Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of similar to the condition of the classical Enestr？m-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers. 1. Introduction A nonzero complex number is called positively algebraic if it is a root of a polynomial all of whose coefficients are positive rational numbers. In 2005, Kuba [1] conjectured that a necessary condition for an algebraic number to be positively algebraic is that none of its conjugates is a positive real number. This conjecture was confirmed affirmatively by Dubickas [2], in 2007, through the following result. Theorem 1 (see [2]). A nonzero complex number is a root of a polynomial with positive rational coefficients if and only if is an algebraic number such that none of its conjugates is a positive real number. In 2009, Brunotte [3] gave an elementary proof of Dubickas-Kuba theorem based on the following lemma ([3, Lemma 2]), which is originated from [4] (see also [5]): if is a polynomial having no nonnegative roots, then there exists such that has only positive coefficients. See also [6], where a bound for the degree of the polynomial with rational positive coefficients was given. We fix the following notation and terminology throughout. Denote by the set of positive real numbers. For , let Some basic properties, which are needed in our work here, modified with the same proofs for from [7], are in the following lemma. Lemma 2. Let and . Then (1) , ;？ , ;(2) , where ;(3) ;(4) and for some has a unique positive zero ; this zero is simple and all other roots have absolute values ;(5)let . Then for all ;(6)([7, Lemma 2]) are positive real numbers, and , where ;(7)([7, Lemma 3]) is a complex number which is not real positive is a root of a polynomial in for any . Our first main result is a certain quantitative improvement of Dubickas-Kuba theorem (Theorem 1 above). Theorem 3. Let be a nonzero algebraic number of degree (over ), let be all its conjugates, and let Then all conjugates of are in if and only if there exists such that . The nontrivial half of Theorem 3 is its necessity part, and its main difficulty is to show the existence of a polynomial in all of whose coefficients are rational numbers. Should we need only a