Injectivity of polynomial maps and foliations in the real plane

Abstract

We develop tools to count the connected components of the fibers of a polynomial submersion in two real variables $p$. As a consequence, we get a necessary condition for a real number to be a bifurcation value of $p$. We further present new methods to verify that $p$ has no Jacobian mates. These results are applied to prove that a polynomial local self-diffeomorphism of the real plane having one coordinate function with degree less than 6 is globally injective. As a byproduct, we completely classify the foliations defined by polynomial submersions of degree less than 6.