Parallels between quaternionic and matrix Nullstellensätze

Cimprič, Jaka (author)

We prove a new quaternionic and a new matrix Nullstellensatz. We also show that both theories are intertwined. For every $g_1, \ldots, g_m, f \in {\mathbb H}[x_1, \ldots, x_d]$ (where $x_1, \ldots, x_d$ are central), we show that the following are equivalent: (a) For every $a \in {\mathbb H}^d$ whose components pairwise commute and which satisfies $g_1(a) = \cdots = g_m(a) = 0$, we have $f(a) = 0$. (b) $f$ belongs to the smallest semiprime left ideal containing $g_1, \ldots, g_m$. On the other hand, for every $G_1, \ldots, G_m, F \in M_n({\mathbb k}[x_1, \ldots, x_d])$, where ${\mathbb k}$ is an algebraically closed field, we show that the following are equivalent (where $I$ is the left ideal generated by $G_1, \ldots, G_m$): (a) For every $a \in {\mathbb k}^d$ and $v \in {\mathbb k}^n$ such that $G_1(a)v = \ldots = G_m(a)v = 0,$ we have $F(a)v = 0$. (b) For every $A \in M_n({\mathbb k})$ there exists $N \in \mathbb{N}_0$ such that $(AF)^N \in I + I(AF) + \ldots + I(AF)^N.$

• Language: English
• Keywords: Hilbert's Nullstellensatz, matrix polynomials, quaternionic polynomials, one-sided ideals, free modules
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: Str. 92-108
• Numbering: Vol. 682
• PID: 20.500.12556/RUL-175183
• UDC: 512
• ISSN on article: 0021-8693
• DOI: 10.1016/j.jalgebra.2025.05.022
• COBISS.SI-ID: 240581635
• Publication date in RUL: 20.10.2025

Record is a part of a journal

• Title: Journal of algebra
• Shortened title: J. algebra
• Publisher: Elsevier
• ISSN: 0021-8693
• COBISS.SI-ID: 1310986

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• License: CC BY-NC 4.0, Creative Commons Attribution-NonCommercial 4.0 International
• Link: http://creativecommons.org/licenses/by-nc/4.0/
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