Progress on the Marcus-de Oliveira Determinantal Conjecture

Abstract

We discuss the well known conjecture of Marcus and de Oliveira of the determinant of the sum of normal matrices. We present the motivation of this conjecture, along with detailed history of its origin. Then we compile several situations in which the conjecture holds. That is the matrices satisfying the equation: \begin{equation} \Delta(A,B) \subseteq \co \{\prod_{i=1}^n (\alpha_i + \beta_{\sigma(i)}) : \sigma \in S_n\} \end{equation} We also display further relations and links between Marcus and de Oliveira conjecture and other topics and conjectures, and indicate the cases that will be implied from them. Moreover, we provide some consequential results in various aspects regarding this conjecture.