Searcharxiv⌕ Search

arXiv subjects

Francesco Battistoni

Publications and source records attributed to Francesco Battistoni.

15 recordsLinked to original sources
arXiv2026

Characterization of regularity via variational stability of alternating projections sequences

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.

math.OC↗
arXiv2024

A note on the Moment Problem for codimension greater than 1

We provide new conditions under which the alternating projection sequence converges in norm for the convex feasibility problem where a linear subspace with finite codimension $N\geq 2$ and a lattice cone in a Hilbert space are considered. The first result holds for any Hilbert lattice, assuming that the orthogonal of the linear subspace admits a basis made by disjoint vectors with respect to the lattice structure. The second result is specific for $\ell^2(\mathbb{N})$ and is proved when only one vector of the basis is not in the cone but the sign of its components is definitively constant and its support has finite intersection with the supports of the remaining vectors.

math.OC↗
arXiv2024

A one parameter family of Volterra-type operators

For every $α\in (0,+\infty)$ and $p,q \in (1,+\infty)$ let $T_α$ be the operator $L^p[0,1]\to L^q[0,1]$ defined via the equality $(T_αf)(x) := \int_0^{x^α} f(y) d y$. We study the norms of $T_α$ for every $p$, $q$. In the case $p=q$ we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case $p=q=2$ we determine the point spectrum and eigenfunctions for $T^*_αT_α$, where $T^*_α$ is the adjoint operator.

math.FA↗
arXiv2024

Generalized Pohst inequality and small regulators

Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over an hypercube. In this paper, we provide new and effective upper bounds for the case of fields with one complex embedding and degree between five and nine: this is done by adapting the strategy we have adopted to study the totally real case, but for this new setting several new computational issues had to be overcome. As a consequence, we detect the four number fields of signature (6,1) with smallest regulator; we also expand current lists of number fields with small regulator in signatures (3,1), (4,1) and (5,1).

math.NT↗
arXiv2022

On the typical rank of elliptic curves over ${\mathbb Q}(T)$

We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (arXiv:2009.08622v2) in the case $m,n\leq2$.

math.NT↗
arXiv2021

On the trace of the integers of a number field

Let $Tr$ denote the trace $\mathbb{Z}$-module homomorphism defined on the ring $\mathcal{O}_{L} $ of the integers of a number field $L.$ We show that $Tr(\mathcal{O}_{L})\varsubsetneq \mathbb{Z}$ if and only if there is a prime factor $p$ of the degree of $L$ such that if $\wp _{1}^{e_{1}}...\wp _{s}^{e_{s}}$ is the prime factorization of the ideal $p\mathcal{O}_{L}$ in $\mathcal{O}_{L},$ then $p$ divides all powers $e_{1},...,e_{s}.$ Also, we prove that the equality $Tr(\mathcal{O}_{L})=\mathbb{Z}$ holds when $L$ is the compositum of certain number fields.

math.NT↗
arXiv2021

Ranks of elliptic curves over $\mathbb{Q}(T)$ of small degree in $T$

We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the factorization and other simple properties of certain polynomials. Finally, we give sharp estimates for the ranks of the considered families and we present several applications, among which there are lists of rational points, generic families with maximal rank and generalizations of former results.

math.NT↗
arXiv2021

Arithmetic equivalence for non-geometric extensions of global function fields

In this paper we study couples of finite separable extensions of the function field $\mathbb{F}_q(T)$ which are arithmetically equivalent, i.e. such that prime ideals of $\mathbb{F}_q[T]$ decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of $\mathbb{F}_q(T)$, which are fields such that their field of constants may be bigger than $\mathbb{F}_q$. In the second part, we explicitly produce examples of non-geometric extensions of $\mathbb{F}_2(T)$ which are equivalent and non-isomorphic over $\mathbb{F}_2(T)$ and non-equivalent over $\mathbb{F}_4(T)$, solving a particular Inverse Galois Problem.

math.NT↗
arXiv2021

A conjectural improvement for inequalities related to regulators of number fields

An inequality proved firstly by Remak and then generalized by Friedman shows that there are only finitely many number fields with a fixed signature and whose regulator is less than a prescribed bound. Using this inequality, Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposo succeeded to detect all fields with small regulators having degree less or equal than 7. In this paper we show that a certain upper bound for a suitable polynomial, if true, can improve Remak-Friedman's inequality and allows a classification for some signatures in degree 8 and better results in degree 5 and 7. The validity of the conjectured upper bound is extensively discussed.

math.NT↗
arXiv2021

An elementary proof for a generalization of a Pohst's inequality

Let $P_n(y_1,\ldots,y_n):= \prod_{1\leq i<j\leq n}\left( 1 -\frac{y_i}{y_j}\right) $ and $P_n:= \sup_{(y_1,\ldots,y_n)}P_n(y_1,\ldots,y_n) $ where the supremum is taken over the $n$-ples $(y_1,\ldots,y_n)$ of real numbers satisfying $0 <|y_1| < |y_2|< \cdots < |y_n|$. We prove that $P_n \leq 2^{\lfloor n/2\rfloor}$ for every $n$, i.e., we extend to all $n$ the bound that Pohst proved for $n\leq 11$. As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.

math.NT↗
arXiv2019

On Locally GCD Equivalent Number Fields

Local GCD Equivalence is a relation between extensions of number fields which is weaker than the classical arithmetic equivalence. It was originally studied by Lochter with Weak Kronecker Equivalence. Among the many results he got, Lochter discovered that number fields extensions of degree $\leq 5$ which are locally GCD equivalent are in fact isomorphic. This fact can be restated saying that number fields extensions of low degree are uniquely characterized by the splitting behaviour of a restricted set of primes: in particular, also extensions of degree 3 and 5 are uniquely determined by their inert primes, just like the quadratic fields. The goal of this note is to present this rigidity result with a different proof, which insists especially on the densities of sets of prime ideals and their use in the classification of number fields up to isomorphism. Alongside Chebotarev's Theorem, no harder tools than basic Group and Galois Theory are required.

math.NT↗
arXiv2019

On small discriminants of number fields of degree 8 and 9

We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature $(3,3)$ and small discriminant which was not previously known.

math.NT↗
arXiv2019

Discriminants of number fields and surjectivty of trace homomorphism on rings of integers

In this note we give a brief survey of the most elementary criteria used to determine the surjectivity of the trace operator on the ring of integers of a number field $K$. Furthermore, we introduce an easy to state yet unknown surjectivity criterion depending only on the prime factorization of the degree $n$ of $K$ and on the squarefree part of the discriminant $d_K$.

math.NT↗
arXiv2018

The minimum discriminant of number fields of degree 8 and signature (2,3)

In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2,3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko-Poitou-Serre method.

math.NT↗