Foliations raised by quadratic-like polynomial submersions on the real plane and a sharp result on the real Jacobian conjecture

Abstract

We study the planar foliations whose leaves are the connected components of the fibers of polynomial submersion functions having degree 2 in one of the variables. As an application of this geometric study, we prove the following result on the so called real Jacobian conjecture in $R^2$: it is true if one of the coordinate functions has degree 2 in one of the variables. This is sharp because of an existing counterexample with one of the coordinate functions having degree 3 in one of the variables.