Английская Википедия:Harmonic polynomial
In mathematics, in abstract algebra, a multivariate polynomial Шаблон:Math over a field such that the Laplacian of Шаблон:Math 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 Шаблон:Fact 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-<math>d</math> univariate polynomial <math>p(x) := \textstyle\sum_{k=0}^d a_k x^k</math>. In order to be harmonic, this polynomial must satisfy
<math display="block">0 = \tfrac{\partial^2}{\partial x^2} p(x) = \sum_{k=2}^d k(k-1) a_k x^{k-2}</math>
at all points <math>x \in \mathbb{R}</math>. In particular, when <math>d=2</math>, we have a polynomial <math>p(x) = a_0 + a_1 x + a_2 x^2</math>, which must satisfy the condition <math>a_2 = 0</math>. Hence, the only harmonic polynomials of one (real) variable are affine functions <math>x \mapsto a_0 + a_1 x</math>.
In the multivariable case, one finds nontrivial spaces of harmonic polynomials. Consider for instance the bivariate quadratic polynomial
<math display="block">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, </math>
where <math>a_{0,0}, a_{1,0}, a_{0,1}, a_{1,1}, a_{2,0}, a_{0,2}</math> are real coefficients. The Laplacian of this polynomial is given by
<math display="block">\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}).</math>
Hence, in order for <math>p(x,y)</math> to be harmonic, its coefficients need only satisfy the relationship <math>a_{2,0} = -a_{0,2}</math>. Equivalently, all (real) quadratic bivariate harmonic polynomials are linear combinations of the polynomials
<math display="block"> 1, \quad x, \quad y, \quad xy, \quad x^2 - y^2. </math>
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:
<math> \begin{array}{rcl} \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{array} </math>
See also
References
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite arXiv
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Cf. Corollary 1.8 of Шаблон:Citation
- Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963) Шаблон:Doi
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