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Is there anything that is known about what is the maximal number of incidences between quadratic forms and points? I looked at the internet and I haven't found anything that works for something that is not a sphere or has a dimension more than 3, but since I have rather "simple" kind of quadratic forms, I wondered if anything is known about such forms. Basically, if we take $6$ variables $x_1,.....,x_6$, then the quadratic forms are of the form $a(x_2x_6-x_3x_5)+b(x_1x_6-x_3x_4)+c(x_1x_5-x_2x_4)=k$. I wondered if there is a literature where such form come as a special case of some theorem.

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What?${}{}{}{}$ –  Will Jagy Apr 26 '13 at 23:45
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-1. This question needs much more explanation to be clear and useful. See mathoverflow.net/howtoask. –  MTS Apr 27 '13 at 1:20

2 Answers 2

Your terse question leaves many possible interpretations, and forgive me if this is a misinterpretation. The 2012 paper by Micha Sharir, Adam Sheffer, and Joshua Zahl,

"Incidences between points and non-coplanar circles." arXiv:1208.0053 (math.CO)

showed that the number of incidences between $m$ points and $n$ arbitrary circles in three dimensions—in the situation when the circles are "truly three-dimensional," in the sense that there exists a $q < n$ so that no sphere or plane contains more than $q$ of the circles—then the number of incidences is

$$O^*\big(m^{3/7}n^{6/7} + m^{2/3}n^{1/2}q^{1/6} + m^{6/11}n^{15/22}q^{3/22} + m + n\big).$$

The complexity of this result illustrates the complexity of the topic!


   alt text
(Figure from Boris Aronov, Vladlen Koltun, Micha Sharir, "Incidences between Points and Circles in Three and Higher Dimensions," Discrete & Computational Geometry. February 2005, Volume 33, Issue 2, pp 185-206. (Springer link).)

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Well, it seems the Kővári–Sós–Turán theorem should give you a bound on the number of incidences, though this bound is likely not sharp. The Kővári–Sós–Turán theorem says that if you have a collection of surfaces with the properties that:

1) Any $t$ surfaces have at most $O(1)$ points common to all $t$ of them
(2) For any collection of $s$ points, there are at most $O(1)$ surfaces that contain all $s$ of the points

Then if you have $m$ points and $n$ such surfaces, the total number of incidences is $O(\operatorname{min}(mn^{1/s},\ m^{1/t}n))$.

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