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 standard separation of variables theorem[citation needed] 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- univariate polynomial . In order to be harmonic, this polynomial must satisfy
at all points . In particular, when , we have a polynomial , which must satisfy the condition . Hence, the only harmonic polynomials of one (real) variable are affine functions .
In the multivariable case, one finds nontrivial spaces of harmonic polynomials. Consider for instance the bivariate quadratic polynomial
where are real coefficients. The Laplacian of this polynomial is given by
Hence, in order for to be harmonic, its coefficients need only satisfy the relationship . Equivalently, all (real) quadratic bivariate harmonic polynomials are linear combinations of the polynomials
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:
^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.
^Cf. Corollary 1.8 of Axler, Sheldon; Ramey, Wade (1995), Harmonic Polynomials and Dirichlet-Type Problems
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