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The aim of this note is to give a clearer and more direct proof of the main result of another paper of the author. Moreover, we give some complementary results related to R-complete algebraic foliations with R a rational function of type ℂ*.

Let (F₁,..., Fₙ): ℂⁿ → ℂⁿ be a locally invertible polynomial map. We consider the canonical pull-back vector fields under this map, denoted by ∂/∂F₁,...,∂/∂Fₙ. Our main result is the following: if n-1 of the vector fields $\partial /\partial F_j$ have complete holomorphic flows along the typical fibers of the submersion $(F₁,...,F_{j-1},F_{j+1},...,Fₙ)$, then the inverse map exists. Several equivalent versions of this main hypothesis are given.