Uniform $l^2$-decoupling in $\mathbb R^2$ for Polynomials

TL;DR

This paper establishes a uniform $l^2$-decoupling inequality for polynomial phases in two-dimensional space, using a novel geometric partition approach tailored to each phase function.

Contribution

It introduces a new geometric partition method for polynomial phases, leading to a uniform decoupling inequality in $\

Findings

Proves a uniform $l^2$-decoupling inequality for all polynomial phases up to degree $d$.

Uses a novel partition based on the geometry of individual phase functions.

Connects the result to previous work but with a different geometric approach.

Abstract

For each positive integer $d$, we prove a uniform $l^2$-decoupling inequality for the collection of all polynomials phases of degree at most $d$. Our result is intimately related to \cite{MR4078083}, but we use a different partition that is determined by the geometry of each individual phase function.