The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic polynomials are important in mathematical physics.
The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomials.
- Walsh, J. L. (1927). "On the Expansion of Harmonic Functions in Terms of Harmonic Polynomials". Proceedings of the National Academy of Sciences. 13 (4): 175–180. doi:10.1073/pnas.13.4.175. PMC 1084921. PMID 16577046.
- Helgason, Sigurdur (2003). "Chapter III. Invariants and Harmonic Polynomials". Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. Mathematical Surveys and Monographs, vol. 83. American Mathematical Society. pp. 345–384.
- Felder, Giovanni; Veselov, Alexander P. (2001). "Action of Coxeter groups on m-harmonic polynomials and KZ equations". arXiv:math/0108012.
- Sobolev, Sergeĭ Lʹvovich (2016). Partial Differential Equations of Mathematical Physics. International Series of Monographs in Pure and Applied Mathematics. Elsevier. pp. 401–408. ISBN 9781483181363.
- Whittaker, Edmund T. (1903). "On the partial differential equations of mathematical physics". Mathematische Annalen. 57 (3): 333–355. doi:10.1007/bf01444290.
- Byerly, William Elwood (1893). "Chapter VI. Spherical Harmonics". An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. Dover. pp. 195–218.
- Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963) doi:10.2307/2373130
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