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Proper Polynomial Maps - The Real Case

Gamboa Mutuberria, José Manuel and Ueno, Carlos (1992) Proper Polynomial Maps - The Real Case. Lecture Notes In Mathematics, 1524 . pp. 240-256. ISSN 0075-8434

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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.

Item Type:Article
Additional Information:Mathematics
Uncontrolled Keywords:semialgebraically proper map; semialgebraically closed map; polynomial mappings; Nash map
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:15431
Deposited On:30 May 2012 09:00
Last Modified:30 May 2012 09:00

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