纯态、非负多项式与平方和
Pure states, nonnegative polynomials and sums of squares
摘要：
近年来，人们对证明基本闭半代数集上多项式严格或非严格正性的多项式恒等式进行了系统的研究。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。大多数重要结果需要阿基米德条件，这意味着半代数集必须是紧的。本文将纯态技术引入到交换代数中。我们表明，这种技术允许的方法，最近的阿基米德Stellensaetze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。他们是第一个允许多项式有任意的，不一定离散，零的半代数集。
英文摘要：
In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial on a basic closed semialgebraic set. 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 the semialgebraic set 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 Stellensaetze 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 the polynomial to have arbitrary, not necessarily discrete, zeros on the semialgebraic set.
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