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Cinzia Bisi

Publications and source records attributed to Cinzia Bisi.

At least 19 recordsLinked to original sources
arXiv2026

(Semi-)Models for Slice Regular Functions on Real Division Algebras

We study slice regular functions on the real division algebras $\mathbb{H}$ and $\mathbb{O}$ from the point of view of equivalence under automorphism and conjugation actions. Motivated by recent orbit-theoretic descriptions in terms of the invariants $(Tr,N,cdiv)$, and by the quaternionic theory of $*$-conjugation by semiregular functions, we investigate whether a slice regular function can be replaced by a canonical representative in its equivalence class. The normalization considered in this paper is algebraic: we look for representatives whose isolated nonreal zeros are aligned in a single complex slice. We call such a representative a model when it belongs to the strong-equivalence class of the original function. We prove that purely vectorial functions always admit models, while in general strong equivalence is too rigid and such representatives need not exist. We therefore introduce semi-models, which preserve the symmetrization and the real and spherical part of the zero set, but are allowed to leave the original strong-equivalence class. Our main result proves that every slice regular function $f$ with $N(f)\not\equiv 0$ admits a semi-model. The construction clarifies the different roles of the invariants $(Tr,N,cdiv)$ and the distinction between strong equivalence, weak equivalence, and semiregular conjugacy.

math.CV↗
arXiv2026

Slice regular functions on alternative *-algebras : prescribing zeroes and values on discrete sets and related extension problems

Slice regular functions are a generalization of holomorphic functions where alternative real $*$-algebras are considered instead of the field of complex numbers. For such functions we show that zero sets and function values on suitable subsets may be prescribed. As a consequence, we show that for any axially symmetric domain there exist slice regular functions which (due to the nature of its zero set) can not be extended to a larger such domain.

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arXiv2026

Reconstruction of a slice regular function from some of its real components

A basic property of holomorphic functions $f:D\to \mathbb{C}$ defined on domains $D$ of $\mathbb{C}$ is that $f$ is uniquely determined by its real part up to an additive constant. The same is true for slice regular functions defined on circular slice domains of the division algebra of quaternions or octonions. The aim of this paper is to extend the latter result to more general classes of algebras, including, among many others, the Clifford algebras $\mathbb{R}_{p,q}$ and the split octonions $\mathbb{S}\mathbb{O}$. New phenomena appear, as well as unexpected connections with graph theory and the theory of slice-Nash functions.

math.CV↗
arXiv2026

Multipoint Schwarz-Pick Lemma for the quaternionic case

Following ideas by Beardon, Minda and Baribeau, Rivard, Wegert in the context of the complex Schwarz-Pick Lemma, we use iterated hyperbolic difference quotients to prove a quaternionic multipoint Schwarz-Pick Lemma, in the context of the theory of slice regular functions. As applications, we obtain quaternionic Dieudonné and Goluzin estimates. Finally, an algorithm for the construction of (Nevanlinna-Pick) interpolating slice regular functions with real nodes is provided as a byproduct of the quaternionic multipoint Schwarz-Pick Lemma.

math.CV↗
arXiv2025

Forms and Complex Manifolds

We study the intersection form $F_X$ on the second cohomology group $H^2(X, \mathbb{Z})$ of a compact Kähler manifold $X$ of dimension $n$. Although the structure of $F_X$ is relatively well understood in dimensions two and three, much less is known for $n \geq 4$. We investigate the fundamental properties of $F_X$ in higher dimensions and discuss several applications to birational geometry. Finally, we present a number of open problems concerning the relationship between birational invariants and topological invariants of Kähler manifolds.

