A Characterization of Two Related Classes of Salem Numbers

Abstract We investigate certain classes of Salem numbers which arise naturally in the study of Salem′s construction of these numbers. Let P * denote the reciprocal of the polynomial P . It is known that all Salem numbers satisfy equations of the form zP ( z ) + P *( z ) = 0, where P can be chosen to be a polynomial with all zeros outside the unit circle (class A) or else P has exactly one zero outside the unit circle and the rest strictly inside (class B). The classes A q and B q considered here are defined by specifying, in addition, that the value of | P (0)| should equal q . We give an intrinsic characterization of these classes which enables one to demonstrate that a given Salem number τ is in A q or B q . The characterization immediately implies that A q ⊂ B q − 2 for q ≥ 2. It had been shown previously by extensive computation that all but two of the known Salem numbers less than 13/10 lie in B 0 . From the results of that computation and the characterization proved here, it is now known that all but six of these numbers lie in A 2 .