Matrix Fej\'er-Riesz type theorem for a union of an interval and a point

The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In the previous work of the second-named author this was extended to the characterization on arbitrary closed semialgebraic sets $K$ in $\mathbb{R}$ by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when $K$ is the whole line, an unbounded interval, a union of two unbounded intervals, and it was conjectured also when $K$ is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem (TMMP) on a union of a bounded interval and a point. The presented technique for solving the corresponding TMMP can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets $K$.