On the corners of certain determinantal ranges

AbstractLet 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(A∘Q):Q∈SO(n)}, where ∘ denotes the Hadamard product of matrices. For a permutation σ on {1,…,n}, define zσ=dσ(A)=∏i=1naiσ(i). It is shown that if the equation zσ=det(A∘Q) 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 zσ1zσ^zσ2, 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