arXiv subjects

Amedeo Altavilla

Publications and source records attributed to Amedeo Altavilla.

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

Complex Quaternions and Superminimal Surfaces in Four-Space

We develop a quaternionic approach to conformal superminimal surfaces in Euclidean four-space. The starting point is the classical Weierstrass representation: every conformal minimal immersion $X\colon M\to\mathbb{R}^4$ is recovered as $X = c + Re\int\Phi dz$, where $\Phi$ is a holomorphic null curve in $\mathbb{C}^4$, identified with the algebra of complex quaternions $\mathbb{H}\otimes\mathbb{C}$. The multiplicativity of the quaternionic symmetrized norm makes it natural to factor null curves as $\Phi=ALB$, where $A$ and $B$ are holomorphic maps of unit symplectic norm and $L$ is a holomorphic null element. We show that on simply connected domains the null factor can always be taken constant. The main result is an explicit quaternionic reformulation of the superminimality condition -- the requirement that the curvature ellipse be a circle at every point. In the fixed-null gauge $L=1+\sqrt{-1} e_1$, superminimality is equivalent to the vanishing of a product of two holomorphic functions built from the left and right Maurer--Cartan forms of $A$ and $B$. On a connected domain, this forces one of the two components of the generalized Gauss map $[\Phi]\colon M\to Q^2\simeq\mathbb{CP}^1\times\mathbb{CP}^1$ to be constant, recovering in spinorial terms the classical ruling condition on the projective null quadric. We further provide a first-order ODE parametrization of the superminimal $ALB$-data, analyse the residual gauge freedom, prove a fixed-gauge rigidity statement for polynomial spinorial factors, and illustrate the theory with explicit examples.

math.DG↗

arXiv2026

The Flat CR Twistor Model $Q^{2,2}$ and Its Algebraic Sections

We study the flat CR twistor model $Q^{2,2}\subset \mathbb{CP}^3$ by explicit projective methods. Using the anti-holomorphic involution $j$ associated with the twistor fibration, we classify the projective lines contained in $Q^{2,2}$ into twistor fibres and transverse lines, and relate the latter to round $2$-spheres in $S^3$ through an explicit incidence--tangency correspondence. We classify hyperplane sections under the twistor-compatible symmetry group $PSp(1,1)$ and describe the induced CR geometries on $S^3$. For smooth $j$-invariant quadric sections, we obtain a complete relative classification in terms of Coxeter's inversive distance and show that, in the disjoint case, the construction yields an explicit one-parameter family of globally defined real-analytic non-spherical Levi-nondegenerate CR structures on $S^3$.

math.DG↗

arXiv2026

Geometry of the Donaldson-Friedman Pushout: Twistor degenerations and instanton charge

We study the Donaldson-Friedman semistable twistor degeneration by combining the scheme-theoretic geometry of its Ferrand pushout with the logarithmic topology of its Kato-Nakayama realisation. For the central fibre $ Z_0=\widetilde Z_1\cup_Q\widetilde Z_2,$ the Ferrand description yields an explicit equaliser presentation of the operational Chow ring and a componentwise specialisation formula, with sharp restrictions on surfaces that glue across the exceptional quadric. The logarithmic structure of the same normal-crossing fibre retains data not visible in ordinary intersection theory: after fixing the phase of the smoothing parameter, the Kato-Nakayama space over $Q$ is the unit circle bundle of the normal line bundle, and over a ruling fibre its anti-diagonal quotient is diffeomorphic to $\mathbb{RP}^3$. Restriction to curves in $Q$ gives a log-topological refinement of the algebraic intersection data. For bundles $E_0$ on $Z_0$ obtained by gluing Ward or Hartshorne-Serre data from the two components, we prove additivity of the second Chern cycle $c_2(E_0)\cap[Z_0]$. If $E_0$ extends over the semistable smoothing, its polarised charge is the sum of the component charges; under the usual reality and triviality conditions of the Ward correspondence, the bundle on a smooth fibre determines an anti-self-dual $SU(2)$-instanton on the connected sum of the underlying four-manifolds.

math.AG↗

arXiv2026

Jordan-Segmentable Masks: A Topology-Aware definition for characterizing Binary Image Segmentation

