An algebraic geometry version of the Kakeya problem

详细信息    查看全文

• 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 5000970&_mathId=si1.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=60559360121b0d220e758ac6c81562de" title="Click to view the MathML source">f,g∈F_q0[x,y]$f,g\in {\mathbb{F}}_{q_0}[x,y]$ and any 5000970&_mathId=si2.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=0f8f2383ce92b28d1ce40d80001f83ce" title="Click to view the MathML source">F_q/F_q0${\mathbb{F}}_q/{\mathbb{F}}_{q_0}$, the image of the map 5000970&_mathId=si3.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=22ae774bd2e0c088fbc96c34d382e148">5000970-si3.gif">${\mathbb{F}}_q^3\to {\mathbb{F}}_q^3$ given by 5000970&_mathId=si4.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=1870e9e5d5bb42c3386304bdcea4a77c" title="Click to view the MathML source">(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 5000970&_mathId=si5.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=9d4c2396d6c5435589c8bb00bfbf11ff">5000970-si5.gif">$\frac{q^3}{4}-O(q^{5/2})$ and prove the special case when 5000970&_mathId=si6.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=270311714cc21f79322e74a7eced239c" title="Click to view the MathML source">f=f(x),g=g(y)$f=f(x),g=g(y)$. We also prove it in the case 5000970&_mathId=si7.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=e3d9a8396634704dc27b4d29d4eb354f" title="Click to view the MathML source">f=f(y),g=g(x)$f=f(y),g=g(x)$ under the additional assumption 5000970&_mathId=si431.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=3e0410b36059a2eebd810a408d04efed" title="Click to view the MathML source">f^′(0)g^′(0)≠0$f^{\prime }(0)g^{\prime }(0)\ne 0$ when 429" class="mathmlsrc">5000970&_mathId=si429.gif&_user=111111111&_pii=S1071579715000970&_rdoc=1&_issn=10715797&md5=6e813d5b51cbc002c284698e9a511f74" title="Click to view the MathML source">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.