Matrix Fejér-Riesz theorem with gaps

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• 作者：Aljaž Zalar ^{aljaz.zalar@imfm.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14P10 ; 13J30 ; 47A56
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：220
• 期：7
• 页码：2533-2548
• 全文大小：421 K

文摘

The matrix Fejér–Riesz theorem characterizes positive semidefinite matrix polynomials on the real line R$\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets K⊆R$K\subseteq \mathbb{R}$ by the use of matrix preorderings from real algebraic geometry. In the compact case a denominator-free characterization exists, while in the non-compact case there are counterexamples. However, there is a weaker characterization with denominators in the non-compact case. At the end we extend the results to algebraic curves.