A Model Theoretic Perspective on Matrix Rings
Igor Klep, Marcus Tressl
TL;DR
The paper analyzes the first-order theory of fixed-size matrix rings $M_n(K)$ with two added unary operations, trace and transpose, establishing precise conditions for quantifier elimination over formally real fields that are intersections of real closed fields. It shows that QE holds for these rings in an expanded language, provided a Specht-property-type invariant condition is satisfied, and it proves that such a definable, natural expansion fails for $M_n( olinebreak \mathbb{C})$ without incorporating simultaneous-conjugacy invariants. The work links QE to invariant theory via the simultaneous conjugacy problem and demonstrates that, in the dimension-free setting (structures capturing matrices of all sizes), undecidability prevails by embedding the universal Horn theory of finite groups. Together, these results delineate the boundaries between definability, quantifier elimination, and decidability in matrix-ring contexts, illuminating how trace and transpose interact with invariants to govern eliminability in algebraic structures.
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 \textit{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.