Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions

TL;DR

This paper extends polynomial partitioning techniques to algebraic varieties of codimension two and applies it to bound point-hypersurface incidences in four-dimensional space, advancing combinatorial geometry methods.

Contribution

It introduces a polynomial partitioning theorem for points on codimension two varieties and derives new incidence bounds in four dimensions.

Findings

Polynomial partitioning theorem for codimension two varieties.

Bound on point-hypersurface incidences in 4D.

Application of semi-algebraic set component bounds.

Abstract

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matou\v{s}ek, Sharir and Safernov\'a. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components of a semi-algebraic set by Barone and Basu.