The de Oliveira-Marcus Conjecture, Bilinear Forms, and Cones

We view the well known determinantal conjecture of de Oliveira and Marcus as a special case of an assertion concerning certain families of bilinear forms. While indicating how this idea might prove valuable for establishing the Oliveira-Marcus conjecture for Hermitian-normal pairs of 4x4 matrices, we show that this approach will completely fail in general. We then formulate the conjecture as a statement about cones and report on some numeric experiments. In this setting we also find that the conjecture would be implied by establishing the emptyness of a certain semialgebraic set defined by a system of quadratic and linear relationsAvailable from Departamento de Matematica, Universidade de Coimbra, 3000 Coimbra, Portugal / FCT - Fundação para o Ciência e a TecnologiaSIGLEPTPortuga