NEGATIVE PISOT AND SALEM NUMBERS AS ROOTS OF NEWMAN POLYNOMIALS

A Newman polynomial has all its coefficients in {0, 1} and constant term 1. It is known that every root of a Newman polynomial lies in the slit annulus {z ∈ C : τ−1 < |z| < τ} \ R+, where τ denotes the golden ratio, but not every polynomial having all of its conjugates in this set divides a Newman polynomial. We show that every negative Pisot number in (−τ,−1) with no positive conjugates, and every negative Salem number in the same range obtained by using Salem's construction on small negative Pisot numbers, is satisfied by a Newman polynomial. We also construct a number of polynomials having all their conjugates in this slit annulus, but that do not divide any Newman polynomial. Finally, we determine all negative Salem numbers in (−τ,−1) with degree at most 20, and verify that every one of these is satisfied by a Newman polynomial.