The Marcus-de Oliveira conjecture, bilinear forms, and cones

Abstract

The well-known determinantal cinjecture of de Oliveira and Marcus (OMC) confines the determinant det (X + Y) of the sum of normaln × n matricesX,Y to a certain region in the complex plane. Even the subconjecture obtained by specializing it ton = 4,X Hermitian andY normal is still open. We view the subconjecture as a special case of an assertion concerning a certain family of bilinear forms ofR16 ×C16 and give a method that may prove useful for establishing it for many of such matrix pairs, independent of their spectrum; in particular we apply it successfully in the case of a prominent unitary similarity of Drury's threatening OMC. Unfortunately we find the assertion, extended naturally to pairs of complex arguments to be false and the ideas outlined inapplicable for the general OMC(n=4) case. We also report on some computer experiments, formulate OMC(n=4) as a statement about cones, and find it would be implied by establishing the emptiness of certain semialgebraic sets defined by systems of quadratic and linear relations.