# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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## The spectra of nonnegative integer matrices via formal power seriesHTML articles powered by AMS MathViewer

by Ki Hang Kim, Nicholas S. Ormes and Fred W. Roush
J. Amer. Math. Soc. 13 (2000), 773-806
DOI: https://doi.org/10.1090/S0894-0347-00-00342-8
Published electronically: June 21, 2000

## Abstract:

We characterize the possible nonzero spectra of primitive integer matrices (the integer case of Boyle and Handelman’s Spectral Conjecture). Characterizations of nonzero spectra of nonnegative matrices over ${\mathbb Z}$ and ${\mathbb Q}$ follow from this result. For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum $(\lambda _1,\lambda _2,\ldots ,\lambda _d)$ to factoring the polynomial $\prod _{i=1}^d (1-\lambda _it)$ as a product $(1-r(t))\prod _{i=1}^n (1-q_i(t))$ where the $q_i$’s are polynomials in $t{\mathbb Z}_+[t]$ satisfying some technical conditions and $r$ is a formal power series in $t{\mathbb Z}_+[[t]]$. To obtain the factorization, we present a hierarchy of estimates on coefficients of power series of the form $\prod _{i=1}^d (1-\lambda _it)/\prod _{i=1}^n (1-q_i(t))$ to ensure nonpositivity in nonzero degree terms.
References
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Bibliographic Information
• Ki Hang Kim
• Affiliation: Mathematics Research Group, Alabama State University, Montgomery, Alabama 36101-0271 and Korean Academy of Science and Technology
• Email: kkim@gmail.alasu.edu
• Nicholas S. Ormes
• Affiliation: Department of Mathematics, C1200, University of Texas, Austin, Texas 78712
• Address at time of publication: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
• MR Author ID: 620777
• Email: ormes@math.utexas.edu
• Fred W. Roush
• Affiliation: Mathematics Research Group, Alabama State University, Montgomery, Alabama 36101-0271
• Email: froush@gmail.alasu.edu
• Received by editor(s): August 19, 1998
• Received by editor(s) in revised form: February 1, 2000
• Published electronically: June 21, 2000