On Roots of Polynomials\with Positive Coefficients

Toufik Za\'{\i}mi

Let $\alpha $ be an algebraic number with no nonnegative conjugates over the field of the rationals. Settling a recent conjecture of Kuba, Dubickas proved that the number $\alpha$ is a root of a polynomial, say $P$, with positive rational coefficients. We give in this note an upper bound for the degree of $P$ in terms of the discriminant, the degree and the Mahler measure of $\alpha$; this answers a question of Dubickas.