Fibers of polynomial mappings over ℝn

Abstract

We find sufficient conditions on a polynomial mapping f = (p1,…,pn): ℝn → ℝn to be surjective. One such a condition is the existence of a non-trivial solution of the induced homogeneous system of the equations Σj=1n(pj)αjgij=0, i = 1,…,n. Here αj∈ℤ+,gij∈ℝ[X1,…,Xn] and det(gij) never vanishes on ℝn. A conclusion that follows is that if Πj=1n(degpj) is an odd integer, then surjectivity f(ℝ)n = ℝn follows if the homogeneous system p1¯=…=pn¯=0 (p¯ is the highest homogeneous component of p) has only the trivial solution. We also investigate mappings f for which the determinant of their Jacobian matrix, det J(f) never vanishes on ℝn. These polynomial mappings are in the core of the Real Jacobian Conjecture. One conclusion is that for such a local polynomial diffeomorphism the system pj¯∂pj¯∂Xi=0,, i = 1,…,n must have non-trivial solutions, and for any j = 1,…,n. Also, such a local diffeomorphism is surjective if the induced homogeneous system of ∑j=1nαj(pj)αj−1∂pj∂Xi=0, i = 1,…n, has only the trivial (zero) solution. These last two theorems give a new point of view on S. Pinchuk’s solution of the Real Jacobian Conjecture. Other obvious applications of our results are for the existence of solutions of the corresponding polynomial equations in n unknowns over the real field, ℝ.