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arXiv · 1503.06034

Matrix Fej\'er-Riesz theorem with gaps

Aljaž Zalar
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Abstract

The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line $\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets $K\subseteq \mathbb{R}$ 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.

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Aljaž Zalar. 2015-03-20. Matrix Fej\'er-Riesz theorem with gaps. https://doi.org/10.1016/j.jpaa.2015.11.018

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