The spectra of nonnegative integer matrices via formal power series

Kim, Ki¹ ; Ormes, Nicholas^2, 3 ; Roush, Fred⁴

¹ Mathematics Research Group, Alabama State University, Montgomery, Alabama 36101-0271 and Korean Academy of Science and Technology
² Department of Mathematics, C1200, University of Texas, Austin, Texas 78712
³ Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
⁴ Mathematics Research Group, Alabama State University, Montgomery, Alabama 36101-0271

Journal of the American Mathematical Society, Tome 13 (2000) no. 4, pp. 773-806

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-00-00342-8

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