Local sign changes of polynomials

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 $-Δf = k^2 f$ 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 $- Δϕ= λϕ$ in $Ω\subset \mathbb{R}^n$ has a root in each $B(x, α_n λ^{-1/2})$ ball: the positive and negative mass in each $B(x,β_n λ^{-1/2})$ ball cancel when integrated against $\|x-y\|^{2-n}$.

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