Francesco Amoroso

On fields with Property (B)

Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L^* (outside the set of roots of unity) is bounded from below by an absolute constant. We discuss some feature of Property (B): stability by algebraic extensions, relations with field arithmetic. As a as a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free.

UNE MINORATION POUR L'EXPOSANT DU GROUPE DES CLASSES D'UN CORPS ENGENDRÉ PAR UN NOMBRE DE SALEM

In this article we extend the main result of [2] concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number α. If k denotes the field fixed by α ↦ α-1 we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent eK/k and the relative size hK/k of the class group of K with respect to the class group of k.

Bounded height in pencils of finitely generated subgroups

In this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell-Lang Theorem by replacing finiteness by emptyness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the results is as follows. Consider the torus scheme G r m/C over a curve C defined over Q, and let Γ be a subgroup-scheme generated by finitely many sections (satisfying some necessary conditions). Further, let V be any subscheme. Then there is a bound for the height of the points P ∈ C(Q) such that, for some γ ∈ Γ which does not generically lie in V , γ(P) lies in the fiber VP. We further offer some direct diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.

Overdetermined Systems of Sparse Polynomial Equations

In this paper, we show that for a system of univariate polynomials given in sparse encoding we can compute a single polynomial defining the same zero set in quasilinear time in the logarithm of the degree. In particular, it is possible to decide whether such a system of polynomials has a zero in quasilinear time in the logarithm of the degree. The underlying algorithm relies on a result of Bombieri and Zannier on multiplicatively dependent points in subvarieties of an algebraic torus. We also present the following conditional partial extension to the higher-dimensional setting. Assume that the effective Zilber conjecture holds. Then, for a system of multivariate polynomials given in sparse encoding we can compute a finite collection of complete intersections outside hypersurfaces that defines the same zero set in quasilinear time in the logarithm of the degree.

Unlikely intersections and multiple roots of sparse polynomials

We present a structure theorem for the multiple non-cyclotomic irreducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents. Roughly speaking, this result shows that the multiple non-cyclotomic irreducible factors of a sparse polynomial, are also sparse. To prove this, we give a variant of a theorem of Bombieri and Zannier on the intersection of a fixed subvariety of codimension 2 of the multiplicative group with all the torsion curves, with bounds having an explicit dependence on the height of the subvariety. We also use this latter result to give some evidence on a conjecture of Bolognesi and Pirola.

Mahler measure on Galois extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.