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Asymptotic distribution and symmetric means of algebraic numbers

Igor E. Pritsker (2015)

Acta Arithmetica

Schur introduced the problem on the smallest limit point for the arithmetic means of totally positive conjugate algebraic integers. This area was developed further by Siegel, Smyth and others. We consider several generalizations of the problem that include questions on the smallest limit points of symmetric means. The key tool used in the study is the asymptotic distribution of algebraic numbers understood via the weak* limits of their counting measures. We establish interesting properties of the...

Class Number Two for Real Quadratic Fields of Richaud-Degert Type

Mollin, R. A. (2009)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the Euler-Rabinowitsch polynomial for real quadratic fields. As well, we complete the list of Richaud-Degert types given in [16] and show how the behaviour of the Euler-Rabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination...

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

We consider a rational function f which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number ℓ of terms. Then we look at the possible decompositions f ( x ) = g ( h ( x ) ) , where g , h are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of g is bounded only in terms of ℓ (and we provide explicit bounds). This supports and quantifies the intuitive...

Concordant sequences and integral-valued entire functions

Jonathan Pila, Fernando Rodriguez Villegas (1999)

Acta Arithmetica

A classic theorem of Pólya shows that the function 2 z is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.

Decomposition of primes in number fields defined by trinomials

P. Llorente, E. Nart, N. Vila (1991)

Journal de théorie des nombres de Bordeaux

In this paper we deal with the problem of finding the prime-ideal decomposition of a prime integer in a number field K defined by an irreducible trinomial of the type X p m + A X + B ∈ ℤ [ X ] , in terms of A and B . We also compute effectively the discriminant of K .

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