AI 中文总结
该研究定义了半正定均值多项式生成的锥，证明其相关预素强生成性，构造了分离实例，并提出基于该锥的多项式优化收敛下界层级。
AI 中文摘要
我们研究非负均值多项式构成的锥$\boldsymbol{\textit{M}}_{n,2d}$，即可表示为加权幂均值$M_{q,p}(Y,w)$（其中$q>p$）的$n$元$2d$次实型多项式。该锥同时推广了平方和锥$\boldsymbol{\textit{Σ}}_{n,2d}$与非负回路多项式和锥$\boldsymbol{\textit{C}}_{n,2d}$。我们证明任意多项式的平方均属于均值多项式预素$T_{\text{mean}}$，且$T_{\text{mean}}$是强生成的，因此在紧半代数集上严格正定的每个多项式都可通过均值多项式证书表示。我们以Robinson型$\boldsymbol{\textit{\textit{Ř}}}$为例，其属于$\boldsymbol{\textit{M}}_{4,4}$但不属于$\text{SOSONC}_{4,4}$，作为分离实例。最后，我们概述了基于均值多项式锥的多项式优化下界收敛层级，并讨论了通过符号规划实现的易处理深度截断近似。
英文摘要
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 $Σ_{n,2d}$ 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.
