An improved bound on the number of point-surface incidences in three dimensions

Abstract

We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree in ${\mathbb{R}}^3$ satisfying suitable nondegeneracy conditions can have at most $O(m^{\frac{2k}{3k-1}}{n}^{\frac{3k-3}{3k-1}}+m+n)$ incidences, provided that any collection of $k$ points have at most O(1) surfaces passing through all of them, for some $k\ge 3$. In the case where the surfaces are spheres and no three spheres meet in a common circle, this implies there are $O((mn{)}^{3/4}+m+n)$ point-sphere incidences. This is a slight improvement over the previous bound of $O((mn{)}^{3/4}\beta (m,n)+m+n)$ for $\beta (m,n)$ an (explicit) very slowly growing function. We obtain this bound by using the discrete polynomial ham sandwich theorem to cut ${\mathbb{R}}^3$ into open cells adapted to the set of points, and within each cell of the decomposition we apply a Turan-type theorem to obtain crude control on the number of point-surface incidences. We then perform a second polynomial ham sandwich decomposition on the irreducible components of the variety defined by the first decomposition. As an application, we obtain a new bound on the maximum number of unit distances amongst $m$ points in ${\mathbb{R}}^3$.

17 pages, revised based on referee comments

References in corpus (1)

• On the Erdos distinct distance problem in the plane

Cited by in corpus (5)

• Polynomial partitioning for a set of varieties
• Distinct distances in the complex plane
• An incidence theorem in higher dimensions
• Highly incidental patterns on a quadratic hypersurface in ${\mathbb{R}}^4$
• Refined bounds on the number of connected components of sign conditions on a variety