Searcharxiv⌕ Search

arXiv subjects

Giulia Sarfatti

Publications and source records attributed to Giulia Sarfatti.

At least 19 recordsLinked to original sources
arXiv2026

A note on irreducible slice algebraic sets

In this short note we prove that if $I$ is a right radical and quasi prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization $\mathbb S_{V_c(I)}$ is an irreducible algebraic set, where $V_c(I)$ is the set of common zeros with commuting components of polynomials in $I$. Combining this fact with the results proved in our previous paper [3], we obtain that for $I$ radical, $V_c(I)$ is irreducible if and only if $I$ is quasi prime.

math.AG↗
arXiv2025

Quaternionic Carleson measures

In this paper we provide a general construction of a quaternionic Banach space of slice regular functions from a given Banach space of holomorphic functions, which we call its quaternionic lift. To the best of our knowledge, this construction encompasses all known examples of quaternionic Banach spaces of slice regular functions in the literature. Our main result is a characterization of Carleson and vanishing Carleson measures for such quaternionic Banach function spaces in terms of the corresponding Carleson measures of the underlying holomorphic function space. This offers a unified approach to a problem that so far has been treated on a case-by-case basis.

math.FA↗
arXiv2025

A Strong Version of the Hilbert Nullstellensatz for slice regular polynomials in several quaternionic variables

In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring $\mathbb H[q_1,\ldots,q_n]$ of slice regular polynomials in several quaternionic variables. Our proof deeply depends on a detailed analysis of the common zeros of slice regular polynomials which belong to an ideal in $\mathbb H[q_1,\ldots,q_n]$. This study motivates the introduction of a new notion of algebraic set in the quaternionic setting, which allows us to define a Zariski-type topology on $\mathbb H^n$.

math.CV↗
arXiv2025

On the irreducibility of slice algebraic sets

In the present paper we investigate the relations between irreducible slice algebraic sets in $\mathbb{H}^n$ and quasi prime right ideals of the ring of slice regular polynomials in $n$ quaternionic variables. We provide algebraic conditions on right ideals of slice regular polynomials which guarantee the irreducibility of the corresponding slice algebraic sets and show that radical ideals associated with irreducible slice algebraic sets are quasi prime. Furthermore we establish that this correspondence is an equivalence in the case of principal right ideals.

math.AG↗
arXiv2023

Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc

We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz'ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter potential theory on the bidisc and prove a Strong Capacitary Inequality. In order to do so, we have to overcome the obstacle that the Maximum Principle fails in the bi-parameter theory.

math.CV↗
arXiv2019

On compact affine quaternionic curves and surfaces

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface S^3 x S^1. As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

math.DG↗
arXiv2019

Quaternionic inner and outer functions

We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.

math.CV↗
arXiv2018

Shift invariant subspaces of slice $L^2$ functions

In this paper we characterize the closed invariant subspaces for the ($*$-)multiplier operator of the quaternionic space of slice $L^2$ functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity.

math.CV↗
arXiv2018

Configurations of flags in orbits of real forms

In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of $\mathrm{PGL}(4,\mathbb{C})$.

math.AG↗
arXiv2017

Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in $\mathbb{CP}^3$

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{gensalsto} and developed in \cite{AAtwistor}. To any slice-polynomial function $P$ we associate its {\em companion} $P^\vee$ and its {\em extension} to the real axis $P_\mathbb{R}$, that are quaternionic functions naturally related to $P$. Then, using the theory of twistor spaces, we are able to show that for any quaternion $q$ the {cardinality of simultaneous} pre-images of $q$ via $P$, $P^\vee$ and $P_\mathbb{R}$ is generically constant, giving a notion of degree. With the brand new tool of slice-polynomial functions, we {compute} the twistor discriminant locus of a cubic scroll $\mathcal{C}$ in $\mathbb{CP}^3$ and we conclude by giving some qualitative results on the complex structures induced by $\mathcal{C}$ via the twistor projection.

math.CV↗
arXiv2016

Quaternionic toric manifolds

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted on by $m$ copies of the group ${\rm Sp}(1)$ of unit quaternions. These manifolds are quaternionic regular and can be endowed with a $4$-plectic structure and a generalized moment map. Convexity properties of the image of the moment map are studied. Quaternionic toric manifolds appear to be a large enough class of examples where one can test and study new results in quaternionic geometry.

math.DG↗
arXiv2016

A direct approach to quaternionic manifolds

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on $\mathbb{H}^n$, in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.

math.CV↗
arXiv2016

Ideals of regular functions of a quaternionic variable

In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides with the entire ring of slice regular functions. The proof required the study of the non-commutative syzygies of a vector of regular functions, that manifest a different character when compared with their complex counterparts.

math.CV↗