Foliations and Polynomial Diffeomorphisms of R³

Original · EN

Let Y=(f,g,h):R³ → R³ be a C² map and let (Y) denote the set of eigenvalues of the derivative DYₚ, when p varies in R³. We begin proving that if, for some ε>0, (Y)∩ (-ε,ε)=, then the foliation F(k), with k∈ {f,g,h}, made up by the level surfaces {k= constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of Rⁿ.