The Cone Generated by Positive Semidefinite Mean Polynomials

Abstract

We study the cone ${\mathcal{M}}_{n,2d}$ of nonnegative mean polynomials---real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares ${\Sigma }_{n,2d}$ and the cone of sums of nonnegative circuit polynomials ${\mathcal{C}}_{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in ${\mathcal{M}}_{4,4}$ but outside ${\mathrm{SOSONC}}_{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.