An algebraic geometry version of the Kakeya problem
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866909214098587648 |
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| author | Slavov, Kaloyan |
| author_facet | Slavov, Kaloyan |
| contents | We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials $f,g\in\F_{q_0}[x,y]$ and any $\F_q/\F_{q_0}$, the image of the map $\F_q^3\to\F_q^3$ given by $(s,x,y)\mapsto (s,sx+f(x,y),sy+g(x,y))$ has size at least $\frac{q^3}{4}-O(q^{5/2})$ and prove the special case when $f=f(x), g=g(y).$ We also prove it in the case $f=f(y), g=g(x)$ under the additional assumption $f'(0)g'(0)\neq 0$ when $f,g$ are both linearized. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1410_3701 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | An algebraic geometry version of the Kakeya problem Slavov, Kaloyan Algebraic Geometry We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials $f,g\in\F_{q_0}[x,y]$ and any $\F_q/\F_{q_0}$, the image of the map $\F_q^3\to\F_q^3$ given by $(s,x,y)\mapsto (s,sx+f(x,y),sy+g(x,y))$ has size at least $\frac{q^3}{4}-O(q^{5/2})$ and prove the special case when $f=f(x), g=g(y).$ We also prove it in the case $f=f(y), g=g(x)$ under the additional assumption $f'(0)g'(0)\neq 0$ when $f,g$ are both linearized. 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. |
| title | An algebraic geometry version of the Kakeya problem |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/1410.3701 |