VENEZIANO F.

Given a polynomial of even degree D(t) with complex coefficients, we consider the continued fraction expansion of root D(t). In this setting, it has been shown by Zannier that the sequence of the degrees of the partial quotients of the continued fraction expansion of root D(t) is eventually periodic, even when the expansion itself is not. In...

We study the quadratic integral points-that is, (S-)integral points defined over any extension of degree two of the base field-on a curve defined in ℙ3 by a system of two Pell equations. Such points belong to three families explicitly described, or belong to a finite set whose cardinality may be explicitly bounded in terms of the base field,...

These are notes from the minicourse given by Umberto Zannier (Scuola Normale Superiore di Pisa). The notes were worked out by Laura Capuano, Peter Jossen,1 Christina Karolus, and Francesco Veneziano. Most of the material of these lectures, except for the numerical examples which were added by us, is already available in [45], The authors wish...

Pisot numbers are real algebraic integers bigger than 1, whose other conjugates all have modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called...

The Torsion Anomalous Conjecture states that an irreducible variety V embedded in a semi-abelian variety contains only finitely many maximal V -torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the Néron-Tate height of all...

We prove a sharp lower bound for the essential minimum of a nontranslate variety in certain abelian varieties. This uses and generalises a result of Galateau. Our bound is a new step in the direction of an abelian analogue by David and Philippon of a toric conjecture of Amoroso and David and has applications in the framework of anomalous intersections.

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are...

We discuss an elementary problem, initially proposed for the Romanian Mathematical Olympiad, which leads to interesting remarks of various nature. We relate the problem to the theory of linear recurrence sequences with non-constant coefficients and their p-adic behaviour. Our considerations can be applied to a larger set of similarlydefined recurrences.

In this paper, we prove a periodicity theorem for certain continued fractions with partial quotients in the ring of integers of a fixed quadratic field. This theorem generalizes the classical theorem of Lagrange to a large set of continued fraction expansions.As an application we consider the beta-continued fractions and show that for any...

Adipose tissue-derived stem cells (ASCs) are a promising tool for the treatment of bone diseases or skeletal lesions, thanks to their ability to potentially repair damaged tissue. One of the major limitations of ASCs is represented by the necessity to be isolated and expanded through in vitro culture; thus, a strong interest was generated by...