Harmonic polynomial

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
 whose <a href="/facts/Laplacian/kov9u3PT">Laplacian</a> is zero is termed a harmonic polynomial.
The harmonic polynomials form a subspace of the <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of polynomials over the given <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. In fact, they form a <a href="/facts/Graded_algebra/AIUjkSaP">graded subspace</a>. For the <a href="/facts/Real_number/R02gw5Pb">real field</a> (
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
), the harmonic polynomials are important in mathematical physics.
The Laplacian is the sum of second-order partial derivatives with respect to each of the variables, and is an <a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariant</a> <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a> under the action of the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> via the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of rotations.
The standard <a href="/facts/Separation_of_variables_theorem/oJdkw5xq">separation of variables theorem</a> 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 <a href="/facts/Free_module/o6rC6yoJ">free module</a> over the ring of radial polynomials.

Harmonic polynomial open-in-new

Harmonic polynomial