Let \(\mathcal{P}_l\) denote the homogeneous polynomials on \(\mathbb{R}^d\). The Laplacian \(\Delta\colon \mathcal{P}_l\to \mathcal{P}_{l-2}\) is surjective [1, Corollary 17.14].

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• proof

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1. B. Hall, Quantum Theory for Mathematicians. New York, NY: Springer New York, 2013. doi:10.1007/978-1-4614-7116-5