On the characteristic polynomial of a supertropical adjoint matrix

2015/10/07 by Yaroslav Shitov, Shitov, Yaroslav
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Optical Network Technologies #Polynomial and algebraic computation #math.CO

paper · pdf · doi:10.48550/arxiv.1510.02143

note, 3 pages

arxiv created 2015/10/07 · openalex publication_date 2015/10/07 · arxiv updated 2015/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let χ(A) denote the characteristic polynomial of a matrix A over a field; a standard result of linear algebra states that χ(A^-1) is the reciprocal polynomial of χ(A). More formally, the condition χⁿ(X) χ^k(X^-1)=χ^n-k(X) holds for any invertible n× n matrix X over a field, where χⁱ(X) denotes the coefficient of λ^n-i in the characteristic polynomial det(λI-X). We confirm a recent conjecture of Niv by proving the tropical analogue of this result.

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