## 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

Zbl 1075.11064
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