[Paper Review] A short proof of Rudnev's point-plane incidence bound

Frank de Zeeuw|arXiv (Cornell University)|Dec 8, 2016

Limits and Structures in Graph Theory4 references20 citations

TL;DR

This paper presents a simplified, direct geometric proof of Rudnev's point-plane incidence bound in three-dimensional space over arbitrary fields, bypassing the complex Klein quadric construction used in the original proof. By mapping point-plane incidences to line-line intersections in 3D space via a natural parametrization, the authors derive the bound $ I(P,Q) = O(|P|^{1/2}|Q| + t|P| + s|Q|) $, where $ s $ and $ t $ control collinear points and co-planar lines, improving on the original by weakening the dependency on collinearity.

ABSTRACT

In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection points of lines. We describe a simple geometric map that directly sends point-plane incidences to line-line intersections in space, allowing us to reprove Rudnev's theorem with fewer technicalities.

Motivation & Objective

• To provide a shorter, more intuitive proof of Rudnev's point-plane incidence bound in $\mathbb{F}^3$ over arbitrary fields.
• To eliminate the need for the four-dimensional Klein quadric construction used in the original proof.
• To strengthen the incidence bound by replacing the $k$-collinear-point condition with a joint constraint on collinear points and co-planar planes.
• To simplify technical machinery by staying within three-dimensional geometry and using a direct incidence-to-intersection correspondence.

Proposed method

• Define a geometric map $\varphi$ that sends each point $p$ to a line $\varphi(p)$ in $\mathbb{F}^3$, representing all lines through $p$ intersecting a fixed $z$-axis and a plane $x=1$.
• Define a dual map $\psi$ that sends each plane $q$ to a line $\psi(q)$, representing lines in $q$ intersecting the same fixed $z$-axis and plane.
• Show that a point-plane incidence $p \in q$ is equivalent to the intersection of lines $\varphi(p)$ and $\psi(q)$ in $\mathbb{F}^3$.
• Use a variant of the Guth–Katz line intersection bound to count the number of such line-line intersections, which equals the number of incidences.
• Control exceptional configurations by assuming no line contains $s$ points of $P$ and lies in $t$ planes of $Q$, preventing a single quadric from containing $s$ lines from $\varphi(P)$ and $t$ lines from $\psi(Q)$.
• Apply degree-based counting arguments on algebraic surfaces to bound intersection points, distinguishing between ruled and non-ruled components.

Experimental results

Research questions

• RQ1 Can Rudnev's point-plane incidence bound be reproven with fewer technical steps and without embedding into the Klein quadric?
• RQ2 What is the minimal geometric condition on point-line and plane-line configurations that still yields a strong incidence bound?
• RQ3 How can the dependence on collinearity be refined to reflect both the number of collinear points and their co-planarity?
• RQ4 Can a direct 3D mapping from incidences to line intersections yield a bound matching or improving the original?

Key findings

• The paper establishes a new incidence bound: $ I(P,Q) = O(|P|^{1/2}|Q| + t|P| + s|Q|) $, where $ s $ is the maximum number of collinear points in $ P $ that lie in $ t $ planes of $ Q $.
• The bound improves upon Rudnev's original result by weakening the condition from 'no line contains $ k $ points' to 'no line contains $ s $ points and lies in $ t $ planes', making it more sensitive to actual geometric configurations.
• The proof avoids the four-dimensional Klein quadric by using a direct 3D parametrization of lines through a fixed axis and a fixed plane, reducing technical complexity.
• The authors show that if a quadric contains $ s $ lines from $ \varphi(P) $ and $ t $ lines from $ \psi(Q) $, then the corresponding points and planes must lie on a common line in $ \mathbb{F}^3 $, contradicting the assumption unless $ s $ or $ t $ is small.
• The method yields a tight bound that is optimal for $ s \geq |P|^{1/2} $, matching the extremal case where $ s-1 $ points lie on a line contained in $ t $ planes.
• The bound holds over arbitrary fields, including positive characteristic fields, provided $ |P| = O(p^2) $ to avoid degeneracies in positive characteristic.