# Incidences of quadratic forms and points

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
-1. This question needs much more explanation to be clear and useful. See mathoverflow.net/howtoask. – MTS Apr 27 '13 at 1:20

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!

(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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