Subcritical paucity and $\ell^p$-improving estimates for finite-type polynomial curves

TL;DR

This paper establishes new bounds for $ ext{l}^p$-improving estimates along certain polynomial curves, utilizing a novel paucity estimate and elimination techniques to handle degenerate cases.

Contribution

It introduces a new paucity estimate for inhomogeneous equations and applies it to derive subcritical $ ext{l}^p$-improving bounds for finite-type polynomial curves.

Findings

Established subcritical $ ext{l}^p$-improving bounds for polynomial curves.

Developed a new paucity estimate using Wooley's elimination method.

Extended understanding of $ ext{l}^p$-improving phenomena in degenerate settings.

Abstract

I prove new subcritical bounds for the $\ell^p$-improving problem along restricted subsets of a degenerate curve. The key input is a new paucity estimate for associated inhomogeneous equations which is proven using an elimination method due to Wooley and Parsell--Wooley.