| Naslov: | Parallels between quaternionic and matrix Nullstellensätze |
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| Avtorji: | ID Cimprič, Jaka (Avtor) |
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| Datoteke: |  PDF - Predstavitvena datoteka, prenos (743,39 KB)
MD5: F4EAECCF6EED05C52B883CF67091D0B1
 URL - Izvorni URL, za dostop obiščite https://www.sciencedirect.com/science/article/pii/S0021869325003278
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| Jezik: | Angleški jezik |
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| Tipologija: | 1.01 - Izvirni znanstveni članek |
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| Organizacija: |  IMFM - Inštitut za matematiko, fiziko in mehaniko
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| Povzetek: | 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 $(AF)^N \in I + I(AF) + \ldots + I(AF)^N.$ |
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| Ključne besede: | Hilbert's Nullstellensatz, matrix polynomials, quaternionic polynomials, one-sided ideals, free modules |
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| Status publikacije: | Objavljeno |
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| Verzija publikacije: | Objavljena publikacija |
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| Datum objave: | 01.11.2025 |
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| Leto izida: | 2025 |
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| Št. strani: | str. 92-108 |
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| Številčenje: | Vol. 682 |
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| PID: | 20.500.12556/DiRROS-23903  |
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| UDK: | 512 |
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| ISSN pri članku: | 0021-8693 |
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| DOI: | 10.1016/j.jalgebra.2025.05.022 |
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| COBISS.SI-ID: | 240581635 |
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| Opomba: |
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| Datum objave v DiRROS: | 21.10.2025 |
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| Število ogledov: | 135 |
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| Število prenosov: | 73 |
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| Metapodatki: |    |
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