On the product of the conjugates outside the unit circle of an algebraic number

Auxiliary polynomials for some problems regarding Mahler's measure

“…The preceding analysis establishes these inequalities in the case that the noncyclotomic part of f has no common factor with the appropriate auxiliary polynomial F (x n ). We need to establish (12) under the weaker hypothesis that f merely contain a noncyclotomic factor.…”

“…A result of Schinzel [12] implies that if α is totally real and α ∈ {−1, 0, 1} then h(α) ≥ (log γ)/2, where γ denotes the golden ratio, γ = (1 + √ 5)/2; a simple proof of this fact appears in [8]. Bombieri and Zannier [2] extended Schinzel's result to local fields.…”

Minimum essentiel et degrés d'obstruction des translatés de sous-tores

“…Par ailleurs, on sait par des travaux de Schinzel [Sch73] que lorsque K est un corps totalement réel ou CM (c'est-à-dire une extension imaginaire quadratique d'un corps totalement réel) et α ∈ K × satisfait |α| = 1, alors…”

Heights and totally p-adic numbers

“…In the same paper Bombieri and Zannier proved that all fields K tv as above have this property. More examples of fields with property (B) are: the maximal totally real field extension Q tr of the rationals (proved by Schinzel [25]), any abelian extension of a given number field (proved by Amoroso and Zannier [1]), and Q(E tors ) where E is an elliptic curve defined over Q (proved by Habegger [15]). In the last three decades the study of height functions associated to a rational function f ∈ Q(x) of degree at least two has raised increasing interest.…”