Image segmentation plays a central role in computer vision. However, widely used evaluation metrics, whether pixel-wise, region-based, or boundary-focused, often struggle to capture the structural and topological coherence of a segmentation. In many practical scenarios, such as medical imaging or object delineation, small inaccuracies in boundary, holes, or fragmented predictions can result in high metric scores, despite the fact that the resulting masks fail to preserve the object global shape or connectivity. This highlights a limitation of conventional metrics: they are unable to assess whether a predicted segmentation partitions the image into meaningful interior and exterior regions. In this work, we introduce a topology-aware notion of segmentation based on the Jordan Curve Theorem, and adapted for use in digital planes. We define the concept of a \emph{Jordan-segmentatable mask}, which is a binary segmentation whose structure ensures a topological separation of the image domain into two connected components. We analyze segmentation masks through the lens of digital topology and homology theory, extracting a $4$-curve candidate from the mask, verifying its topological validity using Betti numbers. A mask is considered Jordan-segmentatable when this candidate forms a digital 4-curve with $\beta_0 = \beta_1 = 1$, or equivalently when its complement splits into exactly two $8$-connected components. This framework provides a mathematically rigorous, unsupervised criterion with which to assess the structural coherence of segmentation masks. By combining digital Jordan theory and homological invariants, our approach provides a valuable alternative to standard evaluation metrics, especially in applications where topological correctness must be preserved.

cs.CV↗

arXiv2025

A Complete Characterization of Pythagorean Hodograph Preserving Mappings

We fully characterize the mappings $\Phi$ that send every Pythagorean-hodograph (PH) curve to a PH curve. We prove that in any dimension, such mappings are precisely the conformal functions whose dilation is the square of a real rational function. In the planar case, this implies (up to conjugation) that $\partial\Phi/\partial z = \Psi^{2}$, where $\Psi$ is meromorphic and satisfies $\operatorname{Res}(\Psi^{2}) = 0$ at every pole. In higher dimensions, PH preservation forces $\Phi$ to be a conformal map; for $n \ge 3$, Liouville's theorem then implies that any local diffeomorphism with this property is (anti-)M\"obius. These results subsume the previously known ``(scaled) PH-preserving'' constructions of mappings $\mathbb{R}^2 \to \mathbb{R}^3$ and align with Ueda's conformal viewpoint on isothermal and spherical geometries. At the level of examples, we demonstrate how PH-preserving mappings relate to the construction of rational PH curves and minimal surfaces.

math.DG↗

arXiv2025

$q$-Fock Space of $q$-Analytic Functions and its realization in $L^{2}(\mathbb{C}; e^{-z\bar z} \,\mathrm{d}x\,\mathrm{d}y)$

We introduce a $q$-deformation of the Fock space of holomorphic functions on $\mathbb{C}$, based on a geometric definition of $q$-analyticity. This definition is inspired by a standard construction in complex differential geometry. Within this framework, we define $q$-analytic monomials $z_q^n$ and construct the associated $q$-Fock space as a Hilbert space with orthonormal basis $\{z_q^n/\sqrt{[n]_q!]}\}_{n\ge 0}$. The reproducing kernel of this space is computed explicitly, and $q$-position and $q$-momentum operators are introduced, satisfying $q$-deformed commutation relations. We show that the $q$-monomials $z_q^n$ can be expanded in terms of complex Hermite polynomials, thereby providing a realization of the $q$-Fock space as a subspace of $L^2(\mathbb{C}; e^{-|z|^2}\,\mathrm{d}x\,\mathrm{d}y)$. Finally, we define a $q$-Bargmann transform that maps suitable $q$-Hermite functions into our $q$-Fock space and acts as a unitary isomorphism. Our construction offers a geometric and analytic approach to $q$-function theory, complementing recent operator-theoretic models.

math.CV↗

arXiv2025

Minimal Surfaces via Complex Quaternions

Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstra\ss-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics.

math.CV↗

arXiv2025

Conjugate $(1/q, q)$-harmonic Polynomials in $q$-Clifford Analysis

We consider the problem of constructing a conjugate $(1/q, q)$-harmonic homogeneous polynomial $V_k$ of degree $k$ to a given $(1/q, q)$-harmonic homogeneous polynomial $U_k$ of degree $k.$ The conjugated harmonic polynomials $V_k$ and $U_k$ are associated to the $(1/q, q)$-mono\-genic polynomial $F = U_k + \overline{e}_0V. $ We investigate conjugate $(1/q, q)$-harmonic homogeneous polynomials in the setting of $q$-Clifford analysis. Starting from a given $(1/q, q)$-harmonic polynomial $U_k$ of degree $k$, we construct its conjugate counterpart $V_k$, such that the Clifford-valued polynomial $F = U_k + e_0 V_k$ is $(1/q, q)$-monogenic, i.e., a null solution of a generalized $q$-Dirac operator. Our construction relies on a combination of Jackson-type integration, Fischer decomposition, and the resolution of a $q$-Poisson equation. We further establish existence and uniqueness results, and provide explicit representations for conjugate pairs, particularly when $U_k$ is real-valued.

