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 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.

In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring H[q(1), ... , 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 H[q(1), ... , 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 H-n. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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.

In the present paper we investigate the relations between irreducible slice algebraic sets in 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.

We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the quaternionic setting.

We introduce a non-commutative resultant, for slice regular polynomials in two quaternionic variables, defined in terms of a suitable Dieudonn & eacute; determinant. We use this tool to investigate the existence of common zeros and common factors of slice regular polynomials and we give a kinematic interpretation of our results.

Bi-parameter potential theory and Carleson measures for the Dirichlet space on the bidisc, Discrete Analysis 2023:22, 58 pp. Carleson measures arise naturally when considering harmonic or holomorphic extensions from the boundary of a domain to the interior of the domain. For instance, suppose one has an $L^p$ function $f$ on the real line for some $1 \leq p < \infty$, and let $u$ be its standard harmonic extension to the upper half-plane, given by convolution with the Poisson kernel. The function $u$ will not be expected to be an $L^p$ function on the upper half-plane with respect to the usual Lebesgue measure, but turns out to be an $L^p(\mu)$ function for other measures $\mu$ on the upper half-plane. Indeed, the celebrated Carleson embedding theorem asserts that a measure $\mu$ has this property if and only if it is what is now known as a "Carleson measure", which means that for every interval $[x-r,x+r]$ on the real line, the measure $\mu( [x-r,x+r] \times [0,r])$ assigned to the rectangle $[x-r,x+r] \times [0,r]$ is bounded by a constant times the length of the interval. Analogous results hold when $f$ is an $H^p$ function on the circle (roughly speaking, an $L^p$ function with a holomorphic extension to the unit disc); this theorem has many applications in complex analysis and harmonic analysis, for instance to the corona problem of determining the spectrum of the Hardy space $H^\infty$ (viewed as a Banach algebra), and in the theory of functions of bounded mean oscillation (BMO). They are also instrumental in describing multipliers of Hardy spaces $H^p$: holomorphic functions $g$ with the property that multiplication by $g$ is a bounded operation on $H^p$, though the description is a bit more complicated, in which the intervals $[x-r,x+r]$ have to be replaced with finite unions of intervals, and the notion of length replaced with the more complicated notion of "Bessel capacity" from potential theory. The original Carleson embedding theorem can be extended to higher dimensions without much difficulty (intervals get replaced by balls, and certain exponents get adjusted accordingly). However, when dealing with holomorphic functions of several complex variables, defined on the polydisc (the product of several copies of the unit disc), the situation becomes more delicate, even for functions of two complex variables on the bidisc, basically because intervals get replaced by axis-parallel rectangles of arbitrary eccentricity, which can no longer be interpreted as single-parameter balls in a metric space, but are instead genuinely bi-parameter objects. Many of the classical harmonic analysis techniques that can handle the geometry of single-parameter metric balls will fail in the bi-parameter setting if adapted naively, but over time many authors have come up with ingenious substitutes for the classical theory that can handle bi-parameter or multi-parameter settings. Often, even just stating the right generalizations correctly is a significant part of the problem. In this paper, the authors state and prove the analogue of the Carleson embedding theorem and the multiplier characterization for the bidisc, where in both cases the characterization involves the Bessel capacity of finite unions of rectangles. This is achieved by first discretizing the problem to an analogous problem on the bitree (the product of two infinite dyadic trees), and then by carefully developing a bi-parameter capacity theory first on the bitree, and then on the bidisc. There are many technical subtleties, as some (but not all) of the classical one-parameter techniques are known to fail in the multi-parameter setting.

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.

A natural question is whether and in which sense the definition of a holomorphic function depends on the choice of the two vectors {1, i} that form a basis of C over R. In fact these two vectors determine both the form of the Cauchy-Riemann operator, and the splitting of a holomorphic function in its harmonic real and imaginary components. In this paper we consider the basis {1, e(i theta)} of C over R, and define as theta-holomorphic the functions that belong to the kernel of a Cauchy-Riemann type operator determined by this basis. We study properties of these functions, and discuss the relation between them and classical holomorphic functions. This analysis will lead us to discover the special role that theta = pi/2 plays, that renders the theory of holomorphic functions special among this family of theories.

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 $\mathrm{PGL}(4,\mathbb{C})$.

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 S3xS1. 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.

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.

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.

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 Gentili et al. (J Eur Math Soc 16(11):2323–2353, 2014) and developed in Altavilla (J Geom Phys 123:184–208, 2018). To any slice-polynomial function P we associate its companion\(P^\vee \) and its 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.

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 innerouter factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity.

The theory of slice regular functions of a quaternionic variable, introduced in 2006 by Gentili and Struppa, extends the notion of holomorphic function to the quaternionic setting. This fast growing theory is already rich of many results and has interesting applications. In this setting, the present paper is devoted to introduce and study the quaternionic counterparts of Hardy spaces of holomorphic functions of one complex variable. The basic properties of the theory of quaternionic Hardy spaces are investigated, and in particular a Poisson-type representation formula, the notions of outer function, singular function and inner function are given. A quaternionic (partial) counterpart of the classical $H^p$-factorization theorem is proved. This last result assumes a particularly interesting formulation for a large subclass of slice regular functions, where it is obtained in terms of an outer function, a singular function and a quaternionic Blaschke product.

We prove a version of the classical Mittag-Leffler Theorem for regular functions over quaternions. Our result relies upon an appropriate notion of principal part, that is inspired by the recent definition of spherical analyticity.

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 , in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.