We show that m points and n smooth algebraic surfaces of bounded degree
in R3 satisfying suitable nondegeneracy conditions can have at most
O(m3k−12kn3k−13k−3+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 R3 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 R3.Comment: 17 pages, revised based on referee comment