We analyze the number of zeros of det(F(a)), where F(a) is the matrix exponent of a Markov Additive Process (MAP) with one-sided jumps. The focus is on the number of zeros in the right half of the complex plane, where F(a) is analytic. In addition, we also consider the case of a MAP killed at an independent exponential time. The corresponding zeros can be seen as the roots of a generalized Cramér–Lundberg equation. We argue that our results are particularly useful in fluctuation theory for MAPs, which leads to numerous applications in queueing theory and finance.