A Model Theoretic Perspective on Matrix Rings

TL;DR

This paper investigates conditions for quantifier elimination in matrix rings with added functions, linking model theory, invariant theory, and matrix conjugacy problems, and establishes undecidability in certain matrix structures.

Contribution

It provides necessary and sufficient conditions for quantifier elimination in matrix rings with trace and transposition, and relates definability issues to the conjugacy problem.

Findings

Quantifier elimination is achieved for matrix rings over intersections of real closed fields.

Finding a definable expansion with quantifier elimination for complex matrices relates to the conjugacy problem.

Certain matrix structures are undecidable, preventing quantifier elimination.

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.