Local sign changes of polynomials

The trigonometric monomial $\cos (⟨k,x⟩)$ on ${\mathbb{T}}^d$, a harmonic polynomial $p:{\mathbb{S}}^{d-1}\to \mathbb{R}$ of degree $k$ and a Laplacian eigenfunction $-\mathrm{\Delta }f=k^{2}f$ have root in each ball of radius $\sim \mathrm{\parallel }k{\mathrm{\parallel }}^{-1}$ 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 $d\in \{2,3\}$ 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 $-\mathrm{\Delta }\phi =\lambda \phi$ in $\mathrm{\Omega }\subset {\mathbb{R}}^n$ has a root in each $B(x,{\alpha }_n{\lambda }^{-1\mathrm{/}2})$ ball: the positive and negative mass in each $B(x,{\beta }_n{\lambda }^{-1\mathrm{/}2})$ ball cancel when integrated against $\mathrm{\parallel }x-y{\mathrm{\parallel }}^{2-n}$.