An algebraic geometry version of the Kakeya problem

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• 作者：Kaloyan Slavov ^{kaloyan7@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14N20 ; 14G15 ; 51E99
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：37
• 期：Complete
• 页码：158-178
• 全文大小：442 K

文摘

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials f,g∈F_q0[x,y]$f,g\in {\mathbb{F}}_{q_0}[x,y]$ and any F_q/F_q0${\mathbb{F}}_q/{\mathbb{F}}_{q_0}$, the image of the map ${\mathbb{F}}_q^3\to {\mathbb{F}}_q^3$ given by (s,x,y)↦(s,sx+f(x,y),sy+g(x,y))$(s,x,y)\mapsto (s,sx+f(x,y),sy+g(x,y))$ has size at least $\frac{q^3}{4}-O(q^{5/2})$ and prove the special case when f=f(x),g=g(y)$f=f(x),g=g(y)$. We also prove it in the case f=f(y),g=g(x)$f=f(y),g=g(x)$ under the additional assumption f^′(0)g^′(0)≠0$f^{\prime }(0)g^{\prime }(0)\ne 0$ when f,g$f,g$ are both affine polynomials. Our approach is based on a combination of Cauchy–Schwarz and Lang–Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials.