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On roots of polynomials with positive coefficients. (English) Zbl 1172.11036

The paper under review establishes a sufficient and necessary condition for an algebraic number \(\alpha\) to be a root of a polynomial with positive rational coefficients. G. Kuba observed in [Arch. Math. 85, No. 1, 70–78 (2005; Zbl 1075.11064)], Lemma 1 that if \(\alpha\) or one of its conjugates is a positive real number, then it cannot be the root of a polynomial with positive rational coefficients. The author of the present paper shows that this is indeed a sufficient condition: the following is the main result of the paper.
Theorem: A complex number \(\alpha\) is a root of a polynomial with positive rational coefficients if and only if \(\alpha\) is an algebraic number over \(\mathbb{Q}\) such that note of its conjugates over \(\mathbb{Q}\) is a nonnegative real number.
Observe that here no attention is paid on the irreducibility of the polynomial and the statement is not about the coefficients of the minimal polynomial for \(\alpha\), but only about the existence of a polynomial fulfilling the wished condition.
The proof of the theorem consists in modifying the minimal polynomials of \(\alpha\) over \(\mathbb{Q}\) in a way that all the coefficients become positive. This is done through the key Lemma 2: this lemma states that for every \(\omega\in\mathbb{C}\) such that \(\operatorname{Im}(\omega)>0\), there exists a polynomial \(T(X)\in\mathbb{Q}[X]\) such that \((X-\omega)(X-\bar{\omega})T(X)\) has positive coefficients. The result is analytic in nature, as it holds for all \(\omega\), no matter if they are algebraic or transcendental. It is proved by combining a classical epsilon-delta argument with the uniform distribution in \([0,1)\) of the fractional parts of \(\{m\theta/2\pi\}\) for \(m\in\mathbb{N}\) and \(0<\theta<\pi/2\). Unfortunately, this last statement is ineffective in the sense that it does not provide for a bound (depending on \(\theta\)) on \(m\) such that the fractional part of \(m\theta/2\pi\) is at least \(1/3\), which is what the author needs. This reflects on the degree of \(T(X)\), so that this degree cannot be controlled in terms of \(\omega\), and the paper ends with an open question about an algorithm which, given \(\alpha\) such that none of its conjugates is real and positive, produces a polynomial with real positive coefficients having \(\alpha\) as a root and of minimal possible degree.

MSC:

11R09 Polynomials (irreducibility, etc.)
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
11R04 Algebraic numbers; rings of algebraic integers

Citations:

Zbl 1075.11064
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References:

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