Nearly-polynomial inverse theorem for the U^d norm in degree d+1
This result addresses a foundational problem in additive combinatorics and theoretical computer science, providing incremental progress by extending inverse theorems to higher-degree polynomials.
The paper tackles the problem of establishing an inverse theorem for the Gowers U^d norm over finite fields, specifically for polynomials of degree d+1, achieving a nearly polynomial bound. This extends recent work that solved the case for degree d with a fully polynomial bound.
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 MiliÄeviÄ and RandeloviÄ with a fully polynomial bound. We moreover provide a nearly polynomial inverse theorem for homogeneous polynomials of any degree smaller than $2d$.