A Hyper-Geometric Approach to the BMV-Conjecture

We prove positivity of the BMV measure in dimension d = 3 in several non-trivial cases by combinatorial methods. The second author was supported by the Wittgenstein prize program Z-36. Much of this work has been done, while he was visiting the university Paris 9. He thanks I. Ekeland and E. Jouini for their hospitality. The third author had the opportunity to spend one month at the university Paris 7 and three weeks at CREST. He thanks in particular for the kind hospitality of Laurence Carassus and Nizar Touzi. The initial motivation for our research on the problem happened during conversations with Peter Michor and Martin Feldbacher. The authors are indebted to M. Fannes and R. Werner who kindly offered us a preprint on known results concerning the BMV-conjecture. The authors also thank the anonymous referee for her/his comments.