M\"obius function is strongly orthogonal to polynomial phases over $\mathbb{F}_p[t]$

TL;DR

This paper establishes strong orthogonality between the Möbius function and polynomial phases over finite fields, providing power-saving bounds and new approximation techniques in the context of function fields.

Contribution

It introduces novel bounds for Möbius-polynomial phase correlations and develops approximation results for phases of biased multilinear forms in function fields.

Findings

Power-saving bounds for Möbius and polynomial phase correlation

New approximation results for phases of multilinear forms

Polynomial bounds in Gowers uniformity inverse theorem

Abstract

In this paper, we prove power-saving bounds for the corelation of the M\"obius function with polynomial phases of degree $k$ in function fields $\mathbb{F}_p[t]$, when $p > k$. The proof relies on a new approximation result for phases of biased multilinear forms and the recently established strong bounds for the problem of finding bounded codimension varieties inside the dense ones. Along the way, we also obtain polynomial bounds in the inverse theorem for Gowers uniformity norms in the special case of polynomial phases in finite vector spaces.