Local sign changes of polynomials

TL;DR

This paper extends known results about the roots of harmonic and trigonometric polynomials to linear combinations and eigenfunctions, showing they have roots in balls whose radius depends on the inverse of their frequencies.

Contribution

It generalizes root existence results to broader classes of functions, including linear combinations and eigenfunctions, with explicit radius bounds based on frequencies.

Findings

Roots exist in balls of radius proportional to inverse frequencies

Eigenfunctions have roots and mass cancellation properties in specific balls

Results apply to trigonometric polynomials, polynomials on spheres, and Laplacian eigenfunctions

Abstract

The trigonometric monomial $\cos(\left\langle k, x \right\rangle)$ on $\mathbb{T}^d$, a harmonic polynomial $p: \mathbb{S}^{d-1} \rightarrow \mathbb{R}$ of degree $k$ and a Laplacian eigenfunction $-\Delta f = k^2 f$ have root in each ball of radius $\sim \|k\|^{-1}$ or $\sim k^{-1}$, respectively. We extend this to linear combinations and show that for any trigonometric polynomials on $\mathbb{T}^d$, any polynomial $p \in \mathbb{R}[x_1, \dots, x_d]$ restricted to $\mathbb{S}^{d-1}$ and any linear combination of global Laplacian eigenfunctions on $ \mathbb{R}^d$ with $d \in \left\{2,3\right\}$ the same property holds for any ball whose radius is given by the sum of the inverse constituent frequencies. We also refine the fact that an eigenfunction $- \Delta \phi = \lambda \phi$ in $\Omega \subset \mathbb{R}^n$ has a root in each $B(x, \alpha_n \lambda^{-1/2})$ ball: the positive and negative mass in each $B(x,\beta_n \lambda^{-1/2})$ ball cancel when integrated against $\|x-y\|^{2-n}$.