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A note on irreducible slice algebraic sets

Anna Gori, Giulia Sarfatti, Fabio Vlacci

TL;DR

This note completes the characterization of irreducible slice algebraic sets associated with radical ideals in the quaternionic setting by showing that for a quasi prime radical right ideal $I$, the symmetrized zero set $\mathbb{S}_{\mathcal{V}_c(I)}$ is irreducible, and hence $\mathcal{V}_c(I)$ is irreducible iff $I$ is quasi prime. The authors remove the previously required irreducibility assumption on $\mathbb{S}_{\mathcal{V}_c(I)}$ via a key lemma, leveraging the relation $\mathbb{S}_{\mathcal{V}_c(I)}=\mathcal{V}_c(\mathcal{S}(I))$. They also discuss the necessity of the radical hypothesis and provide examples showing that radical ideals need not be quasi prime, while exact implications fail without radicality. The result sharpens the correspondence between algebraic properties of right ideals and geometric irreducibility for slice algebraic sets in quaternionic polynomial theory.

Abstract

In this short note we prove that if $I$ is a right radical and quasi prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization $\mathbb S_{V_c(I)}$ is an irreducible algebraic set, where $V_c(I)$ is the set of common zeros with commuting components of polynomials in $I$. Combining this fact with the results proved in our previous paper [3], we obtain that for $I$ radical, $V_c(I)$ is irreducible if and only if $I$ is quasi prime.

A note on irreducible slice algebraic sets

TL;DR

This note completes the characterization of irreducible slice algebraic sets associated with radical ideals in the quaternionic setting by showing that for a quasi prime radical right ideal , the symmetrized zero set is irreducible, and hence is irreducible iff is quasi prime. The authors remove the previously required irreducibility assumption on via a key lemma, leveraging the relation . They also discuss the necessity of the radical hypothesis and provide examples showing that radical ideals need not be quasi prime, while exact implications fail without radicality. The result sharpens the correspondence between algebraic properties of right ideals and geometric irreducibility for slice algebraic sets in quaternionic polynomial theory.

Abstract

In this short note we prove that if is a right radical and quasi prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization is an irreducible algebraic set, where is the set of common zeros with commuting components of polynomials in . Combining this fact with the results proved in our previous paper [3], we obtain that for radical, is irreducible if and only if is quasi prime.
Paper Structure (3 sections, 5 theorems, 22 equations)

This paper contains 3 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result

Key Result

Proposition 2.2

Let ${ \textbf{a}}=(a_1,\ldots,a_n)\in \mathbb H^n$ be such that $a_la_m=a_ma_l$ for any $1\leq l,m\leq n$ and let $P,Q \in \mathbb H[q_1,\ldots,q_n]$. Then

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 2.2: Nul2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: aryapoor
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 10 more