Sparsity of the intersection of polynomial images of an interval

• Autorzy: Mei-Chu Chang
• Języki publikacji: EN

Abstrakty

EN

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 $f(x),g(x) ∈ 𝔽_{p}[x]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies
$p^{1/E(1 + κ/(1-κ)} > M > p^ε$,
where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then
$|f([0,M]) ∩ g([0, M])| ≲ M^{1-ε}$.

Kategorie tematyczne

• 11D79: Congruences in many variables
• 11P21: Lattice points in specified regions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 165
• Numer: 3
• Strony: 243-249

Daty

wydano

2014

Twórcy

autor

Mei-Chu Chang

• Department of Mathematics, University of California, Riverside, CA 92521, U.S.A.

Identyfikatory

DOI

10.4064/aa165-3-3