arXiv subjects

Francesco Amoroso

Publications and source records attributed to Francesco Amoroso.

6 recordsLinked to original sources

arXiv2026

Pencils of norm form equations and a conjecture of Thomas, II

We continue our studies on parametric norm forms $F_t({\bf x})$, with ${\bf x}=(x_0,x_1,\ldots,x_{d-1})$ lying in some parametric linear subvariety $W_t$ and integers $t$ sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions $\bf x$ of $F_t({\bf x})=1$. Here we modify our techniques to treat $F_t({\bf x})=q$ for an arbitrary integer $q$. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all $\bf x$ are polynomially bounded in terms of $|q|$ and $t$. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with $x_2=\cdots=x_{d-1}=0$. We present an example with eight variables.

math.NT↗

arXiv2025

Bogomolov property and Galois representations

In 2013 P. Habegger proved the Bogomolov property for the field generated over Q by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions fixed by the kernel of some modular Galois representations.

math.NT↗

arXiv2023

Equidistribution for sets which are not necessarily Galois stable: On a theorem of Mignotte

An important result of Bilu deals with the equidistribution of the Galois orbits of a sequence $(α_n)_n$ in $\overline{\mathbb{Q}}^*$. Here, we prove a quantitative equidistribution theorem for a sequence of finite subsets in $\overline{\mathbb{Q}}^*$ which are not necessarily stable by Galois action. We follow a method of Mignotte.

math.NT↗

arXiv2018

Factorization of bivariate sparse polynomials

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a variant of the toric Bertini's theorem due to Zannier and Fuchs, Mantova and Zannier.

math.AC↗

arXiv2018

A remark on a recent proof of Lehmer's Conjecture [arXiv:1709.03771]

Over one year ago, a very long preprint posted on arXiv [arXiv:1709.03771] and HAL announced a proof of Lehmer's Conjecture (and of other related results). Unfortunately, as was remarked by several specialists, this proof contains a (at least one) fatal error. The aim of this very short note is to inform the mathematical community, which could be aware of this.

math.NT↗

arXiv2014

Overdetermined systems of sparse polynomial equations

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 time quasi-linear in the logarithm of the degree. In particular, it is possible to determine whether such a system of polynomials does have a zero in time quasi-linear 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 time quasi-linear in the logarithm of the degree.

math.AG↗