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) not equal 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 R(0)  = 2 and of Mahler's measure M(alpha), we prove the inequality vertical bar alpha alpha' - 1 vertical bar > (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 alpha is a cubic Pisot number, then the disc vertical bar z vertical bar   (19 alpha(2))(-1).
pisot number algebraic number non-reciprocal polynomial resultant rouche's theorem