Regular and positive noncommutative rational functions:

Call a noncommutative (nc) rational function r regular if it has no singularities, that is, r(X) is defined for all tuples of self-adjoint matrices X. In this paper, regular nc rational functions r are characterized via the properties of their (minimal size) linear systems realizations r=b∗ L^-1c. It is shown that r is regular if and only if L = A₀ +σ_j A_jx_j is free elliptic. Roughly speaking, a linear pencil L is free elliptic if, after a finite sequence of basis changes and restrictions, the real part of A0 is positive definite and the other Aj are skew-adjoint. The second main result is a solution to an nc version of Hilbert's 17th problem: a positive regular nc rational function is a sum of squares.