On the concentration of points of polynomial maps and applications
On the concentration of points of polynomial maps and applications
复制标题
论多项式映射的点集中及其应用
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
I. Shparlinski
中科院分区：
文献类型：
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作者：
J. Cilleruelo;M. Garaev;Alina Ostafe;I. Shparlinski
对于多项式$${fin{mathbb {F}}_p[X]}$$，我们得到了（x，f（x））模素数p属于边长为H的任意正方形的点数的上界.我们的结果特别是基于维诺格拉多夫中值定理。利用这些估计，我们得到了由非线性多项式生成的动力系统中轨道展开的结果，并得到了曲线$${f（x）equiv y，（{
m mod}，p）}$$，其中$${fin{mathbb {F}}_p[X]}$$是次数d ≥ 2的多项式。我们还使用算术组合学的一些最新结果和技术来研究更一般集合中的值（x，f（x））。
For a polynomial $${fin{mathbb {F}}_p[X]}$$ , 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 $${f(x)equiv y, ({
m mod}, p)}$$ , where $${fin{mathbb {F}}_p[X]}$$ 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.