proof of Minkowski’s bound

proof of Minkowski’s bound

The proof of Minkowski’s bound will rely on Minkowski’s

lattice point

theorem (

http://planetmath.org/MinkowskisTheorem

), but we first need to establish some lemmas.

Lemma 1

.

Let

M

be a

real number

and suppose that for every non-zero ideal

a

of the

ring of integers

O

K

there exists a non-zero

x

∈

a

with norm

N

⁡

(

x

)

≤

M

⁢

N

⁡

(

a

)

.

Then, every

ideal class

of

O

K

has a representative

a

satisfying

N

⁡

(

a

)

≤

M

.

Proof.

Let

[

𝔟

]

be an ideal class represented by the ideal

𝔟

. Choosing a non-zero

x

∈

𝔟

then

x

⁢

𝔟

-

1

is an ideal of

𝒪

K

and, by the condition of the lemma, contains a non-zero

y

satisfying

N

⁡

(

y

)

≤

M

⁢

N

⁡

(

x

⁢

𝔟

-

1

)

.
Then,

𝔞

≡

x

-

1

⁢

y

⁢

𝔟

is an ideal representing

[

𝔟

]

and

N

⁡

(

𝔞

)

=

N

⁡

(

y

)

/

N

⁡

(

x

⁢

𝔟

-

1

)

≤

M

.
∎

If the

real embeddings

of

K

are denoted by

σ

k

:

K

→

ℝ

(

k

=

1

,

…

,

r

1

) and the complex embeddings are

τ

k

:

K

→

ℂ

together with their

complex conjugates

τ

¯

k

(

k

=

1

,

…

,

r

2

), then we define

j

:

K

→

ℝ

r

1

×

ℂ

r

2

,

j

⁢

(

x

)

=

(

σ

1

⁢

(

x

)

,

…

,

σ

r

1

⁢

(

x

)

,

τ

1

⁢

(

x

)

,

…

,

τ

r

2

⁢

(

x

)

)

.

Also note that

ℝ

r

1

×

ℂ

r

2

is

isomorphic

as a

real vector space

to

ℝ

r

1

+

2

⁢

r

2

=

ℝ

n

given by the

isomorphism

f

:

ℝ

r

1

×

ℂ

r

2

→

ℝ

n

,

f

⁢

(

x

1

,

…

,

x

r

1

,

y

1

,

…

,

y

r

2

)

=

(

x

1

,

…

,

x

r

1

,

ℜ

⁡

(

y

1

)

,

…

,

ℜ

⁡

(

y

r

2

)

,

ℑ

⁡

(

y

1

)

,

…

,

ℑ

⁡

(

y

r

2

)

)

.

As

f

and

j

are

linear maps

(with respect to the

field of rationals

ℚ

), the

combination

f

∘

j

gives a

ℚ

-linear map from

K

to

ℝ

n

. The image will be a

lattice

, and we can compute its volume.

Lemma 2

.

If

a

is a non-zero ideal of

O

K

, then

Γ

=

f

∘

j

⁢

(

a

)

is a lattice in

R

n

(

http://planetmath.org/LatticeInMathbbRn

). Its fundamental mesh has volume

vol

⁡

(

Γ

)

=

2

-

r

2

⁢

|

D

K

|

⁢

N

⁡

(

𝔞

)

.

Proof.

The proof of this is to be added.
∎

Lemma 3

.

For any

L

>

0

, let

S

be the set in

R

r

1

×

C

r

2

consisting of points

(

x

1

,

…

,

x

r

1

,

y

1

,

y

r

2

)

satisfying

∑

k

=

1

r

1

|

x

k

|

+

2

⁢

∑

k

=

1

r

2

|

y

k

|

≤

L

.

Then,

f

⁢

(

S

)

has volume

(

2

r

1

-

r

2

⁢

π

r

2

/

n

!

)

⁢

L

n

.

Proof.

The proof of this is to be added.
∎

Proof of Minkowski’s bound

For an ideal

𝔞

and any constant

b

>

1

, let

L

>

0

be given by

2

r

1

-

r

2

⁢

π

r

2

n

!

⁢

L

n

=

2

n

⁢

b

⁢

2

-

r

2

⁢

|

D

K

|

⁢

N

⁡

(

𝔞

)

.

Letting

S

be the set given in Lemma

3

and

Γ

=

f

∘

j

⁢

(

𝔞

)

, Lemmas

2

and

3

give

vol

⁡

(

S

)

>

2

n

⁢

vol

⁡

(

Γ

)

. As

S

is convex and

symmetric about

the origin, Minkowski’s theorem tells us that there is a non-zero

x

∈

𝔞

with

f

∘

j

⁢

(

x

)

∈

S

.

As the

geometric mean

is always bounded above by the

arithmetic mean

, we get the

inequality

N

⁡

(

x

)

=

∏

k

=

1

r

1

|

σ

k

⁢

(

x

)

|

⁢

∏

k

=

1

r

2

|

τ

k

⁢

(

x

)

|

2

≤

n

-

n

⁢

(

∑

k

=

1

r

1

|

σ

k

⁢

(

x

)

|

+

2

⁢

∑

k

=

1

r

2

|

τ

k

⁢

(

x

)

|

)

n

≤

n

-

n

⁢

L

n

=

b

⁢

M

K

⁢

|

D

K

|

⁢

N

⁡

(

𝔞

)

where

M

K

=

(

n

!

/

n

n

)

⁢

(

4

/

π

)

r

2

. If we choose

b

such that

b

⁢

M

K

⁢

|

D

K

|

⁢

N

⁡

(

𝔞

)

is less than the smallest

integer

greater than

M

K

⁢

|

D

K

|

⁢

N

⁡

(

𝔞

)

, then this gives

N

⁡

(

x

)

≤

M

K

⁢

|

D

K

|

⁢

N

⁡

(

𝔞

)

and Minkowski’s bound follows from Lemma

1

.

Title

proof of Minkowski’s bound

Canonical name

ProofOfMinkowskisBound

Date of creation

2013-03-22 18:33:41

Last modified on

2013-03-22 18:33:41

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Proof

Classification

msc 11R29

Classification

msc 11H06

Related topic

MinkowskisTheorem

Related topic

MinkowskisConstant

Related topic

IdealClass

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