Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The spectra of nonnegative integer matrices via formal power series


Authors: Ki Hang Kim, Nicholas S. Ormes and Fred W. Roush
Journal: J. Amer. Math. Soc. 13 (2000), 773-806
MSC (1991): Primary 15A18; Secondary 15A36, 58F03, 58F20
Published electronically: June 21, 2000
MathSciNet review: 1775737
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 15A18, 15A36, 58F03, 58F20

Retrieve articles in all journals with MSC (1991): 15A18, 15A36, 58F03, 58F20


Additional 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
Email: ormes@math.utexas.edu

Fred W. Roush
Affiliation: Mathematics Research Group, Alabama State University, Montgomery, Alabama 36101-0271
Email: froush@gmail.alasu.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00342-8
PII: S 0894-0347(00)00342-8
Keywords: Spectrum of nonnegative matrix, zeta function of subshift of finite type
Received by editor(s): August 19, 1998
Received by editor(s) in revised form: February 1, 2000
Published electronically: June 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society