math.CV↗

arXiv2023

Twistor fibers in hypersurfaces of the flag threefold

We study surfaces of bidegree (1,d) contained in the flag threefold in relation to the twistor projection. In particular, we focus on the number and the arrangement of twistor fibers contained in such surfaces. First, we prove that there is no irreducible surface of bidegree (1,d) containing d+2 twistor fibers in general position. On the other hand, given any collection of (d+1) twistor fibers satisfying a mild natural constraint, we prove the existence of a surface of bidegree (1,d) that contains them. We improve our results for d=2 or d=3, by removing all the generality hypotheses.

math.AG↗

arXiv2023

$*$-Logarithm for Slice Regular Functions

In this paper, we study the (possible) solutions of the equation $\exp_{*}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $\mathbb{H}$ and $\exp_{*}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $\exp_{*}(f)=g$ is called a $*$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the ``vector'' part $g_{v}$ of $g$, for the existence of a $*$-logarithm of $g$, under a natural topological condition on the domain $Ω$. By the way, we prove an existence result if $g_{v}$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_{v}$ are finite, the domain is either the unit ball, or $\mathbb{H}$, or $\mathbb{D}$ and a further condition on the ``real part'' $g_{0}$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_{v}$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.

math.CV↗

arXiv2023

The $*$-exponential as a covering map

We employ tools from complex analysis to construct the $*$-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the $*$-exponential; we establish sufficient conditions for the $*$-product of two $*$-exponentials to also be a $*$-exponential; we calculate the slice derivative of the $*$-exponential of a regular function.

math.CV↗

arXiv2022

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $π$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $\mathbb{CP}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in $\mathbb{CP}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest type of surfaces in $\mathbb{F}$ of bidegree $(1,1)$. These surfaces define orthogonal complex structures on specified dense open subsets of $\mathbb{CP}^2$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $(1,1)$ are determined, and bounds given on the number of twistor fibres that are contained in more general algebraic surfaces in $\mathbb{F}$.

math.DG↗

arXiv2022

Slice regular functions as covering maps and global $\star$-roots

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of $k$-th $\star$-roots of a slice regular function. Moreover, we are also able to compute all the solutions which, quite surprisingly, in the most general case, are in number of $k^2$. The last part is devoted to compute the monodromy and to present a technique to compute all the $k^2$ roots starting from one of them.

math.CV↗

arXiv2022

Surfaces in the flag threefold containing smooth conics and twistor fibers

We study smooth integral curves of bidegree $(1,1)$, called \textit{smooth conics}, in the flag threefold $\mathbb{F}$. The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection $\mathbb{F}\to\mathbb{CP}^{2}$. We give a bound on the maximum number of smooth conics contained in a smooth surface $S\subset\mathbb{F}$. Then, we show qualitative properties of algebraic surfaces containing a prescribed number of smooth conics. Lastly, we study surfaces containing infinitely many twistor fibers. We show that the only smooth cases are surfaces of bidegree $(1,1)$. Then, for any integer $a>1$, we exhibit a method to construct an integral surface of bidegree $(a,a)$ containing infinitely many twistor fibers.

math.AG↗

arXiv2021

Spherical coefficients of slice regular functions

Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of $f$ with those of its slice derivative $\partial_{c}f$ obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.

math.CV↗

arXiv2020

Equivalence of slice semi-regular functions via Sylvester operators

The aim of this paper is to study some features of slice semi-regular functions $\mathcal{RM}(Ω)$ on a circular domain $Ω$ contained in the skew-symmetric algebra of quaternions $\mathbb{H}$ via the analysis of a family of linear operators built from left and right $*$-multiplication on $\mathcal{RM}(Ω)$; this class of operators includes the family of Sylvester-type operators $\mathcal{S}_{f,g}$. Our strategy is to give a matrix interpretation of these operators as we show that $\mathcal{RM}(Ω)$ can be seen as a $4$-dimensional vector space on the field $\mathcal{RM}_{\mathbb{R}}(Ω)$. We then study the rank of $\mathcal{S}_{f,g}$ and describe its kernel and image when it is not invertible. By using these results, we are able to characterize when the functions $f$ and $g$ are either equivalent under $*$-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. We also provide a complete classification of idempotents and zero divisors on product domains of $\mathbb{H}$.

math.CV↗

arXiv2020

Canonical models on strongly convex domains via the squeezing function

We prove that if a holomorphic self-map $f\colon Ω\to Ω$ of a bounded strongly convex domain $Ω\subset \mathbb C^q$ with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball $\mathbb B^k$. We also obtain the dual result for a holomorphic self-map $f\colon Ω\to Ω$ with a boundary repelling fixed point. Both results are obtained by rescaling the dynamics of $f$ via the squeezing function.

math.CV↗