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A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial

A nonzero complex number

A nonzero complex number

In 2009, Brunotte [

We fix the following notation and terminology throughout. Denote by

Let

let

([

([

Our first main result is a certain quantitative improvement of Dubickas-Kuba theorem (Theorem

Let

The nontrivial half of Theorem

Combining Theorem

The proof of Theorem

Let

Let

Let

Since

Observe that the proof of this last lemma remains true when

Let

Writing

The next lemma proves Theorem

Let

Assume that the nonzero algebraic number

Conversely, assume that there exists

We are now ready to prove Theorem

Assume that all conjugates of

On the other hand, assume that there exists

It is worth noting that in [

Let

all conjugates of

there exists

there exists

there exist

Assertions (i) and (ii) are equivalent by Theorem

That assertion (iv) implies that assertion (i) is again an immediate consequence of Lemma

To proceed further, we need a new notion.

For

Part (1) of Lemma

Let

In this section, we derive a proposition yielding conditions which are necessary for a product of two polynomials to be in

Let

Let

The next proposition indicates the significance of the upper and lower Eneström-Kakeya quotients.

Let

The upper Eneström-Kakeya quotient

The lower Eneström-Kakeya quotient

Part (i) follows from Lemma

In passing, from the definition of Eneström-Kakeya quotients, it seems natural to ask whether one quotient can be a reciprocal of the other. This is indeed the case when the polynomial is self-reciprocal. Let

Let

Writing

If

(i) If

In this section, we aim to show that the bound

Let

Proposition

Let

For a nonzero algebraic number

Let

If

We have

Assertion (i) follows at once from the preceding remarks.

(ii) If

Theorem

Let

Since all conjugates of

It may be of interest to look at the growth factors and the Eneström-Kakeya quotients for positively minimal polynomials of small degrees.

Let

If

If

for

for

for

If

Part (1) is trivial. We now consider part (2), that is,

Finally consider part (3), that is,

We split our consideration into three cases depending on the maximum absolute value of the conjugates.

Case

If

Case

If

If

If

If

yielding

Consider now

If

If

If

We end this paper with another class of algebraic numbers for which the bound in Theorem

Let

Assume on the contrary that there is

If

The authors declare that there is no conflict of interests regarding the publication of this paper.