Bounding the number of remarkable values via Jouanolou's theorem

Guillaume Chèze (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the size of a Newton polygon associated to the vector field. We prove that this bound is almost reached.

Mots clés: remarkable values; spectrum; Jouanolou's theorem

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

Guillaume Chèze. Bounding the number of remarkable values via Jouanolou's theorem. Journal of Differential Equations, 2015, ⟨10.1016/j.jde.2015.01.027⟩. ⟨hal-00869805v2⟩