Local sign changes of polynomials

The trigonometric monomial cos(〈k, x〉)on \(\mathbb{T}^{d}\) $${\mathbb{T}}^d$$ , a harmonic polynomial \(p:\mathbb{S}^{d-1}\rightarrow\mathbb{R}\) $$p:{\mathbb{S}}^{d-1}\to \mathbb{R}$$ of degree k and a Laplacian eigenfunction −Δf = k²f have a root in each ball of radius ∼ ∥k∥⁻¹ or ∼ k⁻¹, respectively. We extend this to linear combinations and show that for any trigonometric polynomials on \(\mathbb{T}^{d}\) $${\mathbb{T}}^d$$ , any polynomial p ∈ ℝ[x₁,…,x_d] restricted to \(\mathbb{S}^{d-1}\) $${\mathbb{S}}^{d-1}$$ and any linear combination of global Laplacian eigenfunctions on ℝ^d with d ∈ {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 −Δφ = λφ in Ω ⊂ ℝⁿ has a root in each B(x, α_nλ^−1/2) ball: the positive and negative mass in each B(x, β_nλ^−1/2) ball cancel when integrated against ∥x − y∥²⁻ⁿ.