Local sign changes of polynomials

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 $\|x-y\|^{2-n}$.