# Concentration estimates for algebraic intersections

## TL;DR

This paper introduces a new approach over arbitrary fields to bound the degree of algebraic intersections based on concentration on lower codimension sets, extending degree-reduction methods to higher dimensions and degrees.

## Contribution

It extends degree-reduction techniques to higher-dimensional and high-degree varieties, providing sharp bounds and insights into relevant varieties for incidence problems.

## Key findings

- Sharp bounds for algebraic intersections in arbitrary fields.
- Identification of only a few relevant varieties for incidence questions.
- Extension of degree-reduction methods to complex algebraic settings.

## Abstract

We present an approach over arbitrary fields to bound the degree of intersection of families of varieties in terms of how these concentrate on algebraic sets of smaller codimension. This provides in particular a substantial extension of the method of degree-reduction that enables it to deal efficiently with higher-dimensional problems and also with high-degree varieties. We obtain sharp bounds that are new even in the case of lines in $\mathbb{R}^n$ and show that besides doubly-ruled varieties, only a certain rare family of varieties can be relevant for the study of incidence questions.