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On the geometry of zero sets of central quaternionic polynomials II

Gil Alon, Adam Chapman, Elad Paran

TL;DR

The work analyzes zero sets of left ideals in the quaternionic polynomial ring and introduces the algebraic hull $\mathcal{H}(v)$ as the enveloping multisphere $B(v)$. It proves $\mathcal{H}(v)=B(v)$ and uses multisphere geometry to give a new proof of the central-zeros theorem, while also showing the result does not extend to arbitrary division algebras via a constructed counterexample. The methodology combines slice-regularity, multi-affine behavior on multispheres, and explicit left-ideal constructions to illuminate the structure of central zero sets and delineate the theorem's limitations. Overall, the paper clarifies the geometry of central zero sets in quaternionic polynomials and delineates the boundaries of the main result beyond quaternions.

Abstract

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and prove it is a product of spheres. Using this description we give a new proof to a conjecture of Gori, Sarfatti and Vlacci. We also show that the main result of [2] does not extend to general division algebras.

On the geometry of zero sets of central quaternionic polynomials II

TL;DR

The work analyzes zero sets of left ideals in the quaternionic polynomial ring and introduces the algebraic hull as the enveloping multisphere . It proves and uses multisphere geometry to give a new proof of the central-zeros theorem, while also showing the result does not extend to arbitrary division algebras via a constructed counterexample. The methodology combines slice-regularity, multi-affine behavior on multispheres, and explicit left-ideal constructions to illuminate the structure of central zero sets and delineate the theorem's limitations. Overall, the paper clarifies the geometry of central zero sets in quaternionic polynomials and delineates the boundaries of the main result beyond quaternions.

Abstract

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and prove it is a product of spheres. Using this description we give a new proof to a conjecture of Gori, Sarfatti and Vlacci. We also show that the main result of [2] does not extend to general division algebras.
Paper Structure (6 sections, 15 theorems, 22 equations)

This paper contains 6 sections, 15 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multispheres
  4. The algebraic hull
  5. A new proof of Theorem \ref{['thm:central_zeros']}
  6. A counterexample

Key Result

Theorem 1.1

Let $I$ be a left ideal of $R=\mathbb{H}[x_1,\dotsc,x_n]$. If a polynomial $f\in R$ vanishes on $\mathcal{V}_c(I)$, then $f$ vanishes on $\mathcal{V}(I)$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 18 more