polynomial function is a proper map

polynomial function is a proper map

Assume that

𝕂

is either the

field of real

numbers or the field of

complex numbers

and let

W

:

𝕂

→

𝕂

be a polynomial function in one

variable

over

𝕂

with

positive

degree.

Proposition

. Polynomial function

W

:

𝕂

→

𝕂

is a

proper map

, i.e. for any

compact subset

K

⊆

𝕂

the

preimage

W

-

1

⁢

(

K

)

is

compact

.

Proof

. Assume that

W

⁢

(

x

)

=

∑

k

=

0

m

a

k

⋅

x

k

,

where

m

=

deg

⁢

(

W

)

≥

1

is the degree of

W

.

Recall that

K

⊆

𝕂

is compact if and only if

K

is closed and

bounded

. 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

{

x

n

}

n

=

1

∞

⊆

K

such that

lim

n

→

∞

⁡

∥

x

n

∥

=

+

∞

,

where

∥

x

∥

denotes the

Euclidean norm

of

x

∈

𝕂

.

Recall that for any

x

,

y

∈

𝕂

we have

∥

x

+

y

∥

≥

∥

x

∥

-

∥

y

∥

. Thus we have:

∥

W

⁢

(

x

)

∥

=

∥

∑

k

=

0

m

a

k

⋅

x

k

∥

≥

∥

a

m

⋅

x

m

∥

-

∑

k

=

0

m

-

1

∥

a

k

⋅

x

k

∥

=

∥

a

m

∥

⋅

∥

x

∥

m

-

∑

k

=

0

m

-

1

∥

a

k

∥

⋅

∥

x

∥

k

.

Let

V

⁢

(

x

)

=

∥

a

m

∥

⋅

x

m

-

∑

k

=

0

m

-

1

∥

a

k

∥

⋅

x

k

.

Then

V

is a real polynomial of degree

m

and the

leading coefficient

of

V

is positive, which implies that

lim

x

→

+

∞

⁡

V

⁢

(

x

)

=

+

∞

.

Now for each

n

∈

ℕ

we have

∥

W

⁢

(

x

n

)

∥

≥

V

⁢

(

∥

x

n

∥

)

,

but

V

⁢

(

∥

x

n

∥

)

tends to

infinity

, therefore

∥

W

⁢

(

x

n

)

∥

tends to infinty.

Contradiction

, since for each

n

∈

ℕ

we have that

W

⁢

(

x

n

)

∈

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

:

𝕂

→

𝕂

is closed. Thus (due to proposition) polynomial functions are closed.

□

Corollary 2

. Assume that

W

:

𝕂

→

𝕂

is a polynomial function such that

W

⁢

(

x

)

≠

0

for any

x

∈

𝕂

. Let

f

:

𝕂

→

𝕂

be a map defined by the

formula

f

⁢

(

x

)

=

1

W

⁢

(

x

)

.

Then

f

is bounded.

Proof

. We wish to show that there exists

M

>

0

such that for all

x

∈

𝕂

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

𝕂

∖

Im

⁢

(

W

)

is open and it contains

0

, thus there exists

ϵ

>

0

such that the ball around

0

with radius

ϵ

has empty

intersection

with

Im

⁢

(

W

)

. This means that for all

x

∈

𝕂

we have that

∥

W

⁢

(

x

)

∥

≥

ϵ

>

0

. Now for

M

=

ϵ

-

1

and for any

x

∈

𝕂

we have:

∥

f

⁢

(

x

)

∥

=

∥

1

W

⁢

(

x

)

∥

=

1

∥

W

⁢

(

x

)

∥

≤

1

ϵ

=

M

which

completes

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

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