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

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 $k\geq 3$. 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 $\mathbb{R}^3$.Comment: 17 pages, revised based on referee comment

A Szemeredi-Trotter type theorem in R4\mathbb{R}^4

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.Comment: 50 pages. V3: final version. To appear in Discrete and Computational Geometr

The polynomial method over varieties

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2019, Director: Martín Sombra[en] In 2010, Guth and Katz introduced the polynomial partitioning theorem as a tool in incidence geometry and in additive combinatorics. This allowed the application of results from algebraic geometry (mainly on intersection theory and on the topology of real algebraic varieties) to the solution of long standing problems, including the celebrated Erdős distinct distances problem. Recently, Walsh has extended the polynomial partitioning method to an arbitrary subvariety. This result opens the way to the application of this method to control the point-hypersurface incidences and, more generally, of variety-variety incidences, in spaces of arbitrary dimension. This final project consists in studying Walsh’s paper, to explain its contents and explore its applications to t his kind of incidence problems