2025/02/19 by Leon Chini, Chini, Leon
Engineering · #Control and Dynamics of Mobile Robots
paper · pdf · doi:10.48550/arxiv.2502.13667
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 \mathbbV in every model, we define Tθ:= T ∪ \\text``θ defines a K-endomorphism of \mathbbV''\. We then consider extensions of the form Tθ∪ \∑\nolimitsk\bigcap\nolimitslKer(ρj, k, l[θ]) = ∑\nolimitsk\bigcap\nolimitsl Ker(ηj, k, l[θ]) : j ∈ J\, where all sums and intersections are finite, and all the ρ[θ]'s and η[θ]'s are polynomials over K with θ plugged in. Note that properties such as θ2 - 2Id = 0 or Ker(θn) = Ker(θn+1) can be expressed in such a form. We then parametrize the consistent extensions of this form by a family \TCθ: C ∈ C\ and characterize the existentially closed models of each TCθ. 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 TCθ exists.