On the corners of certain determinantal ranges

Abstract

Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define [Delta](A)={det(AoQ):Q[set membership, variant]SO(n)}, where o denotes the Hadamard product of matrices. For a permutation [sigma] on {1,...,n}, define It is shown that if the equation z[sigma]=det(AoQ) has in SO(n) only the obvious solutions (Q=([epsilon]i[delta][sigma]i,j), [epsilon]i=±1 such that [epsilon]1...[epsilon]n=sgn[sigma]), then the local shape of [Delta](A) in a vicinity of z[sigma] resembles a truncated cone whose opening angle equals , where [sigma]1, [sigma]2 differ from [sigma] by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n×n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.