A note on the Darboux theory of integrability of non-autonomous polynomial differential systems

Abstract

In this work, we unfold some differential algebraic aspects of Darboux first integrals of polynomial vector fields. An interesting improvement is that our approach can be applied both to autonomous and non-autonomous vector fields. We give a sufficient and necessary condition for the existence of a Darboux first integral of a specific form for a polynomial vector field with some known algebraic invariant hypersurfaces. For the autonomous case, the classical result of Darboux is obtained as a corollary. For the non-autonomous case our characterization improves a known criterium.

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

“…For more recent versions of Theorem 1.1 see [37,38] and for some generalizations see [34,7]. Functions of the form (8) are called Darboux functions.…”

Liouvillian integrability of polynomial differential systems

“…By definition the former include the latter as a special one. In [1] Blázquez-Sanz and Pantazi provided a new approach to study the Darboux integrability of polynomial differential systems of degree m, where they replaced the dimension of C m−1 [x] which the cofactors of Darboux polynomials and exponential factors are located in by the rank of a matrix associated to these cofactors. Here C m−1 [x] denotes the linear space formed by polynomials in x ∈ C n of degree no more than m−1.…”

Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

“…As we show in Theorem 7 for polynomial vector fields the invariant algebraic curves and (because of their multiplicity, see [13]) the exponential factors are the basic elements in order to construct first integrals/integrating factors. As we will see the method of Theorem 7 also works for more general expressions of vector fields, curves, exponential factors and cofactors, see also [5,18,19].…”

“…Prelle and Singer [36,39] gave the relation between elementary/Liouvillian first integrals and integrating factors that are constructed by Darboux's method. Additionally, Darboux's ideas have been extended to a particular class of non-autonomous vector fields, see [5,30].…”