## The spectra of nonnegative integer matrices via formal power series

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- 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
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## 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
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**13**(2000), 773-806 - MSC (1991): Primary 15A18; Secondary 15A36, 58F03, 58F20
- DOI: https://doi.org/10.1090/S0894-0347-00-00342-8
- MathSciNet review: 1775737