Pure states, nonnegative polynomials, and sums of squares
Pure states, nonnegative polynomials, and sums of squares
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纯态、非负多项式和平方和
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
M. Schweighofer
中科院分区：
文献类型：
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作者：
Sabine Burgdorf;C. Scheiderer;M. Schweighofer
近年来，人们对证明基本闭集K ∈ R上多项式f的严格或非严格正性的多项式恒等式作了系统的研究。对这些恒等式的兴趣不仅源于它们在多项式优化中的重要性。大多数的重要结果需要阿基米德条件，这意味着K必须是紧的。本文将纯态技术引入交换代数。我们表明，这种技术允许的方法，最近的阿基米德Stellensatze是相当容易和概念比以前的证明。特别是，我们谴责和加强一些最重要的成果，从过去几年。此外，我们建立了几个这样的结果是全新的。它们是第一个允许f在K中有任意的，不一定是离散的零的。
In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial f on a basic closed set K ⊂ R. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that K has to be compact. This paper introduces the technique of pure states into commutative algebra. We show that this technique allows an approach to most of the recent archimedean Stellensatze that is considerably easier and more conceptual than the previous proofs. In particular, we reprove and strengthen some of the most important results from the last years. In addition, we establish several such results which are entirely new. They are the first that allow f to have arbitrary, not necessarily discrete, zeros in K.