On the location of roots of non-reciprocal integer polynomials

Let P be a polynomial of degree d with integer coefficients such that P(0) ≠ 0. Assuming that P has no reciprocal factors we obtain a lower bound on the modulus of the smallest root of P in terms of its degree d, its Mahler measure M(P) and the number of roots of P lying outside the unit circle, say, k. We derive from this that all d roots of P must lie in the annulus R0 < |z| < R1, where R0 = R0(d, k, M(P)) and R1 = R1(d, k, M(P)) are given explicitly. As an application, for non-reciprocal conjugate algebraic numbers α, α′ of degree d ≥ 2 and of Mahler’s measure M(α), we prove the inequality $${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}$$. Some lower bounds on the moduli of the conjugates of a Pisot number are also given. In particular, it is shown that if α is a cubic Pisot number, then the disc |z| ≤ α−1 + 0.1999α−2 contains no conjugates of α. Here the constant 0.1999 cannot be replaced by the constant 0.2. We also show that if α is a Pisot number of degree at least 4 and α′ is its conjugate, then |αα′ − 1| > (19α2)−1.