Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

Vichian Laohakosol,Suton Tadee

doi:10.1155/2014/296828

Vichian Laohakosol, Suton Tadee

Open Access

https://doi.org/10.1155/2014/296828

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Abstract

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial f 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 f similar to the condition of the classical Eneström-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.

Highlights

A nonzero complex number γ 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 γ to be positively algebraic is that none of its conjugates is a positive real number
A nonzero complex number γ is a root of a polynomial with positive rational coefficients if and only if γ is an algebraic number such that none of its conjugates is a positive real number

Summary

Introduction

A nonzero complex number γ is called positively algebraic if it is a root of a polynomial all of whose coefficients are positive rational numbers. A nonzero complex number γ is a root of a polynomial with positive rational coefficients if and only if γ is an algebraic number such that none of its conjugates is a positive real number. Should we need only a polynomial with nonnegative real coefficients, this necessity part follows from Roitman-Rubinstein’s result [7, Lemma 4] but with a bound for r which depends on the conjugates of α and on other roots. We show that the bound |γ| is optimal for a class of algebraic numbers without nonnegative conjugates The investigation in this second part leads us to some interesting connections with the classical result of Enestrom-Kakeya [8], which gives upper and lower bounds for the absolute values of the roots of polynomials with positive coefficients

Proof of Theorem 3

Eneström-Kakeya Theorem

Minimal Polynomials with Positive Coefficients