harmonic polynomial

In mathematics, a polynomial $p$ whose Laplacian is zero is termed a harmonic polynomial.{{cite journal|author=Walsh, J. L.|authorlink=Joseph L. Walsh|title=On the Expansion of Harmonic Functions in Terms of Harmonic Polynomials|journal=Proceedings of the National Academy of Sciences|volume=13|issue=4|year=1927|pages=175–180|pmc=1084921|pmid=16577046|doi=10.1073/pnas.13.4.175|bibcode=1927PNAS...13..175W|doi-access=free}}{{cite book|author=Helgason, Sigurdur|authorlink=Sigurdur Helgason (mathematician)|chapter=Chapter III. Invariants and Harmonic Polynomials|title=Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions|year=2003|publisher=American Mathematical Society|series=Mathematical Surveys and Monographs, vol. 83|pages=345–384|isbn=9780821826737|chapter-url=https://books.google.com/books?id=WuDyBwAAQBAJ&pg=345}}

The harmonic polynomials form a subspace of the vector space of polynomials over the given field. In fact, they form a graded subspace.{{cite arXiv|author=Felder, Giovanni|author2=Veselov, Alexander P.|title=Action of Coxeter groups on m-harmonic polynomials and KZ equations|year=2001|eprint=math/0108012}} For the real field ( $\mathbb{R}$ ), the harmonic polynomials are important in mathematical physics.{{cite book|author=Sobolev, Sergeĭ Lʹvovich|authorlink=Sergei Sobolev|title=Partial Differential Equations of Mathematical Physics|series=International Series of Monographs in Pure and Applied Mathematics|publisher=Elsevier|year=2016|pages=401–408|url=https://books.google.com/books?id=P-xPDAAAQBAJ&pg=PA401|isbn=9781483181363}}{{cite journal|url=https://babel.hathitrust.org/cgi/pt?id=njp.32101080167032;view=1up;seq=351|author=Whittaker, Edmund T.|authorlink=E. T. Whittaker|title=On the partial differential equations of mathematical physics|journal=Mathematische Annalen|volume=57|issue=3|year=1903|pages=333–355|doi=10.1007/bf01444290|s2cid=122153032}}{{cite book|author=Byerly, William Elwood|authorlink=William Elwood Byerly|chapter=Chapter VI. Spherical Harmonics|title=An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics|year=1893|pages=195–218|publisher=Dover|chapter-url=https://books.google.com/books?id=BMQ0AQAAMAAJ&pg=PA195}}

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 {{fact|date=March 2020}} 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.Cf. Corollary 1.8 of {{Citation|title=Harmonic Polynomials and Dirichlet-Type Problems|last1=Axler|first1=Sheldon|last2=Ramey|first2=Wade|year=1995}}

Examples

Consider a degree- $d$ univariate polynomial $p(x) := \textstyle\sum_{k=0}^d a_k x^k$ . In order to be harmonic, this polynomial must satisfyFile:HarmonicPolynomials2D.png

$0 = \tfrac{\partial^2}{\partial x^2} p(x) = \sum_{k=2}^d k(k-1) a_k x^{k-2}$

at all points $x \in \mathbb{R}$ . 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

$p(x,y) := a_{0,0} + a_{1,0} x + a_{0,1} y + a_{1,1} x y + a_{2,0} x^2 + a_{0,2} y^2,$

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

$\Delta p(x,y) = \tfrac{\partial^2}{\partial x^2} p(x,y) + \tfrac{\partial^2}{\partial y^2} p(x,y) = 2(a_{2,0} + a_{0,2}).$

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

$1, \quad x, \quad y, \quad xy, \quad x^2 - y^2.$

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:

$\begin{align}
\phi_{0} (x,y) &= 1 \\
\phi_{1,1}(x,y) &= x & \phi_{1,2}(x,y) &= y \\
\phi_{2,1}(x,y) &= x y & \phi_{2,2}(x,y) &= x^2 - y^2 \\
\phi_{3,1}(x,y) &= y^3 - 3 x^2 y & \phi_{3,2}(x,y) &= x^3 - 3 x y^2 \\
\phi_{4,1}(x,y) &= x^3 y - x y^3 & \phi_{4,2}(x,y) &= -x^4 + 6 x^2 y^2 - y^4 \\
\phi_{5,1}(x,y) &= 5 x^4 y - 10 x^2 y^3 + y^5 & \phi_{5,2}(x,y) &= x^5 - 10 x^3 y^2 + 5 x y^4 \\
\phi_{6,1}(x,y) &= 3 x^5 y - 10 x^3 y^3 + 3 x y^5 & \phi_{6,2}(x,y) &= -x^6 + 15 x^4 y^2 - 15 x^2 y^4 + y^6
\end{align}$

References

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}}

Category:Abstract algebra

Category:Polynomials

harmonic polynomial

Examples

See also

References