Abstract
ABSTRACT In this note it is proved that certain level sets of some real proper polynomial maps are nothing but spheres. As an application of this, we provide new proofs of Theorems 1.1, 1.2 and of the fundamental theorem of algebra. In addition, we show that every strictly convex (concave) polynomial map is proper. The latter implies that every real polynomial map g(x): R n → R n , whose Jacobian matrix is symmetric and has nonzero eigenvalues of the same sign, is a homeomorphism of R n onto R n .
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