Model Theory of Generic Vector Space Endomorphisms

Abstract

This paper deals with the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we define $T_θ:= T \cup \{\text{``$θ$definesa$K$-endomorphismof$\mathbb{V}$''}\}$. We then consider extensions of the form $$T_{\theta }\cup \{{\sum }_k{\bigcap }_l\mathrm{Ker}({\rho }_{j,k,l}[\theta ])={\sum }_k{\bigcap }_l\mathrm{Ker}({\eta }_{j,k,l}[\theta ]):j\in \mathcal{J}\},$$ where all sums and intersections are finite, and all the $\rho [\theta ]$'s and $\eta [\theta ]$'s are polynomials over $K$ with $\theta$ plugged in. Note that properties such as ${\theta }^2-2\mathrm{Id}=0$ or $\mathrm{Ker}({\theta }^n)=\mathrm{Ker}({\theta }^{n+1})$ can be expressed in such a form. We then parametrize the consistent extensions of this form by a family $\{T_{\theta }^C:C\in \mathcal{C}\}$ and characterize the existentially closed models of each $T_{\theta }^C$. We also present a sufficient criterion, which depends only on $T$, for when these characterizations are first-order expressible, i.e., for when a model companion of each $T_{\theta }^C$ exists.

arXiv comment56 pages