Injective morphisms of real algebraic varieties

A recent note of D. J. Newman 3 shows that for polynomial maps of the real plane into itself injectivity implies surjectivity. The present note combines two independently obtained corrections and generalizations of Newman's proof. We need some preliminaries. Take the complex numbers as universal domain and define a real algebraic set to be the set of real points of an algebraic set that is defined over the reals. A real algebraic set V is Zariski-dense in its Zariski closure V, which is an algebraic set defined over the reals, and V is the set of real points of V. Any algebraic subset of V meets V in a real algebraic subset. V is irreducible (in its Zariski topology), i.e. V is a real algebraic variety, if and only if V is irreducible (over the complex numbers), we define dim V =dim V, and we call PC V simple if P is a simple point of V; such points P exist, and there exist uniformizing parameters that are defined over the reals for 7 at such a point P, hence real local power series expansions, so that at each of its simple points V, in its ordinary topology, is locally a real analytic manifold of dimension dim V. For real algebraic sets V, W define a rational map f: V-*W to be the restriction to VX W of a rational map f: V--W that is defined over the reals; the rational map f: V-*W is a morphism if f is defined at each point of V. Supposing the morphism of real algebraic varieties f: V-*W to be such that f(V) is Zariski-dense in W, a simple point PE V may be found such that f(P) is simple on Wand df has the correct rank dim V-dim W at P, implying that, for the real analytic structure of V, W at P, f(P) respectively, f is locally a projection onto a direct factor; in particular, if f is finite-to-one, then dim V=dim W