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 Aq and Bq 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 Aq or Bq. The characterization immediately implies that Aq ⊂ Bq − 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 B0. From the results of that computation and the characterization proved here, it is now known that all but six of these numbers lie in A2.