arXiv subjects

Antonio Carbone

Publications and source records attributed to Antonio Carbone.

10 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

Differentiable approximation of continuous locally definable maps that preserves the image

Recently, we showed that continuous definable maps defined on compact definable sets can be uniformly approximated by continuous definable maps of class $\mathcal{C}^p$ without changing their image. The aim of this paper is to extend the previous result, this time taking into account the (strong) Whitney topology, to continuous locally definable maps defined on locally compact locally definable sets. The argument is an interplay between o-minimal and PL geometry and makes essential use of Paw\l{}ucki's desingularization techniques as well as our aforementioned result for the compact case.

math.AG↗

arXiv2026

$K$-holomorphic functions with definable real part

Let $R$ be a real closed field and $K:=R(i)$ its algebraic closure. Let $U\subset K^n$ be an open and definable set in a fixed o-minimal structure. In this note, we study the relationship between definability of a $K$-holomorphic function $f=f_1+if_2:U\to K$ and the definability and (strong) $R$-analyticity of its real part $f_1:U\to R$. Our results turn out to be the best possible {in general}, and their precision depends on the considered o-minimal structure. We obtain a complete characterisation in the semialgebraic case.

math.AG↗

arXiv2026

Applications of the Nash double of a Nash manifold with corners

In this work we study some properties and applications of Nash manifolds with corners. Our first main result shows how to `build' a Nash manifold with corners ${\mathcal Q}\subset{\mathbb R}^n$ from a suitable Nash manifold $M\subset{\mathbb R}^n$ (of its same dimension), that contains ${\mathcal Q}$ as a closed subset, by folding $M$ along the irreducible components of a normal-crossings divisor of $M$ (the smallest Nash subset of $M$ that contains the boundary $\partial{\mathcal Q}$ of ${\mathcal Q}$). Our second main results shows that we can choose as the Nash manifold $M$ the Nash `double' $D({\mathcal Q})$ of ${\mathcal Q}$, which is the analogous to the Nash double of a Nash manifold with (smooth) boundary, but $D({\mathcal Q})$ takes into account the peculiarities of the boundary of a Nash manifold with corners. We propose several applications of the previous results: (1) Nash ramified coverings of closed semialgebraic sets, (2) Weak Nash uniformization of closed semialgebraic sets using Nash manifolds with (smooth) boundary, (3) Representation of compact semialgebraic sets connected by analytic paths as images under Nash maps of closed unit balls, (4) Explicit construction of Nash models for compact orientable smooth surfaces of genus $g\geq0$, and (5) Nash approximation of continuous semialgebraic maps whose target spaces are Nash manifolds with corners.

math.AG↗

arXiv2026

Nash approximation of differentiable semialgebraic maps

Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $\mu$-approximation target space} (or a $({\mathcal N},\mu)$-${\tt ats}$ for short) if it has the following universal approximation property: {\em For each $m\in{\mathbb N}$ and each locally compact semialgebraic subset $S\subset{\mathbb R}^m$, the subspace of Nash maps ${\mathcal N}(S,T)$ is dense in the space ${\mathcal S}^\mu(S,T)$ of ${\mathcal C}^\mu$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({\mathcal N},\mu)$-${\tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are $({\mathcal N},\mu)$-${\tt ats}$ for each $\mu\geq0$}. As an application of a stronger version of the previous statement, we show that if two Nash maps $f,g:S\to Q$, where $S$ is a locally compact semialgebraic set of ${\mathbb R}^m$ and $Q$ is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of ${\mathcal S}^0(S,T)$ (and consequently they are (continuous) semialgebraically homotopic), then $f,g$ are Nash homotopic.

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↗

arXiv2025

Holomorphic functions with Nash real part

In this paper we show that a holomorphic function, defined on an open subset $D$ of $\mathbb{C}^n$, is a complex Nash function if and only if its real part (or equivalently its imaginary part) is a real Nash function.

math.CV↗

arXiv2023

Differentiable approximation of continuous definable maps that preserves the image

Recently Paw\l{}ucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class $\mathcal{C}^p$ for each integer $p\geq 1$. In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image after the approximation. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a `surjective definable version' of the finite simplicial approximation theorem that we prove here.

math.AG↗

arXiv2023

Nash uniformization of chessboard sets by Nash manifolds with corners

Bierstone and Parusi\'nski studied the desingularization of $d$-dimensional closed subanalytic sets and in particular of $d$-dimensional closed semialgebraic sets. Their main tools are Hironaka's desingularization of real algebraic sets (to `uniform' the Zariski closure of the closed semialgebraic set) and Hironaka's embedded desingularization of real algebraic subsets of non-singular real algebraic sets (to uniform afterwards the Zariski closure of the boundary of the uniformed closed semialgebraic set). The obtained models in the desingularization process, that we call in the following closed chessboard sets, are the closures of (finite) unions of connected components of the complements of normal-crossings divisors of non-singular real algebraic sets. The local models for $d$-dimensional chessboard sets are unions of (standard) closed orthants of ${\mathbb R}^d$, that is, $\bigcup_{(\varepsilon_1,\ldots,\varepsilon_d)\in{\mathfrak F}}\{\varepsilon_1{\tt x}_1\geq0,\ldots,\varepsilon_d{\tt x}_d\geq0\}\subset{\mathbb R}^d$ for some set ${\mathfrak F}\subset\{-1,1\}^d$. We study the Nash uniformization of $d$-dimensional closed chessboard sets ${\mathcal S}$ using Nash manifolds with corners ${\mathcal Q}$ with the same number of connected components as ${\mathcal S}$ (or equivalently the same number of irreducible components). Nash manifolds with corners are closed chessboard set whose local models are either ${\mathbb R}^d$ or semialgebraic sets of the type $\{{\tt x}_1\geq0,\ldots,{\tt x}_k\geq0\}$ for some $1\leq k\leq d$. More generally, a chessboard set is a semialgebraic set in between a finite union of connected components of the complement of a normal-crossings divisor of non-singular real algebraic set and its closure. We also provide a Nash uniformization result for general chessboard sets ${\mathcal S}$.

math.AG↗

arXiv2023

Surjective Nash maps between semialgebraic sets

In this work we study the existence of surjective Nash maps between two given semialgebraic sets ${\mathcal S}$ and ${\mathcal T}$. Some key ingredients are: the irreducible components ${\mathcal S}_i^*$ of ${\mathcal S}$ (and their intersections), the analytic-path connected components ${\mathcal T}_j$ of ${\mathcal T}$ (and their intersections) and the relations between dimensions of the semialgebraic sets ${\mathcal S}_i^*$ and ${\mathcal T}_j$. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps. The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: {\em a full characterization of the semialgebraic subsets ${\mathcal S}\subset{\mathbb R}^n$ that are the image of the closed unit ball $\overline{\mathcal B}_m$ of ${\mathbb R}^m$ centered at the origin under a Nash map $f:{\mathbb R}^m\to{\mathbb R}^n$}. The necessary and sufficient conditions that must satisfy such a semialgebraic set ${\mathcal S}$ are: {\em it is compact, connected by analytic paths and has dimension $d\leq m$}. Two remarkable consequences of the latter result are the following: (1) {\em pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension $d$ are Nash images of $\overline{\mathcal B}_d$}, and (2) {\em compact semialgebraic sets of dimension $d$ are projections of non-singular algebraic sets of dimension $d$ whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions)}.

math.AG↗