  A univariate trace polynomial is a polynomial in a variable x and formal
trace symbols Tr(x^j). Such an expression can be naturally evaluated on
matrices, where the trace symbols are evaluated as normalized traces. This
paper addresses global and constrained positivity of univariate trace
polynomials on symmetric matrices of all finite sizes. A tracial analog of
Artin's solution to Hilbert's 17th problem is given: a positive semidefinite
univariate trace polynomial is a quotient of sums of products of squares and
traces of squares of trace polynomials.
