98-559 Ki Hang Kim, Nicholas S. Ormes, Fred W. Roush

The spectra of nonnegative integer matrices via formal power series (100K, LATeX 2e) Aug 10, 98

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Abstract. We characterize the $d$-tuples of nonzero complex numbers $\spectrum$ which can occur as the nonzero part of the spectrum of a matrix with nonnegative integer/rational entries. These results follow easily from our main theorem: a characterization of the possible nonzero portions of spectra of primitive integer matrices (the integer case of Boyle and Handelman's Spectral Conjecture). For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum $\spectrum$ 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 $t\integers_+[t]$ satisfying some technical conditions and $r$ is a formal power series in $t\integers_+[[t]]$. 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 non-positivity in nonzero degree terms.

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