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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.13980 |
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Table of Contents:
- This paper studies the complexity of matrix Putinar's Positivstellens{ä}tz on the semialgebraic set that is given by the polynomial matrix inequality. \rev{When the quadratic module generated by the constrained polynomial matrix is Archimedean}, we prove a polynomial bound on the degrees of terms appearing in the representation of matrix Putinar's Positivstellens{ä}tz. Estimates on the exponent and constant are given. As a byproduct, a polynomial bound on the convergence rate of matrix sum-of-squares relaxations is obtained, which resolves an open question raised by Dinh and Pham. When the constraining set is unbounded, we also prove a similar bound for the matrix version of Putinar--Vasilescu's Positivstellens{ä}tz by exploiting homogenization techniques.