Negative Pisot and Salem numbers as roots of Newman polynomials

Abstract

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 $\{z\in\c: \tau^{-1} \lt |{z}| \lt \tau\}\setminus\r^+$, where $\tau$ 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 $(-\tau,-1)$ 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 $(-\tau,-1)$ with degree at most $20$, and verify that every one of these is satisfied by a Newman polynomial.