Darboux theory of integrability in the sparse case

Guillaume Chèze (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context the bound is optimal.

Mots clés: Darboux integrability; Invariant algebraic hypersurface; Rational first integral; bound

Domaines: Analyse classique [math.CA]; Systèmes Solubles et Intégrables [nlin.SI]

Guillaume Chèze. Darboux theory of integrability in the sparse case. Journal of Differential Equations, 2014, ⟨10.1016/j.jde.2014.04.012⟩. ⟨hal-00746773⟩