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

We show thatm points and n smooth algebraic surfaces of bounded degree in R satisfying suitable nondegeneracy conditions can have at most O(m 2k 3k−1 n 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 ≥ 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)3/4β(m,n)+m+n) for β(m,n) an (explicit) very slowly growing function. We obtain this bound by using the discrete polynomial ham sandwich theorem to cut R 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 R.