A model theoretic perspective on matrix rings

Klep, Igor (avtor), Tressl, Marcus (avtor)

In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings $M_n(K)$ in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This is used together with invariant theory to prove quantifier elimination when $K$ is an intersection of real closed fields. On the other hand, it is shown that finding a natural definable expansion with quantifier elimination of the theory of $M_n({\mathbb C})$ is closely related to the infamous simultaneous conjugacy problem in matrix theory. Finally, for various natural structures describing dimension-free matrices it is shown that no such elimination results can hold by establishing undecidability results.

• Jezik: Angleški jezik
• Ključne besede: model theory, quantifier elimination, matrix rings, trace, decidability, free analysis, simultaneous conjugacy problem
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2025
• Št. strani: 20 str.
• Številčenje: Vol. 309, iss. 3, art. 45
• PID: 20.500.12556/RUL-167034
• UDK: 512
• ISSN pri članku: 0025-5874
• DOI: 10.1007/s00209-024-03671-w
• COBISS.SI-ID: 225046275
• Datum objave v RUL: 05.02.2025

Gradivo je del revije

• Naslov: Mathematische Zeitschrift
• Skrajšan naslov: Math. Z.
• Založnik: Springer Nature
• ISSN: 0025-5874
• COBISS.SI-ID: 25915904

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.