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
Z
{\mathbb Z}
and
Q
{\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
(
λ
1
,
λ
2
,
…
,
λ
d
)
(\lambda _1,\lambda _2,\ldots ,\lambda _d)
to factoring the polynomial
∏
i
=
1
d
(
1
−
λ
i
t
)
\prod _{i=1}^d (1-\lambda _it)
as a product
(
1
−
r
(
t
)
)
∏
i
=
1
n
(
1
−
q
i
(
t
)
)
(1-r(t))\prod _{i=1}^n (1-q_i(t))
where the
q
i
q_i
’s are polynomials in
t
Z
+
[
t
]
t{\mathbb Z}_+[t]
satisfying some technical conditions and
r
r
is a formal power series in
t
Z
+
[
[
t
]
]
t{\mathbb Z}_+[[t]]
. To obtain the factorization, we present a hierarchy of estimates on coefficients of power series of the form
∏
i
=
1
d
(
1
−
λ
i
t
)
/
∏
i
=
1
n
(
1
−
q
i
(
t
)
)
\prod _{i=1}^d (1-\lambda _it)/\prod _{i=1}^n (1-q_i(t))
to ensure nonpositivity in nonzero degree terms.