TY - JOUR

T1 - On the concentration of points of polynomial maps and applications

AU - Cilleruelo, Javier

AU - Garaev, Moubariz Z.

AU - Ostafe, Alina

AU - Shparlinski, Igor E.

PY - 2012

Y1 - 2012

N2 - For a polynomial f ε F p[X], we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on the Vinogradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve f(x) ≡ y (mod p), where f ε F p[X] is a polynomial of degree d ≥ 2. We also use some recent results and techniques from arithmetic combinatorics to study the values (x, f (x)) in more general sets.

AB - For a polynomial f ε F p[X], we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on the Vinogradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve f(x) ≡ y (mod p), where f ε F p[X] is a polynomial of degree d ≥ 2. We also use some recent results and techniques from arithmetic combinatorics to study the values (x, f (x)) in more general sets.

UR - http://www.scopus.com/inward/record.url?scp=84869201326&partnerID=8YFLogxK

U2 - 10.1007/s00209-011-0959-7

DO - 10.1007/s00209-011-0959-7

M3 - Article

AN - SCOPUS:84869201326

VL - 272

SP - 825

EP - 837

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3-4

ER -