Inférence et exploitation de conditions nécessaires pour l'existence de polynômes de Darboux

Dynamical systems can be used to model the time dependence of phenomena acting according to some law (such as physical laws), but do not generally have closed-form solutions. These systems can still be (partially) solved when they possess first integrals, functions that are constant on any solution of the system. Several important classes of first integrals can be constructed by combining sufficiently many polynomials, called Darboux polynomials. One can show the non-existence of such first integrals by exhaustive enumeration of Darboux polynomials, which can only be done up to a certain bound of their degree. In this thesis, we present algorithms that can generate proofs that a given system does not possess any Darboux polynomials. We provide a new, entirely automated proof that the Van der Pol oscillator does not have any Darboux polynomials. Our approach is not limited by the dimension of the system: we show that the Shimizu-Morioka system, of dimension 3, does not have any Darboux polynomial for all valuation of its parameters, which answers an open conjecture. Finally, we show how to speed up existing algorithms that generate Darboux polynomials up to a bounded degree. We experimentally show that our strategy reduces the dependency on the chosen monomial order and is able to generate Darboux polynomials of higher degrees for systems of dimension 3.