math.AG↗
arXiv2025

Division algebras of slice-Nash functions

The purpose of this paper is to introduce the notion of Nash functions in the context of slice regular functions of one quaternionic or octonionic variable. We begin with a detailed analysis of the possible definitions of Nash slice regular functions which leads us to the definition of \textit{slice-Nash} function proposed in this paper (and which we strongly believe to be the natural generalisation of the classical real and complex Nash functions to this context). Once the `correct' definition of slice-Nash functions has been established, we study their properties with particular focus on their finiteness properties. These finiteness properties position this new class of slice-Nash functions as an intermediate class between the class of slice regular functions and the class of slice polynomials, in analogy with the classical real and complex case. We also introduce semiregular slice-Nash functions, in analogy with meromorphic Nash functions, and study their finiteness properties.

math.CV↗
arXiv2024

Invariants and Automorphisms for slice regular functions

Let $A$ be one of the following Clifford algebras : $\mathbb{R}_2 \cong \mathbb{H}$ or $\mathbb{R}_3$. For the algebra $A$, the automorphism group $Aut(A)$ and its invariants are well known. In this paper we will describe the invariants of the automorphism group of the algebra of slice regular functions over $A$.

math.CV↗
arXiv2024

Some interesting birational morphisms of smooth affine quadric $3$-folds

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form $x_1x_4-x_2x_3=$ constant, which seems to have some (among many others) interesting/unexpected characters: a) they are cohomologically hyperbolic, b) their second dynamical degree is an algebraic number but not an algebraic integer, and c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on $\mathbb{C}^4$ preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.

math.AG↗
arXiv2021

On the Quadratic Cone of $\mathbb{R}_3.$

In this paper we study the following type of functions $f: \mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $ \mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra $\mathbb{R}_{3}$. From the fact that it is possible to write the algebra $ \mathbb{R}_{3}$ as a direct sum of quaternions, we get the observation that it is possible to find a clever representation for $ \mathcal{Q}_{\mathbb{R}_3}$. By using this result a slice regular theory was introduced and a Cauchy formula is discussed. Moreover, a detailed study of the zeros is performed. Finally, we find a formula for the determinant of a matrix with entries in $ \mathcal{Q}_{\mathbb{R}_3}$.

math.CV↗
arXiv2021

On a Runge Theorem over $\mathbb{R}_3$

In this paper we investigate a topological characterization of the Runge theorem in the Clifford algebra $ \mathbb{R}_3$ via the description of the homology groups of axially symmetric open subsets of the quadratic cone in $\mathbb{R}_3$.

math.CV↗
arXiv2021

On Brolin's theorem over the quaternions

In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin's theorems for the one slice preserving polynomials and for generic polynomials are also proved.

math.CV↗
arXiv2020

The harmonicity of slice regular functions

In this article we investigate harmonicity, Laplacians, mean value theorems and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well known Representation Formula for slice regular functions over $\mathbb{H}$. Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over $\mathbb{H}$ (analogous to an holomorphic function over $\mathbb{C}$) "harmonic" in some sense, i.e. is it in the kernel of some order-two differential operator over $\mathbb{H}$ ? Finally, some applications are deduced, such as a Poisson Formula for slice regular functions over $\mathbb{H}$ and a Jensen's Formula for semi-regular ones.

math.CV↗
arXiv2020

On a quaternionic Picard theorem

The classical theorem of Picard states that a non-constant holomorphic function $f:\mathbb{C}\to\mathbb{C}$ can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable $f:\mathbb{H}\to\mathbb{H}$ may avoid.

math.CV↗
arXiv2018

Log-biharmonicity and a Jensen formula in the space of quaternions

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.

math.CV↗
arXiv2018

Slice-quaternionic Hopf surfaces

We investigate slice-quaternionic Hopf surfaces. In particular, we construct new structures of slice-quaternionic manifold on $\mathbb{S}^1\times\mathbb{S}^7$, we study their group of automorphisms and their deformations.

math.CV↗
arXiv2018

On Quaternionic Tori and their Moduli Spaces

Quaternionic tori are defined as quotients of the skew field $\mathbb{H}$ of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a fundamental region in a $12$-dimensional real subspace is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable \emph{special} bases of rank-4 lattices, which are studied with respect to the action of the group $GL(4, \mathbb{Z})$, and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms - and all possible groups of their biregular automorphisms - are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.

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