On the polynomial Wolff axioms

TL;DR

This paper proves a conjecture by Guth on the maximum number of tubes near algebraic varieties, using advanced algebraic geometry tools, with implications for harmonic analysis.

Contribution

It confirms Guth's conjecture and extends the result to semialgebraic sets, employing deep geometric theorems in the proof.

Findings

Confirmed Guth's conjecture on $ ext{delta}$-tubes

Extended results to semialgebraic sets with a $ ext{delta}^{- ext{epsilon}}$ factor

Utilized algebraic and differential geometry tools in the proof

Abstract

We confirm a conjecture of Guth concerning the maximal number of $\delta$-tubes, with $\delta$-separated directions, contained in the $\delta$-neighborhood of a real algebraic variety. Modulo a factor of $\delta^{-\varepsilon}$, we also prove Guth and Zahl's generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski's projection theorem and Gromov's algebraic lemma.