Polynomial convolutions in max-plus algebra

Amnon Rosenmann, Franz Lehner, Aljosa Peperko

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

Abstract

Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava and then Marcus studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant and the maximum is the analogue of the expected value. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots.
Originalspracheenglisch
Seiten (von - bis)370-401
Seitenumfang32
FachzeitschriftLinear algebra and its applications
Jahrgang578
DOIs
PublikationsstatusVeröffentlicht - 1 Okt 2019

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Max-plus Algebra
Convolution
Polynomial
Free Probability
Analogue
Interlacing
Characteristic polynomial
Random Matrices
Expected Value
Determinant
Roots
Invariant

Schlagwörter

    Dies zitieren

    Polynomial convolutions in max-plus algebra. / Rosenmann, Amnon; Lehner, Franz; Peperko, Aljosa.

    in: Linear algebra and its applications, Jahrgang 578, 01.10.2019, S. 370-401.

    Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

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    AU - Lehner, Franz

    AU - Peperko, Aljosa

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    AB - Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava and then Marcus studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant and the maximum is the analogue of the expected value. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots.

    KW - Max-plus algebra

    KW - Maxplus polynomial convolution

    KW - Maxplus characteristic polynomial

    KW - Hadamard product

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