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A model theoretic perspective on matrix ringsIgor Klep,
Marcus Tressl, 2025, izvirni znanstveni članek
Povzetek: 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.
Ključne besede: model theory, quantifier elimination, matrix rings, trace, decidability, free analysis, simultaneous conjugacy problem
Objavljeno v DiRROS: 20.10.2025; Ogledov: 282; Prenosov: 138
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