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This is true for quadratic polynomials with constant term +1. Any such polynomial is the determinant of a matrix in SL2 ℤ (e.g. using the companion matrix as indicated by Ben Webster). It's well known that any such matrix is conjugate to a multiple product of [[1,1],[0,1]] and [[1,0],[1,1]] (upper and lower triangular unipotent matrices). I'm not sure the original reference for this fact, but a reference is Proposition 2.1 of this paper.

I believe that your criterion implies that the maximal root of the polynomial is a Perron number. If so, then Lind has shown that every Perron number occurs as the spectral radius of a non-negative integral Perron-Frobenius matrix (and therefore the spectral radius of a recurrent digraph). This only implies that the polynomial divides the characteristic polynomial of the matrix - there might be other factors.

Added comment: The general quadratic case might be possible to work out using Markov partitions of the induced map of a torus.

I forgot about the cyclotomic case, which can occur if the matrix is not Perron-Frobenius. If the polynomial is irreducible, I think the condition implies that the maximal norm roots are complex Perron numbers (or cyclotomic). These crop up in work of Kenyon on self-similar tilings (MR1392326 (97j:52025) ).

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This is true for quadratic polynomials with constant term +1. Any such polynomial is the determinant of a matrix in SL2 ℤ (e.g. using the companion matrix as indicated by Ben Webster). It's well known that any such matrix is conjugate to a multiple product of [[1,1],[0,1]] and [[1,0],[1,1]] (upper and lower triangular unipotent matrices). I'm not sure the original reference for this fact, but a reference is Proposition 2.1 of this paper.

I believe that your criterion implies that the maximal root of the polynomial is a Perron number. If so, then Lind has shown that every Perron number occurs as the spectral radius of a non-negative integral Perron-Frobenius matrix (and therefore the spectral radius of a recurrent digraph). This only implies that the polynomial divides the characteristic polynomial of the matrix - there might be other factors.