All issues

## Acta Arithmetica

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

## 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 $$|f([0,M])\cap g([0, M])|\lesssim M^{1-\varepsilon }.$$

#### Authors

• Mei-Chu ChangDepartment of Mathematics
University of California
Riverside, CA 92521, U.S.A.
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

## Rewrite code from the image 