Proof of a conjecture on polynomials preserving nonnegative matrices

2022/01/31 by Raphael Loewy, Loewy, Raphael
Mathematics · #15A18 #15A29 #15B48 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:15A18 #msc:15A29 #msc:15B48

paper · pdf · doi:10.48550/arxiv.2202.00110

5 pages

arxiv created 2022/01/31 · arxiv updated 2022/02/02

Abstract

We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n nonnegative (element-wise) matrix. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and Paparella conjectured that P(n+1) is strictly contained in P(n). We prove this conjecture.

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