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.
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