On the degree and half-degree principle for symmetric polynomials
On the degree and half-degree principle for symmetric polynomials
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关于对称多项式的一次和半次原理
DOI：
10.1016/j.jpaa.2011.08.012
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发表时间：
2010
影响因子：
0.8
通讯作者：
C. Riener
中科院分区：
文献类型：
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作者：
C. Riener
在这篇笔记中，我们的目的是给出一个新的，初步证明的陈述，首先证明了由Zerofte（2003）[15]。它说n元d次对称真实的多项式F在Rn上（或在R≥ 0 n上）是正的当且仅当它在至多具有max{d/2 <$，2}个不同分量的点的子集上是非负的。我们推导出的原始声明的推论稍微更一般的声明对称优化问题。我们用来证明这个陈述的想法是将它与轨道空间中的线性优化问题联系起来。事实上，对于对称群Snthis的情况下，可以被看作是一个问题，规范化的一元真实的多项式只有真实的根，使我们能够得出结论的定理在一个非常基本的方式。我们希望，这里提出的方法将使人们有可能在其他群体的情况下得出类似的声明。
In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte (2003) [15]. It says that a symmetric real polynomial F of degree d in n variables is positive on Rn(or on R≥0n) if and only if it is non-negative on the subset of points with at most max{⌊d/2⌋,2} distinct components. We deduce Timofte’s original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea that we are using to prove this statement is that of relating it to a linear optimization problem in the orbit space. The fact that for the case of the symmetric group Snthis can be viewed as a question on normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way. We hope that the methods presented here will make it possible to derive similar statements also in the case of other groups.