Semidefinite Representability of the Trace of Totally Positive Laurent Polynomial Matrix Functions

Abstract. The function that maps a positive semidefinite matrix to the trace of one of its nonnegative integer power is semidefinite representable. In this note, we reduce the size of this semidefinite representation from O(kn) linear matrix inequalities of dimension n, where k is the desired power and n× n the size of the matrix to O(log 2 (k)) linear matrix inequalities of dimension 2n. We also propose a variant of our strategy that can deal with traces of negative powers.