Harmonic polynomial explained

p whose Laplacian is zero is termed a harmonic polynomial.^{[1] [2]}

The harmonic polynomials form a subspace of the vector space of polynomials over the given field. In fact, they form a graded subspace.^[3] For the real field (

R ), the harmonic polynomials are important in mathematical physics.^{[4] [5] [6]}

The Laplacian is the sum of second-order partial derivatives with respect to each of 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]

Examples

Consider a degree-

d univariate polynomial p(x) : = style\sum ^{d} _{k =0} a _{k} x ^{k} . In order to be harmonic, this polynomial must satisfy$$0\; =\; \backslash tfrac\; p(x)\; =\; \backslash sum\_^d\; k(k-1)\; a\_k\; x^$$at all points x \in R . In particular, when d =2 , we have a polynomial p(x) = a _{0} + a _{1} x + a _{2} x ^{2} , which must satisfy the condition a _{2} = 0 . Hence, the only harmonic polynomials of one (real) variable are affine functions x \mapsto a _{0} + a _{1} x .

In the multivariable case, one finds nontrivial spaces of harmonic polynomials. Consider for instance the bivariate quadratic polynomial$$p(x,y)\; :=\; a\_\; +\; a\_\; x\; +\; a\_\; y\; +\; a\_\; x\; y\; +\; a\_\; x^2\; +\; a\_\; y^2,$$where

a _{0,0} , a _{1,0} , a _{0,1} , a _{1,1} , a _{2,0} , a _{0,2} are real coefficients. The Laplacian of this polynomial is given by$$\backslash Delta\; p(x,y)\; =\; \backslash tfrac\; p(x,y)\; +\; \backslash tfrac\; p(x,y)\; =\; 2(a\_\; +\; a\_).$$

Hence, in order for

p(x,y) to be harmonic, its coefficients need only satisfy the relationship a _{2,0} = - a _{0,2} . Equivalently, all (real) quadratic bivariate harmonic polynomials are linear combinations of the polynomials$$1,\; \backslash quad\; x,\; \backslash quad\; y,\; \backslash quad\; xy,\; \backslash quad\; x^2\; -\; y^2.$$

Note that, as in any vector space, there are other choices of basis for this same space of polynomials.

A basis for real bivariate harmonic polynomials up to degree 6 is given as follows:

See also

• Harmonic function
• Spherical harmonics
• Zonal spherical harmonics
• Multilinear polynomial

References

1. Walsh, J. L.. Joseph L. Walsh. On the Expansion of Harmonic Functions in Terms of Harmonic Polynomials. Proceedings of the National Academy of Sciences. 13. 4. 1927. 175–180. 1084921. 16577046. 10.1073/pnas.13.4.175. 1927PNAS...13..175W. free.
2. Book: Sigurdur Helgason (mathematician)

   . Helgason, Sigurdur. Sigurdur Helgason (mathematician). Chapter III. Invariants and Harmonic Polynomials. Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. 2003. American Mathematical Society. Mathematical Surveys and Monographs, vol. 83. 345–384. 9780821826737. https://books.google.com/books?id=WuDyBwAAQBAJ&pg=345.

3. Felder, Giovanni. Veselov, Alexander P.. Action of Coxeter groups on m-harmonic polynomials and KZ equations. 2001. math/0108012.
4. Book: Sergei Sobolev

   . Sobolev, Sergeĭ Lʹvovich. Sergei Sobolev. Partial Differential Equations of Mathematical Physics. International Series of Monographs in Pure and Applied Mathematics. Elsevier. 2016. 401–408. 9781483181363.

5. On the partial differential equations of mathematical physics. Whittaker, Edmund T.. E. T. Whittaker. Mathematische Annalen. 57. 3. 1903. 333–355. 10.1007/bf01444290. 122153032.
6. Book: William Elwood Byerly

   . Byerly, William Elwood. William Elwood Byerly. Chapter VI. Spherical Harmonics. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. 1893. 195–218. Dover. https://books.google.com/books?id=BMQ0AQAAMAAJ&pg=PA195.

7. Cf. Corollary 1.8 of

• Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963)