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Southern Regional Algebra Conference

SRAC 2017: Advances in Algebra pp 169–181Cite as

A Survey of the Marcus–de Oliveira Conjecture

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 277))

Abstract

We classify and survey the progress on the famous Marcus–de Oliveira determinantal conjecture (MOC) and related problems. The MOC claims that for two normal matrices A and B with eigenvalues \(a_1,\ldots , a_n\) and \(b_1,\ldots ,b_n\), respectively, the set \(\varDelta (A,B)=\{\det (A+UBU^*): U \text { is unitary}\}\) is in the convex hull of the set \(\{\prod _{i=1}^{n} (a_i+b_{\sigma (i)}):\sigma \in S_n\}\). We review the origin and the motivations of this conjecture from M. Fiedler’s work in the Hermitian case to Marcus and de Oliveira’s independent questions. Then, we survey the major positive cases of the MOC in terms of matrix degree, eigenvalues, and other things. We also describe some known properties of the set \(\varDelta (A,B)\), including compactness, connectivity, simply connectivity, convexity, star shapedness, and boundary and corner properties. Finally, we list some extended results that are related to the MOC. The main goal of this article is to provide a fairly comprehensive and brief overview of the progress of the MOC to interested readers and researchers for further explorations of the subject.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Huajun Huang

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  1. Huajun Huang
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Correspondence to Huajun Huang .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Jörg Feldvoss

  2. Department of Mathematics, Spring Hill College, Mobile, AL, USA

    Lauren Grimley

  3. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Drew Lewis

  4. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Andrei Pavelescu

  5. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Cornelius Pillen

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Huang, H. (2019). A Survey of the Marcus–de Oliveira Conjecture. In: Feldvoss, J., Grimley, L., Lewis, D., Pavelescu, A., Pillen, C. (eds) Advances in Algebra. SRAC 2017. Springer Proceedings in Mathematics & Statistics, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-030-11521-0_8

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