Proper polynomial maps: The real case

Abstract

Let R be a real closed field. The semialgebraic subsets of Rn form the smallest collection of subsets of Rn containing all sets of the form {x 2 Rn| f(x) > 0}, where f 2 R[X1, · · · ,Xn], and closed under complementation and finite union and intersection. The Euclidean topology on Rn is defined by taking the open balls Bn(x, r) := {y 2 Rn| ky −xk < r} to be a basis of open sets. Given semialgebraic setsX Rn and Y Rm, a continuous map f:X !Y is semialgebraic if the graph of f is a semialgebraic set. A semialgebraic map f is semialgebraically closed if f(C) is a closed semialgebraic set for each closed semialgebraic set C X. It is semialgebraically proper if for every semialgebraic map g:Z ! Y , the canonical projection p:X ×Y Z = {(x, z) 2 X × Z| f(x) = g(z)}!Z is semialgebraically closed. Theorem (Delfs, Knebusch): Let f:X !Y be a semialgebraic map; then f is semialgebraically proper iff f is semialgebraically closed and its fibers are closed in Rn and bounded. The present paper gives refinements of this theorem for Nash and polynomial mappings: A semialgebraic map f:U !R, where U is an open semialgebraic subset of Rn, is a Nash function if f has continuous, semialgebraic partial derivatives of all orders. A map f = (f1, · · · , fr):U ! Rm is a Nash map if each fi is a Nash function, and a polynomial map if each fi is a polynomial. Theorem 1: A nonconstant semialgebraically closed Nash map is semialgebraically proper. Theorem 2: Let f:Rn !R be a nonconstant polynomial map. Then f is semialgebraically proper if and only if there exists some M 2 R+ such that the fibers f−1(t) are bounded for every t 2 R with |t| >M.