On the joints problem in finite fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:47:03Z http://mathoverflow.net/feeds/question/90645 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90645/on-the-joints-problem-in-finite-fields On the joints problem in finite fields Cosmin Pohoata 2012-03-09T00:44:30Z 2012-03-09T02:37:49Z <p>The original version of the so-called "joints problem" consists of the following:</p> <p>Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, where by joint we understand a point of $\mathbb{R}^{3}$ lying on three lines from the set $L$ but which do not all lie in the same plane (i.e. they are non-coplanar).</p> <p>The conjectured answer by Sharir was that this number of joints is $\leq C |L|^{\frac{3}{2}}$, for some positive constant $C$; this was proven by Guth and Katz using a rather simple polynomial mathod which easily generalies to $\mathbb{R}^{d}$ in which case the upper bound becomes $C |L|^{\frac{d}{d-1}}$. (references can be found very easily on google; for example, see <a href="http://www.dagstuhl.de/Materials/Files/09/09111/09111.SharirMicha.Other.pdf" rel="nofollow">http://www.dagstuhl.de/Materials/Files/09/09111/09111.SharirMicha.Other.pdf</a>)</p> <p>Now, it seems to me that this polynomial method does not generalize to finite fields; so, my question is, can we get some upper bound in this case? My thoughts for now are to use the graph $G$ having as vertices the lines and to connected them if the lines intersect. Then the number of joints is the number of triangles of $G$ minus $\frac{1}{2}\left(\binom{|L|}{2} - k\right)$, where $k$ is the number of distinct planes determined by the $|L|$ lines... but I feel that I'll be getting really weak bounds if I majorize this (using graph theoretic stuff about the number of triangles and Beck's theorem or related things for $k$).</p> <p>So, any other ideas or knowledge about the finite field case in literature? Thanks.</p> http://mathoverflow.net/questions/90645/on-the-joints-problem-in-finite-fields/90651#90651 Answer by Marina Iliopoulou for On the joints problem in finite fields Marina Iliopoulou 2012-03-09T02:32:15Z 2012-03-09T02:37:49Z <p>I think that Quilodrán's solution to the joints problem in $\mathbb{R}^n$ can be applied to the finite field case, to get the same bounds. This is the paper, "The joints problem in $\mathbb{R}^n$": Abstract: <a href="http://arxiv.org/abs/0906.0555" rel="nofollow">arXiv:0906.0555v3</a>; <a href="http://arxiv.org/pdf/0906.0555.pdf" rel="nofollow">PDF link</a>.</p>