Harmonic polynomial

In mathematics, in abstract algebra, a multivariate polynomial p over a field such that the Laplacian of p is zero is termed a harmonic polynomial.[1][2]

The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace.[3] For the real field, the harmonic polynomials are important in mathematical physics.[4][5][6]

The Laplacian is the sum of second partials with respect to all the variables, and is an invariant differential operator under the action of the orthogonal group via the group of rotations.

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 radial polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials.[7]

See also

References
  1. 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. Bibcode:1927PNAS...13..175W. doi:10.1073/pnas.13.4.175. PMC 1084921. PMID 16577046.
  2. 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. ISBN 9780821826737.
  3. Felder, Giovanni; Veselov, Alexander P. (2001). "Action of Coxeter groups on m-harmonic polynomials and KZ equations". arXiv:math/0108012.
  4. 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.
  5. Whittaker, Edmund T. (1903). "On the partial differential equations of mathematical physics". Mathematische Annalen. 57 (3): 333–355. doi:10.1007/bf01444290. S2CID 122153032.
  6. 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.
  7. Cf. Corollary 1.8 of Axler, Sheldon; Ramey, Wade (1995), Harmonic Polynomials and Dirichlet-Type Problems


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