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 $$T_{\hat{I}}¸\cup \{{\sum }_k{\bigcap }_l\mathrm{Ker}(\ddot{I}{}_{j,k,l}[\hat{I}¸])={\sum }_k{\bigcap }_l\mathrm{Ker}(\hat{I}{\cdot }_{j,k,l}[\hat{I}¸]):j\in \mathcal{J}\},$$ where all sums and intersections are finite, and all the $\ddot{I}[\hat{I}¸]$'s and $\hat{I}\cdot [\hat{I}¸]$'s are polynomials over $K$ with $\hat{I}¸$ plugged in. Note that properties such as $\hat{I}{¸}^2-2\mathrm{Id}=0$ or $\mathrm{Ker}(\hat{I}{¸}^n)=\mathrm{Ker}(\hat{I}{¸}^{n+1})$ can be expressed in such a form. We then parametrize the consistent extensions of this form by a family $\{T_{\hat{I}}^C¸:C\in \mathcal{C}\}$ and characterize the existentially closed models of each $T_{\hat{I}}^C¸$. 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 $T_{\hat{I}}^C¸$ exists.

56 pages

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