Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Question

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of squares of $d$ dimensional Fourier harmonics up to degree $n$.

My question is if $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$ and can be represented as combination of spherical harmonics dimension $d$, then does there exist some spherical harmonic polynomials $g_1,\ldots, g_k$ of degree $n$ such that $p=g_1^2+\cdots g_k^2$ is a sum of squares?

i edit the question after Zach Teitler's comment.

The interval $[-\pi,\pi]^d$ means we concern the trigonometric polynomials positive on frequency domains.

The optimization problems about the polynomials positive on frequency domain $[-\pi,\pi]^d$ can be implemented via SDP approach(Gram matrix Rpresentation).

Given a positive polynomial represented as combination of spherical harmonics dimension $d$, Obviously, it is sum of squares of $d$ dimensional Fourier harmonics. Furthermore, it implies the symmetry relationship between $[-\pi,\pi] \times [0,\pi]$ and $[-\pi,\pi] \times [-\pi,0]$ on 2-sphere as an example. May be there is less information on sphere than cube? So, is it the sum of squares of spherical harmonics?

edit after Zach Teitler's answer

My appoligize, the theroy of positive polynomial on $[-\pi,\pi]^d$ is from Dumitrescu, “Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design,” IEEE Transactions on Signal Processing, 2006 Theorem 1. That is

Given $z=[z_1,\ldots, z_d]$ and $z^k=z_1^{k_1}z_2^{k_2}\ldots z_d^{k_d}$, a Hermitian trigonommetric polynomial of degree $n$ that

$R(z)=\sum_{k=-n}^n r_k z^{-k}, r_{-k}=r_k^*$

is positive on unit $d$-circle, i.e. $z_i=e^{j\theta_i}, \theta_i \in[-\pi,\pi]^d$ , then it is sum of squares.

My question is on the relationship between sum of suqare polynomial and the polynomial that sum of squares spherical harmonics.

Thank you very much again!

Please feel free to provide any advices. Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

Answer 1

This Wikipedia page has many references: https://en.wikipedia.org/wiki/Positive_polynomial, including for example

• Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008.
• B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75-97.

It is not true that positivity on the unit sphere implies representation as a sum of squares of polynomials. For example $M(x,y,z) = x^4 y^2 + x^2 y^4 + z^6 - 3 x^2 y^2 z^2$ is a nonnegative homogeneous form which cannot be written as a sum of squares. (This is called the "Motzkin form". See for example http://www.msri.org/attachments/workshops/327/553_Lecture-notes_week1_Blekherman.pdf.)

Note that for a homogeneous form $p$ the following are equivalent:

• $p \geq 0$ on the unit sphere
• $p \geq 0$ on a neighborhood of the origin
• $p \geq 0$ everywhere (globally)

so there is no difference between assuming that $p \geq 0$ on the unit sphere, or on a cube such as $[-\pi,\pi]^d$. It is the same hypothesis. In either case, it does not imply that $p$ is a sum of squares of polynomials.

If you want to ask about $p$ being a sum of squares of rational functions, or if $p$ is not homogeneous, then that is a different situation.