Displaying similar documents to “Algebraic S-integers of fixed degree and bounded height”

Algebraic Numbers

Yasushige Watase (2016)

Formalized Mathematics

Similarity:

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to...

On the asymptotic behavior of some counting functions

Maciej Radziejewski, Wolfgang A. Schmid (2005)

Colloquium Mathematicae

Similarity:

The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most k different lengths is investigated. It is shown that this constant is positive if k is greater than 1 and that it is also positive if k equals 1 and the class...

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

Peter Bundschuh, Keijo Väänänen (2015)

Acta Arithmetica

Similarity:

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...

Multiplicative dependence of shifted algebraic numbers

Paulius Drungilas, Artūras Dubickas (2003)

Colloquium Mathematicae

Similarity:

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

Reduction of semialgebraic constructible functions

Ludwig Bröcker (2005)

Annales Polonici Mathematici

Similarity:

Let R be a real closed field with a real valuation v. A ℤ-valued semialgebraic function on Rⁿ is called algebraic if it can be written as the sign of a symmetric bilinear form over R[X₁,. .., Xₙ]. We show that the reduction of such a function with respect to v is again algebraic on the residue field. This implies a corresponding result for limits of algebraic functions in definable families.

Note on algebraic interior systems

Ivan Chajda (2005)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

We get an interrelation between an algebraic closure system and its conjugated interior system. We introduce the concept of algebraic interior system and we get its representation.