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On the geometry of zero sets of central quaternionic polynomials

Gil Alon, Elad Paran

Abstract

Let R be the ring of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if a polynomial p in R vanishes at all the common zeros of I in H^n with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in H^n. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

On the geometry of zero sets of central quaternionic polynomials

Abstract

Let R be the ring of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if a polynomial p in R vanishes at all the common zeros of I in H^n with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in H^n. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.
Paper Structure (4 sections, 9 theorems, 14 equations)

This paper contains 4 sections, 9 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Prelinimaries
  3. Embedded spheres
  4. Proof of the main theorem

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 (17)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • ...and 7 more