Sign uncertainty principles and low-degree polynomials

TL;DR

This paper establishes sharp bounds for sign uncertainty principles involving low-degree polynomials multiplied by a Gaussian, revealing limitations on improvements for sublinear degrees in high dimensions.

Contribution

It provides an asymptotically sharp version of sign uncertainty principles for polynomials of sublinear degree, connecting to sphere packing and modular bootstrap bounds.

Findings

Polynomials of sublinear degree cannot outperform degree-three polynomials asymptotically.

The results are relevant for sphere packing bounds and free boson bootstrap.

The bounds are sharp in high-dimensional limits.

Abstract

We prove an asymptotically sharp version of the Bourgain-Clozel-Kahane and Cohn-Gon\c{c}alves sign uncertainty principles for polynomials of sublinear degree times a Gaussian, as the dimension tends to infinity. In particular, we show that polynomials whose degree is sublinear in the dimension cannot improve asymptotically on those of degree at most three. This question arises naturally in the study of both linear programming bounds for sphere packing and the spinless modular bootstrap bound for free bosons.