On the corners of certain determinantal ranges Original Research Article

Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define Δ(A)={det(Aring operatorQ):Qset membership, variantSO(n)}, where ring operator denotes the Hadamard product of matrices. For a permutation σ on {1,…,n}, define image It is shown that if the equation zσ=det(Aring operatorQ) has in SO(n) only the obvious solutions (Q=(εiδσi,j), εi=±1 such that ε1…εn=sgnσ), then the local shape of Δ(A) in a vicinity of zσ resembles a truncated cone whose opening angle equals image, where σ1, σ2 differ from σ 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.