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Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and homogeneous harmonic polynomials $g_1,\ldots, g_k$ of degree $m+d$ such that $|x|^{2m}p=g_1^2+\cdots g_k^2$ is a sum of squares. Here $|x|^2=x_1^2+\cdots x_n^2$ is the standard Euclidean metric.

A theorem of Reznick states that for large $m$, homogenous $g_i$ can always to found, but these are not necessarily harmonic.

Since harmonic polynoimals are $L^2$ on the sphere with the round metric, we have the following geometric interpretation. The map $G=(g_1,\ldots, g_k):S\to {\mathbb{R}}^k$ is an algebraic $L^2$ isometric map in the sense that the pullback $G^* |\bullet |=p$.

REMARK ON EDIT: In the original question, the given $p$ was assumed to be harmonic. After Andrew answer, I removed this assumption and rewrote the question.

REMARK ON DOUBLE EDIT: I edited the question again after Gerry's comment.

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  • $\begingroup$ If $p$ is a polynomial, positive on the unit sphere, then isn't $\sqrt p$ in $L^2$ of the sphere? $\endgroup$
    – Gerry Myerson
    Commented Jun 6, 2011 at 7:09
  • $\begingroup$ Thanks Gerry for that note. I seem to have difficulty with phrasing my problem. The polynomial $\sqrt{p}$ would be an infinite linear combination of harmonic polynomials. What I really wish is to restrict to only finite linear combinations. $\endgroup$
    – user2529
    Commented Jun 6, 2011 at 11:49

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The average of a nonconstant homogeneous harmonic polynomial over the unit sphere (with the center at the origin) is zero and so it cannot be positive everywhere on $S$.

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