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

Mei-Chu Chang Acta Arithmetica 165 (2014), 243-249 MSC: Primary 11P21, 11D79. DOI: 10.4064/aa165-3-3


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


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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image