TITLE: Signed factors of harmonic polynomials
QUESTION [10 upvotes]: Let ${\rm Harm}_n^d$ be the space of real harmonic polynomials in $n$ variables, homogeneous of degree $d$. If $P\in{\rm Harm}_n^d$, then
$$\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2}\right)P=0.$$
A harmonic polynomial is not necessarily irreducible in ${\mathbb R}[X_1,\ldots,X_n]$. For instance every non-zero $P\in{\rm Harm}_2^4$ splits as the product of two quadratic forms; it turns out that none of them is positive definite. Besides, it is not two difficult to show that if $X_1^2+\cdots+X_n^2$ divides a harmonic polynomial $P$, then $P=0$. These observations lead me to me following question:

Is it possible that a non-zero harmonic polynomial (say homogeneous) factorizes $P=QR$ in ${\mathbb R}[X_1,\ldots,X_n]$, with the factor $Q$ being non-constant and  positive definite (i.e. $Q(x)>0$ for every $x\ne0$) ?

I incline toward a negative answer, of course.

REPLY [5 votes]: S. Kharlamov pointed out to me that it is a consequence of the diagonalization of the Laplacian $\Delta_S$ over the unit sphere. Its eigenvalues are the integers $\lambda_d=d(d+n-2)$, and the corresponding eigenspace $E_d$ is given by the trace over $S$ of the harmonic polynomials of degree $d$. Finally, the space of traces of polynomials of degree $\le d$ is the sum of the $E_{d-2k}$ for $k=0,\ldots,[d/2]$. This implies that 
${\rm Harm}_n^d$ is orthogonal to ${\rm Hom}_n^{d-2k}$, the space of homogeneous polynomials of degree $d-2k$, whenever $k=1,\cdots,[d/2]$. The orthogonality refers to the $L^2$-scalar product over $S$:
$$(f,g):=\int_Sf(x)g(x)d\sigma(x).$$
Suppose now that $P$ is harmonic of degree $d$ and it splits as $QR$, with $Q$ positive definite and non constant. Then $Q$ has degree $2k$ for some $k\ge1$. We have seen that $(P,R)=0$, which means
$$\int_SQ(x)R(x)^2d\sigma(x)=0.$$
Because $Q$ has a constant sign, this implies $QR^2\equiv0$, that is $P=0$.
Of course, instead of integrating over the unit sphere, one may integrate against the standard Gaussian measure. A classical trick in several topics, including random matrices theory.