| Title: | A model theoretic perspective on matrix rings |
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| Authors: | ID Klep, Igor (Author)
ID Tressl, Marcus (Author) |
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| Files: |  PDF - Presentation file, download (367,37 KB)
MD5: 19E4713AD70D9254F5E994F5C3041E9D
 URL - Source URL, visit https://link.springer.com/article/10.1007/s00209-024-03671-w
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | model theory, quantifier elimination, matrix rings, trace, decidability, free analysis, simultaneous conjugacy problem |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.03.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 20 str. |
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| Numbering: | Vol. 309, iss. 3, article no. 45 |
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| PID: | 20.500.12556/DiRROS-23892  |
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| UDC: | 512 |
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| ISSN on article: | 0025-5874 |
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| DOI: | 10.1007/s00209-024-03671-w |
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| COBISS.SI-ID: | 225046275 |
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| Publication date in DiRROS: | 20.10.2025 |
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| Views: | 195 |
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| Downloads: | 90 |
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| Metadata: |    |
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