| Title: | Parallels between quaternionic and matrix Nullstellensätze |
|---|
| Authors: | ID Cimprič, Jaka (Author) |
|---|
| Files: |  PDF - Presentation file, download (743,39 KB)
MD5: F4EAECCF6EED05C52B883CF67091D0B1
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0021869325003278
|
|---|
| Language: | English |
|---|
| Typology: | 1.01 - Original Scientific Article |
|---|
| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
|
|---|
| Abstract: | 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 $(AF)^N \in I + I(AF) + \ldots + I(AF)^N.$ |
|---|
| Keywords: | Hilbert's Nullstellensatz, matrix polynomials, quaternionic polynomials, one-sided ideals, free modules |
|---|
| Publication status: | Published |
|---|
| Publication version: | Version of Record |
|---|
| Publication date: | 01.11.2025 |
|---|
| Year of publishing: | 2025 |
|---|
| Number of pages: | str. 92-108 |
|---|
| Numbering: | Vol. 682 |
|---|
| PID: | 20.500.12556/DiRROS-23903  |
|---|
| UDC: | 512 |
|---|
| ISSN on article: | 0021-8693 |
|---|
| DOI: | 10.1016/j.jalgebra.2025.05.022 |
|---|
| COBISS.SI-ID: | 240581635 |
|---|
| Note: |
|
|---|
| Publication date in DiRROS: | 21.10.2025 |
|---|
| Views: | 136 |
|---|
| Downloads: | 76 |
|---|
| Metadata: |    |
|---|
|
:
|
Copy citation |
|---|
| | | | Share: |  |
|---|
Hover the mouse pointer over a document title to show the abstract or click
on the title to get all document metadata. |
