On the complexity of matrix Putinar's Positivstellensatz (2406.13980v1)

Abstract: This paper studies the complexity of matrix Putinar's Positivstellensatz on the semialgebraic set that is given by the polynomial matrix inequality. Under the archimedeanness, we prove a polynomial bound on degrees of terms appearing in the representation of matrix Putinar's Positivstellensatz. 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 (the archimedeanness fails), we also prove a similar bound for the matrix version of Putinar--Vasilescu's Positivstellensatz by exploiting homogenization techniques.