Searcharxiv⌕ Search

arXiv subjects

Riccardo Ghiloni

Publications and source records attributed to Riccardo Ghiloni.

At least 19 recordsLinked to original sources
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

A manifold Fueter-Sce phenomenon in one hypercomplex variable

Fueter's theorem states, in modern terms, that the Laplacian maps slice-regular quaternionic functions into Fueter-regular functions with axial symmetry. This phenomenon is also present in the Clifford setting, where both slice-monogenic functions and generalized partial-slice monogenic functions are mapped by the Laplacian into monogenic functions with axial symmetry. These results are due, respectively, to Sce and Qian and to Xu and Sabadini. The present work puts the Fueter-Sce phenomenon into context for the wider class of strongly $T$-regular functions. It shows that the phenomenon appears over general associative $*$-algebras. Moreover, the symmetry considered here is multi-axial in a sense introduced by Eelbode. Additionally, but more surprisingly, the phenomenon studied by Fueter, Sce, Xu and Sabadini turns out to be the last step in a multi-step process. A new phenomenon in one hypercomplex variable is therefore discovered.

math.CV↗
arXiv2026

A unified theory of regular functions of a hypercomplex variable

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.

math.CV↗
arXiv2025

The Nash-Tognoli theorem over the rationals and its version for isolated singularities

Let $\mathbb{Q}$ be the field of rational numbers and let $X$ be a subset of $\mathbb{R}^n$. We say that $X$ is $\mathbb{Q}$-algebraic if it is the common zero set in $\mathbb{R}^n$ of a family of polynomials in $\mathbb{Q}[\mathtt{x}_1,\ldots,\mathtt{x}_n]$. If $X$ is $\mathbb{Q}$-algebraic and of dimension $d$, then we say that $X$ is $\mathbb{Q}$-nonsingular if, for all $a\in X$, there exist a neighborhood $U$ of $a$ in $\mathbb{R}^n$ and $f_1,\ldots,f_{n-d}\in\mathbb{Q}[\mathtt{x}_1,\ldots,\mathtt{x}_n]$ such that $\nabla f_1(a),\ldots,\nabla f_{n-d}(a)$ are linearly independent and $X\cap U=\{x\in U:f_1(x)=0,\cdots,f_{n-d}(x)=0\}$. The celebrated Nash-Tognoli theorem asserts the following: if $M$ is a compact smooth manifold of dimension $d$ and $ψ:M\to\mathbb{R}^{2d+1}$ is a smooth embedding, then $ψ$ can be approximated by an arbitrarily close smooth embedding $ϕ:M\to\mathbb{R}^{2d+1}$ whose image $ϕ(M)$ is a nonsingular algebraic subset of $\mathbb{R}^{2d+1}$. In this article, we prove that $ϕ$ can be chosen in such a way that $ϕ(M)$ is a $\mathbb{Q}$-nonsingular $\mathbb{Q}$-algebraic subset of $\mathbb{R}^{2d+1}$. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold $M$ can be described both globally and locally by means of finitely many exact data, such as a finite system of generators of the ideal of polynomials in $\mathbb{Q}[\mathtt{x}_1,\ldots,\mathtt{x}_{2d+1}]$ vanishing on $ϕ(M)$. We extend our result to the singular setting by proving that every real algebraic set with finitely many singularities is semialgebraically homeomorphic to a $\mathbb{Q}$-algebraic set with the same number of singularities.

math.AG↗
arXiv2025

Subfield-algebraic geometry

In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study $K$-algebraic subsets of $R^n$, i.e., those subsets of $R^n$ that are the zero loci of polynomials with coefficients in $K$. Real algebraic geometry already covers the case when $K$ is also a real closed field. Our goal is to extend real algebraic geometry to the case when $K$ is not real closed, for example when $K$ is the field $\mathbb{Q}$ of rational numbers. Several new geometric phenomena appear. There is no complex counterpart to this generalized real algebraic geometry. The reason is as follows. If $C|K$ is a field extension with $C$ algebraically closed and $X$ is a $K$-algebraic subset of $C^n$, then Hilbert's Nullstellensatz implies that the ideal of polynomials with coefficients in $C$ that vanish on~$X$ is generated by the ideal of polynomials with coefficients in $K$ that vanish on $X$. In the real realm, this is false in general, for example when we consider field extensions $R|K$ with $R$ real closed and $K=\mathbb{Q}$. This monograph also presents some applications of the theory developed. Here is an example. The celebrated Nash-Tognoli theorem states that every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set $M'$, called algebraic model of $M$. The theory developed here provides the theoretical basis to prove that the algebraic model $M'$ of $M$ can be chosen to be $\mathbb{Q}$-algebraic and $\mathbb{Q}$-nonsingular. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold can be encoded both globally and locally involving only finitely many exact data.

