Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2510.09779

Division algebras of slice-Nash functions

Cinzia Bisi·Antonio Carbone
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.CVmath.AG

Keep this discovery

BibTeXRIS

Cinzia Bisi, Antonio Carbone. 2025-10-10. Division algebras of slice-Nash functions. https://arxiv.org/abs/2510.09779

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The two-dimensional Matkowski--Sut\^o equation with holomorphic and strictly increasing generators

We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.

math.CV↗
arXiv2026

A counterexample to an open problem of Dorff

The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized convex harmonic mapping with bounded image must remain in the same class. We construct a normalized sense-preserving harmonic diffeomorphism that maps the unit disk onto an ellipse; its self-convolution has vanishing Jacobian at some interior point of the unit disk, which provides a negative answer to Dorff's open problem.

math.CV↗
arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a K\"ahler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