有限反射群下多项式不变量非负性的测试集
Test Sets for Nonnegativity of Polynomials Invariant under a Finite Reflection Group
摘要：
集合$S\subset \mathbb{R}^n$是实齐次多项式集合$U$的非负证明，如果$U$中的$F$在$\mathbb{R}^n$上非负当且仅当它在$S$的所有点上非负。我们证明了垂直于根系元素的超平面的并集$Φ\subseteq \mathbb{R}^n$是证明在$Φ$定义的反射群下不变的低次形式的非负性的证据集。证明了对于所有包含乘$-1$的反射群，我们的度界是尖锐的。然后，我们刻画子空间的形式，任意高的程度，这个联盟的超平面是一个非负的证据集。最后，我们提出了有限反射群的半度原理的一个几何推广。
英文摘要：
A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ of real homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ if and only if it is nonnegative at all points of $S$. We prove that the union of the hyperplanes perpendicular to the elements of a root system $Φ\subseteq \mathbb{R}^n$ is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by $Φ$. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by $-1$. We then characterize subspaces of forms of arbitrarily high degree where this union of hyperplanes is a nonnegativity witness set. Finally we propose a conjectural generalization of Timofte's half-degree principle for finite reflection groups.
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