Abstract
The trigonometric monomial cos(〈k, x〉)on \(\mathbb{T}^{d}\), a harmonic polynomial \(p:\mathbb{S}^{d-1}\rightarrow\mathbb{R}\) 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 \(\mathbb{T}^{d}\), any polynomial p ∈ ℝ[x1,…,xd] restricted to \(\mathbb{S}^{d-1}\) and any linear combination of global Laplacian eigenfunctions on ℝd 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, αnλ−1/2) ball: the positive and negative mass in each B(x, βnλ−1/2) ball cancel when integrated against ∥x − y∥2−n.
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Steinerberger, S. Local sign changes of polynomials. JAMA (2024). https://doi.org/10.1007/s11854-024-0344-1
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DOI: https://doi.org/10.1007/s11854-024-0344-1