We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $n\times n$ can be written as a sum of squares $M=Q^TQ$, where $Q$ has size $(n+1)\times n$, which was recently proved by Blekherman-Plaumann-Sinn-Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $M=Q^TQ$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $\det(M)$ as sums of two squares.
In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial $M$ that is positive semidefinite along the real line, is a square, which is known as the matrix Fej\'er-Riesz Theorem.
