Nearly-polynomial inverse theorem for the U^d norm in degree d+1
2026-03-17 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors prove a new mathematical result about the Gowers U^d norm, which measures how 'structured' certain functions are over finite fields. They extend previous work by providing a nearly polynomial inverse theorem specifically for polynomials of degree d+1, whereas earlier results covered degree d. Additionally, they offer a similar theorem for homogeneous polynomials with degrees less than 2d. Their work builds on recent advances and provides stronger bounds in these settings.
Gowers U^d normfinite fieldsinverse theorempolynomialsdegreehomogeneous polynomialsnon-small characteristicpolynomial boundsMilićević and Randelović
Authors
Tomer Milo, Guy Moshkovitz
Abstract
We prove a nearly polynomial inverse theorem for the Gowers $U^d$ norm, over finite fields of non-small characteristic, for polynomials of degree $d+1$. The case of degree $d$ was very recently settled by Milićević and Randelović with a fully polynomial bound. We moreover provide a nearly polynomial inverse theorem for homogeneous polynomials of any degree smaller than $2d$.