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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

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

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