A model theoretic perspective on matrix rings

Klep, Igor (author), Tressl, Marcus (author)

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.

• Language: English
• Keywords: model theory, quantifier elimination, matrix rings, trace, decidability, free analysis, simultaneous conjugacy problem
• 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: 20 str.
• Numbering: Vol. 309, iss. 3, art. 45
• PID: 20.500.12556/RUL-167034
• UDC: 512
• ISSN on article: 0025-5874
• DOI: 10.1007/s00209-024-03671-w
• COBISS.SI-ID: 225046275
• Publication date in RUL: 05.02.2025

Record is a part of a journal

• Title: Mathematische Zeitschrift
• Shortened title: Math. Z.
• Publisher: Springer Nature
• ISSN: 0025-5874
• COBISS.SI-ID: 25915904

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.