Bounding the number of remarkable values via Jouanolou's theorem

Guillaume Chèze (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this note we bound the number of remarkable values of a polynomial vector field. The proof is short and based on a theorem due to Jouanolou about rational first integrals of planar polynomial derivations. We give two different bounds. The first one is given in term of the degree of the vector field. The second one is given in term of the size of a Newton polytope associated to the vector field. In this case we prove that our bound is optimal.

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

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

Guillaume Chèze. Bounding the number of remarkable values via Jouanolou's theorem. 2013. ⟨hal-00869805v1⟩