math.AG↗
arXiv2025

A converse to Cartan's Theorem B: The extension property for real analytic and Nash sets

In 1957 Cartan proved his celebrated Theorem B and deduced that if $Ω\subset{\mathbb R}^n$ is an open set and $X$ is a coherent real analytic subset of $Ω$, then $X$ has the analytic extension property: Each real analytic function on $X$ extends to a real analytic function on $Ω$. The converse implication remains unproven. In the literature only special cases of non-coherent real analytic sets $X\subsetΩ$ without the extension property appear: mainly real analytic sets $X\subsetΩ$ that have a visible `tail'. We prove the converse implication: If $X\subsetΩ$ has the analytic extension property, it is a coherent real analytic subset of $Ω$. Taking advantage of cohomology of sheaves, we provide for each non-coherent real analytic set $X$, `many' failing analytic functions on $X$ (that have no analytic extension to $Ω$), yielding an almost complete description of the possible non-extendability sets. We extend the previous characterization to the Nash case, which is somehow more demanding, because of its finiteness properties and its disappointing behavior with respect to cohomology of sheaves of Nash function germs, and we provide again an almost complete description of the possible non-extendability sets. Let $Ω\subset{\mathbb R}^n$ be an open semialgebraic set and let $X\subsetΩ$: Each Nash function on $X$ extends to a Nash function on $Ω$ if and only if $X\subsetΩ$ is a coherent Nash set. The `if' implication goes back to some celebrated results of Coste, Ruiz and Shiota. If $M\subset{\mathbb R}^n$ is a Nash manifold, ${\mathcal C}^\infty$ semialgebraic functions on $M$ coincide with Nash functions on $M$. As an application of our results, we provide a full characterization of the semialgebraic sets $S\subsetΩ$ for which ${\mathcal C}^\infty$ semialgebraic functions on $S$ coincide with Nash functions on $S$.

math.AG↗
arXiv2024

Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator

In the present paper, we prove a resolvent equation for the $\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent operator in an open neighborhood of any given quaternion belonging to the $\mathcal{S}$-resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the $\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the $\mathcal{S}$-resolvent operator.

math.SP↗
arXiv2023

Quaternionic slice regularity beyond slice domains

After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular function, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function $f:Ω\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $Ω$ (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent $f$ better and better and allow to prove increasingly stronger results.

math.CV↗
arXiv2023

A unified notion of regularity in one hypercomplex variable

We define a very general notion of regularity for functions taking values in an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail.

math.CV↗
arXiv2023

The curved Mimetic Finite Difference method: allowing grids with curved faces

We present a new mimetic finite difference method for diffusion problems that converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of $P_{0}$-consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries.

math.NA↗
arXiv2022

Slice regular functions and orthogonal complex structures over $\mathbb{R}^8$

This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions.

math.CV↗
arXiv2021

Inverting the discrete curl operator: a novel graph algorithm to find a vector potential of a given vector field

We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse theory. We introduce the new concept of complete acyclic matchings and we show that they give the same end result of Gaussian elimination. Basically, instead of doing costly row and column operations on a sparse matrix, we compute equivalent cheap combinatorial operations that preserve the underlying sparsity structure. Currently, the most efficient algorithms proposed in literature to find discrete vector potentials make use of tree-cotree techniques. We show that they compute a special type of complete acyclic matchings. Moreover, we show that the problem of computing them is equivalent to the problem of deciding whether a given mesh has a topological property called collapsibility. This fact gives a topological characterization of well-known termination problems of tree-cotree techniques. We propose a new recursive algorithm to compute discrete vector potentials. It works directly on basis elements of $1$- and $2$-chains by performing elementary Gaussian operations on them associated with acyclic matchings. However, the main novelty is that it can be applied recursively. Indeed, the recursion process allows us to sidetrack termination problems of the standard tree-cotree techniques. We tested the algorithm on pathological triangulations with known topological obstructions. In all tested problems we observe linear computational complexity as a function of mesh size. Moreover, the algorithm is purely graph-based so it is straightforward to implement and does not require specialized external procedures. We believe that our framework could offer new perspectives to sparse matrix computations.

math.NA↗
arXiv2021

On the generators of Clifford semigroups: polynomial resolvents and their integral transforms

This paper deals with generators $\mathsf{A}$ of strongly continuous right linear semigroups in Banach two-sided spaces whose set of scalars is an arbitrary Clifford algebra $\mathit{C}\ell(0,n)$. We study the invertibility of operators of the form $P(\mathsf{A})$, where $P(x)\in\mathbb{R}[x]$ is any real polynomial, and we give an integral representation for $P(\mathsf{A})^{-1}$ by means of a Laplace-type transform of the semigroup $\mathsf{T}(t)$ generated by $\mathsf{A}$. In particular, we deduce a new integral representation for the operator $(\mathsf{A}^2 - 2\mathrm{Re}(q) \,\mathsf{A} + |q|^2)^{-1}$. As an immediate consequence, we also obtain a new proof of the well-known integral representation for the $S$-resolvent operator of $\mathsf{A}$ (also called spherical resolvent operator of $\mathsf{A}$).

