Displaying similar documents to “Nonreciprocal algebraic numbers of small measure”

Mahler measures in a cubic field

Artūras Dubickas (2006)

Czechoslovak Mathematical Journal

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We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E . This extends the result of Schinzel who proved the same statement for every real quadratic field E . A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure. ...

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 .

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,...

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. ...

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.

Nonreciprocal algebraic numbers of small Mahler's measure

Artūras Dubickas, Jonas Jankauskas (2013)

Acta Arithmetica

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We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials 1 + x r ₁ + ⋯ + x r ₅ , where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including 2 r j < r j + 1 for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct...