Ki Hang Kim, Nicholas S. Ormes, Fred W. Roush
The spectra of nonnegative integer matrices via formal power series
(100K, LATeX 2e)
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.