math.FA↗
arXiv2021

Smooth approximations in PL geometry

Let $Y\subset{\mathbb R}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a $\mathscr{C}^r$-approximation target space, or a $\mathscr{C}^r\text{-}\mathtt{ats}$ for short, if it has the following universal approximation property: For each $m\in{\mathbb N}$ and each locally compact subset $X$ of~${\mathbb R}^m$, any continuous map $f:X\to Y$ can be approximated by $\mathscr{C}^r$ maps $g:X\to Y$ with respect to the strong $\mathscr{C}^0$ Whitney topology. Taking advantage of new approximation techniques we prove: if $Y$ is weakly $\mathscr{C}^r$ triangulable, then $Y$ is a $\mathscr{C}^r\text{-}\mathtt{ats}$. This result applies to relevant classes of triangulable sets, namely: (1) every locally compact polyhedron is a $\mathscr{C}^\infty\text{-}\mathtt{ats}$, (2) every set that is locally $\mathscr{C}^r$ equivalent to a polyhedron is a $\mathscr{C}^r\text{-}\mathtt{ats}$, and (3) every locally compact locally definable set of an arbitrary o-minimal structure is a $\mathscr{C}^1\text{-}\mathtt{ats}$ (this includes locally compact locally semialgebraic sets and locally compact subanalytic sets). In addition, we prove: if $Y$ is a global analytic set, then each proper continuous map $f:X\to Y$ can be approximated by proper $\mathscr{C}^\infty$ maps $g:X\to Y$. Explicit examples show the sharpness of our results.

math.DG↗
arXiv2020

Slice-by-slice and global smoothness of slice regular and polyanalytic functions

The concept of slice regular function over the real algebra $\mathbb{H}$ of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let $Ω$ be an open subset of $\mathbb{H}$, which intersects $\mathbb{R}$ and is invariant under rotations of $\mathbb{H}$ around $\mathbb{R}$. A function $f:Ω\to\mathbb{H}$ is slice regular if it is of class $\mathscr{C}^1$ and, for all complex planes $\mathbb{C}_I$ spanned by $1$ and a quaternionic imaginary unit $I$, the restriction $f_I$ of $f$ to $Ω_I=Ω\cap\mathbb{C}_I$ satisfies the Cauchy-Riemann equations associated to $I$, i.e., $\overline{\partial}_I f_I=0$ on $Ω_I$, where $\overline{\partial}_I=\frac{1}{2}\big(\frac{\partial}{\partialα}+I\frac{\partial}{\partialβ}\big)$. Given any positive natural number $n$, a function $f:Ω\to\mathbb{H}$ is called slice polyanalytic of order $n$ if it is of class $\mathscr{C}^n$ and $\overline{\partial}_I^{\,n} f_I=0$ on $Ω_I$ for all $I$. We define global slice polyanalytic functions of order $n$ as the functions $f:Ω\to\mathbb{H}$, which admit a decomposition of the form $f(x)=\sum_{h=0}^{n-1}\overline{x}^hf_h(x)$ for some slice regular functions $f_0,\ldots,f_{n-1}$. Global slice polyanalytic functions of any order $n$ are slice polyanalytic of the same order $n$. The converse is not true: for each $n\geq2$, we give examples of slice polyanalytic functions of order $n$, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations $\overline{\partial}_I^{\,n} f_I=0$ on $Ω_I$ and as solutions of their global version $\overline{\vartheta}^nf=0$ on $Ω\setminus\mathbb{R}$. Our quaternionic results extend to the monogenic case.

math.CV↗
arXiv2020

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several variables ranging in any real alternative $^*$-algebra, including quaternions, octonions and Clifford algebras. This theory is an extension of the classical theory of holomorphic functions in several complex variables.

math.CV↗
arXiv2020

On a class of orientation-preserving maps of $\mathbb R^4$

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian $J_f$ of a slice regular function $f$ proving in particular that $\det(J_f)\geq0$, i.e. $f$ is orientation-preserving. We give a complete characterization of the fibers of $f$ making use of a new notion we introduce here, the one of wing of $f$. We investigate the singular set $N_f$ of $f$, i.e. the set in which $J_f$ is singular. The singular set $N_f$ turns out to be equal to the branch set of $f$, i.e. the set of points $y$ such that $f$ is not a homeomorphism locally at $y$. We establish the quasi-openness properties of $f$. As a consequence we deduce the validity of the Maximum Modulus Principle for $f$ in its full generality. Our results are sharp as we show by explicit examples.

math.CV↗