First integrals and Darboux polynomials of homogeneous linear differential systems

Abstract

This paper studies rational and Liouvillian first integrals of homogeneous linear differential systems Y′=AY over a differential field k. Following [26], our strategy to compute them is to compute the Darboux polynomials associated with the system. We show how to explicitly interpret the coefficients of the Darboux polynomials as functions on the solutions of the system; this provides a correspondence between Darboux polynomials and semi-invariants of the differential Galois groups, which in turn gives indications regarding the possible degrees for Darboux polynomials (particularly in the completely reducible cases). The algorithm is implemented and we give some examples of computations.