Abstract
We characterize (by elementary conditions) those $k$-tuples of complex numbers which are the nonzero portion (including multiplicities) of the spectrum of a nonnegative real matrix. The proof relies on methods and results from symbolic dynamics.
More generally, let S be a unital subring of the reals. We conjecture that certain elementary necessary conditions are sufficient for a $k$-tuple $\Delta$ of complex numbers to be the nonzero portion of the spectrum of a primitive matrix over S. (The general nonnegative case would follow easily from the primitive case-but not conversely.) We verify this under the additional condition that some subtuple of $\Delta$ containing its maximal (real) entry be the nonzero portion of the spectrum of a primitive matrix over S. In particular, if the maximal entry of $\Delta$ is in $\mathrm{S}$ or is quadratic over it, the elementary necessary conditions are sufficient. As one application (with $\mathrm{S} = \mathbf{Z}$), we characterize the zeta functions of mixing finitely presented dynamical systems.
