polynomial function is a proper map

Assume that 𝕂 is either the field of real numbers or the field of complex numbersMathworldPlanetmathPlanetmath and let W:𝕂𝕂 be a polynomial function in one variable over 𝕂 with positivePlanetmathPlanetmath degree.

PropositionPlanetmathPlanetmathPlanetmath. Polynomial function W:𝕂𝕂 is a proper map, i.e. for any compact subset K𝕂 the preimageMathworldPlanetmath W-1(K) is compact.

Proof. Assume that


where m=deg(W)1 is the degree of W.

Recall that K𝕂 is compact if and only if K is closed and boundedPlanetmathPlanetmathPlanetmathPlanetmath. Since polynomial functions are continous, it is sufficient to show that preimage of a bounded set is bounded. So assume that K is bounded and W-1(K) is not bounded. Take a sequence {xn}n=1K such that


where x denotes the Euclidean norm of x𝕂.
Recall that for any x,y𝕂 we have x+yx-y. Thus we have:




Then V is a real polynomial of degree m and the leading coefficient of V is positive, which implies that


Now for each n we have


but V(xn) tends to infinityMathworldPlanetmathPlanetmath, therefore W(xn) tends to infinty. ContradictionMathworldPlanetmathPlanetmath, since for each n we have that W(xn)K and K is bounded.

Corollary 1. Polynomial functions on 𝕂 are closed maps.

Proof. Note that 𝕂 is compactly generated Hausdorff space and therefore every proper and continous map f:𝕂𝕂 is closed. Thus (due to proposition) polynomial functions are closed.

Corollary 2. Assume that W:𝕂𝕂 is a polynomial function such that W(x)0 for any x𝕂. Let f:𝕂𝕂 be a map defined by the formulaMathworldPlanetmathPlanetmath


Then f is bounded.

Proof. We wish to show that there exists M>0 such that for all x𝕂 the inequality f(x)M holds. Since polynomial functions are closed maps, then the image Im(W) of W is a closed subset of 𝕂. Therefore 𝕂Im(W) is open and it contains 0, thus there exists ϵ>0 such that the ball around 0 with radius ϵ has empty intersectionMathworldPlanetmath with Im(W). This means that for all x𝕂 we have that W(x)ϵ>0. Now for M=ϵ-1 and for any x𝕂 we have:


which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title polynomial function is a proper map
Canonical name PolynomialFunctionIsAProperMap
Date of creation 2013-03-22 18:30:49
Last modified on 2013-03-22 18:30:49
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Theorem
Classification msc 12D99
Related topic ProperMap
Related topic PolynomialFunction