Chris Miller : Expansions of the real field by trajectories of linear vector fields

Posted by Matthias Aschenbrenner , part of the Logic Seminar.

At

Aug. 29, 2006, 4 p.m.

In

SEO 427

Abstract

Let $F\colon \mathbb R^n\to \mathbb R^n$ be linear and $\gamma\colon \mathbb R\to \mathbb R^n$ be differentiable such that $\gamma'(t)=F(\gamma(t))$ for all $t\in \mathbb R$. Then the image $ \gamma(\mathbb R)$ is interdefinable over the real field with at least one of: the real exponential function $e^x$; the complex exponential function $e^z$; or a finite set of functions $t\mapsto t^w\colon (0,\infty) \to \mathbb C$, where $w\in\mathbb C$. Moreover, which case(s) hold can be semialgebraically computed from the coefficients of $F$.