Displaying similar documents to “Totally positive algebraic integers of small trace”

Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II

Peter Bundschuh, Keijo Väänänen (2015)

Acta Arithmetica

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This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first...

The mean values of logarithms of algebraic integers

Artūras Dubickas (1998)

Journal de théorie des nombres de Bordeaux

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Let α be an algebraic integer of degree d with conjugates α 1 = α , α 2 , ⋯ , α d . In the paper we give a lower bound for the mean value M p ( α ) = 1 d ∑ i = 1 d | log | α i | | p p when α is not a root of unity and p > 1 .

Complex Analogues of the Rolle's Theorem

Sendov, Blagovest (2007)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 30C10. Classical Rolle’s theorem and its analogues for complex algebraic polynomials are discussed. A complex Rolle’s theorem is conjectured.

Zhang-Zagier heights of perturbed polynomials

Christophe Doche (2001)

Journal de théorie des nombres de Bordeaux

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In a previous article we studied the spectrum of the Zhang-Zagier height [2]. The progress we made stood on an algorithm that produced polynomials with a small height. In this paper we describe a new algorithm that provides even smaller heights. It allows us to find a limit point less than 1 . 289735 i.e. better than the previous one, namely 1 . 2916674 . After some definitions we detail the principle of the algorithm, the results it gives and the construction that leads to this new limit point. ...

Comments on the height reducing property

Shigeki Akiyama, Toufik Zaimi (2013)

Open Mathematics

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A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one,...

Transcendence results on the generating functions of the characteristic functions of certain self-generating sets

Peter Bundschuh, Keijo Väänänen (2014)

Acta Arithmetica

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This article continues two papers which recently appeared in this same journal. First, Dilcher and Stolarsky [140 (2009)] introduced two new power series, F(z) and G(z), related to the so-called Stern polynomials and having coefficients 0 and 1 only. Shortly later, Adamczewski [142 (2010)] proved, inter alia, that G(α),G(α⁴) are algebraically independent for any algebraic α with 0 < |α| < 1. Our first key result is that F and G have large blocks of consecutive zero coefficients....

A generalization of a theorem of Schinzel

Georges Rhin (2004)

Colloquium Mathematicae

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We give lower bounds for the Mahler measure of totally positive algebraic integers. These bounds depend on the degree and the discriminant. Our results improve earlier ones due to A. Schinzel. The proof uses an explicit auxiliary function in two variables.

Algebraic S-integers of fixed degree and bounded height

Fabrizio Barroero (2015)

Acta Arithmetica

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Let k be a number field and S a finite set of places of k containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of S-integers of k. Moreover, we give an asymptotic formula for the number of S̅-integers of bounded height and fixed degree over k, where S̅ is the set of places of k̅ lying above the ones in S.