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Division algebras of slice-Nash functions

Cinzia Bisi, Antonio Carbone

TL;DR

This work defines a natural and robust notion of Nash functions in the context of slice regular hypercomplex analysis, addressing the inadequacy of naive algebraicity over slice polynomials by introducing stem-Nash functions and the induced slice-Nash class. It develops a coherent framework in which slice-Nash functions sit between slice regular functions and slice polynomials, preserving key finiteness and approximation properties akin to the real/complex Nash theory. Central results include characterisations via splitting and real components, stability under slice differentiation, and the algebraic structure of the SN_A(Ω) family, together with a parallel theory for semiregular slice-Nash functions. The paper also establishes global and local finiteness properties: global slice-Nash functions reduce to slice polynomials, zeros are finite in appropriate domains, and growth at infinity is polynomially bounded, which together with the semiregular theory yields a tractable, algebraically tamable analogue of holomorphic/numerical-analytic behavior in the quaternionic and octonionic settings.

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.

Division algebras of slice-Nash functions

TL;DR

This work defines a natural and robust notion of Nash functions in the context of slice regular hypercomplex analysis, addressing the inadequacy of naive algebraicity over slice polynomials by introducing stem-Nash functions and the induced slice-Nash class. It develops a coherent framework in which slice-Nash functions sit between slice regular functions and slice polynomials, preserving key finiteness and approximation properties akin to the real/complex Nash theory. Central results include characterisations via splitting and real components, stability under slice differentiation, and the algebraic structure of the SN_A(Ω) family, together with a parallel theory for semiregular slice-Nash functions. The paper also establishes global and local finiteness properties: global slice-Nash functions reduce to slice polynomials, zeros are finite in appropriate domains, and growth at infinity is polynomially bounded, which together with the semiregular theory yields a tractable, algebraically tamable analogue of holomorphic/numerical-analytic behavior in the quaternionic and octonionic settings.

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.

Paper Structure

This paper contains 33 sections, 47 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Hypercomplex analysis and geometry
  3. Nash geometry
  4. Slice-Nash functions
  5. Preliminaries
  6. Real quaternions and octonions
  7. Slice functions
  8. Slice regular functions
  9. Splitting lemma
  10. Slice product
  11. Slice polynomials
  12. Normal function
  13. Zeros of slice functions
  14. Reciprocals of slice functions
  15. Singularities of slice regular functions
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $X\subset {\mathbb R}^n$ be a compact algebraic set (resp. a compact Nash manifold) and $Y\subset {\mathbb R}^n$ be any algebraic set (resp. Nash set). Then every real analytic map $f:X\to Y$ can be uniformly approximated by real Nash maps $g:X\to Y$.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1: Artin's theorem ku
  • Definition 2.3: Stem function
  • Proposition 2.5
  • Definition 2.6: Slice function
  • Proposition 2.7: Representation formula gp
  • ...and 88 more