Pisot Numbers from {0, 1}-Polynomials

Abstract

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.