Leo Jimenez
The Ohio State University
Title
First integrals of planar vector fields and model theory
Abstract
A first integral of a vector field is a function that is constant along the trajectories of the vector field. In the 80s, Prelle and Singer studied vector fields with elementary first integrals: those that can be obtained using addition, multiplication, and composition of algebraic, exponential, logarithm and trigonometric functions. Even though the depth of composition needed is a priori arbitrary, they show that in fact, an elementary first integral must be of a very specific form: a sum of an algebraic function and logarithms of algebraic functions.
An example is given by the Lotka-Volterra systems: their elementary first integral was recently used by Duan, Eagles and myself to classify algebraic relations between solutions of these systems. In this talk, I will present work in progress, joint with Duan, which attempts to generalize these methods to arbitrary planar systems with an elementary first integral. Along the way, I will present a criterion for minimality of such a system (a key differential transcendence notion coming from model theory), and present a conjecture connecting minimality and elementary first integrals.
