On the geometry of zero sets of central quaternionic polynomials

Gil Alon; Elad Paran

doi:10.1016/j.jalgebra.2024.07.019

On the geometry of zero sets of central quaternionic polynomials

Gil Alon, Elad Paran

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let R be the ring H[x₁,…,x_n] 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 p∈R vanishes at all the common zeros of I in Hⁿ with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hⁿ. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

Original language	English
Pages (from-to)	780-788
Number of pages	9
Journal	Journal of Algebra
Volume	659
DOIs	https://doi.org/10.1016/j.jalgebra.2024.07.019
State	Submitted - 2 Feb 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier Inc.

Keywords

Central polynomials
Nullstellensatz
Quaternionic
Slice regular functions

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10.1016/j.jalgebra.2024.07.019

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AB - Let R be the ring H[x1,…,xn] 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 p∈R vanishes at all the common zeros of I in Hn with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hn. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

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KW - Nullstellensatz

KW - Quaternionic

KW - Slice regular functions

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On the geometry of zero sets of central quaternionic polynomials

Abstract

Bibliographical note

Keywords

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