On the distribution of values and zeros of polynomial systems over arbitrary sets

JOURNAL OF NUMBER THEORY

卷:133

Article

Kerr, Bryce¹  Shparlinski, Igor E.¹ 

[1] Macquarie Univ, Dept Comp, Sydney, NSW 2109, Australia

关键词: Polynomials over finite fields;    Distribution of values;    Distribution of zeros;    Discrepancy;   

DOI  :  10.1016/j.jnt.2013.02.012

来源: Elsevier

【 摘 要 】

Let G(1), ... , G(n) is an element of F-p inverted right perpendicularX(1), ... , X(m)inverted left perpendicular be n polynomials in m variables over the finite field F-p of p elements. A result of E. Fouvry and N.M. Katz shows that under some natural condition, for any fixed epsilon and sufficiently large prime p the vectors of fractional parts ({G(1) (x)/p}, ... , {G(n) (x)/p}), x is an element of Gamma, are uniformly distributed in the unit cube [0, 1](n) for any cube Gamma is an element of [0, p - 1](m) with the side length h >= p(1/2)(log p)(1+epsilon). Here we use this result to show the above vectors remain uniformly distributed, when x runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences. (c) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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