Characterization of matrix-exponential distributions

The class of matrix-exponential distributions can be equivalently defined as the class of all distributions with rational Laplace-Stieltjes transform. An immediate question that arises is: when does a rational Laplace-Stieltjes transform correspond to a matrix-exponential distribution? For a rational Laplace-Stieltjes transform that has a pole of maximal real part that is real and negative, we give a geometric description of all admissible numerator polynomials that give rise to matrix-exponential distributions. Using this approach we give a complete characterization for all matrix-exponential distributions of order three.