On the minimum of the unit lattice

V

OLKER

K

ESSLER

On the minimum of the unit lattice

Journal de Théorie des Nombres de Bordeaux, tome

3, n

o

_{2 (1991),}

p. 377-380

<http://www.numdam.org/item?id=JTNB_1991__3_2_377_0>

© Université Bordeaux 1, 1991, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

On

the

minimum

of the

unit lattice.

PAR VOLKER KESSLER

1. Introduction.

Computations

in lattices often

_require

a lower bound for the minimum

of the

_lattice,

both for

_{practical purposes}

and for a theoretical

analysis

of

the

_algorithms,

_e.g.

_[1]

and

_[2].

In this _paper we recall two results of Dobrowolski

_[3]

and

_Smyth

_[5]

in order to

_get

such a bound for the unit lattice.

2. Lower bound.

Let Ii be a finite extension

_{of Q}

of

_degree

n with maximal order R. For

1 i n we denote

by

the n different

embeddings

of K into the field C of

complex

numbers. The

first rl of those

embeddings

are

_real,

the last

_2r2

embeddings

are non-real

and numbered such that the

_(rl

+ r2 +

_i)th

_embedding

is the

complex-conjugation

of the

_(r,

+

_i)th

_embedding.

Then the

_logarithmic

_map is

given

by

with the unit rank r = _{ri + r2 -} 1 and

The kernel of

_Log

consists

_exactly

of the roots of the

_{unity lying}

in K. We define the minamum

_A(L)

of the unat lattice L :=

Log(R*)

by

where

_{11 II}

denotes the Euclidean

-

THEOREM : A lower bound

_for

the minimum

_A(L)

is

_{given by}

which is "a bit"

_larger

than

Thus the inverse

_1/A(L)

is

_of

the

_magnitude

_0(nl/2+f)

_{for every}

c &#x3E; 0. PROOF. Let E

E R*

be a unit of

degree

m over

Q,

which is no root of

unity.

Without loss of

_generality

we can assume that m =

n, because if

[[Log E~~

is

_larger

than

_p(K’)

for a subfield K’ of li it is also

_larger

than

I~~I~~.

-We are interested in two subsets of the

_conjugates

_{e~,’’’ ,}

Since E is no root of

unity

S is

_non-empty

and therefore T cannot be

_empty

because of

_N(e)

= 1.

We call c

_reciprocal

if E is

_conjugate

i.e. its minimal

_polynomial

f(X)

= X "’ + _{+ ... +}

ao satisfies

If e is

non-reciprocal

we know from the theorem of

[5]

that

But from

_N(E)

= 1 it follows

The value

_If(r+l)1

_I

does not occur in the norm of

Log(E).

But as a consequence of

(3)

it does not matter if r + 1 lies in

S

or in T and so we can assume without restriction that r ₊

1 ~

S. Thus

(The

second

_inequality

follows from the well known norm

equivalence

be-tween 1-norm and Euclidean

_norm.)

For

_reciprocal

E we know

_by

Theorem 1 of

[3] :

We now use the

Taylor

series of the

logarithm

(Iyl

1) :

The

_inequality

follows

_directly

from

_{Lagrange’s representation}

of the resi-due.

_Applying

₍₅₎

to

₍₄₎

_yields

Since E is

reciprocal

the inverses of the

conjugates

of c are also

conjugate

to E. This

_implies

that the numbers of

_conjugates

outside the unit circle

equals

the number of

_conjugates

inside the unit

circle,

i.e

Again

by

(3)

+

1 ~

S

which is

_larger

than

Because of

_{2~~~~~~ ) ~}

0.001178 we thus

proved

the lower bound.

REMARK. If the

_conjecture

of Schinzel and Zassenhaus

_[5]

is correct the

term be substituted

_by

a constant

_independent

of n. This term

₍

g

n )

bound would be

_provable

the best one

(up

to

constants).

REFERENCES

[1]

Buchmann, Zur _Komplexität der _Berechnung von Einheiten und Klassenzahlen

al-gebraicher Zahlkörper, Habilitationsschrift Düsseldorf

_(1987).

[2]

Buchmann, Kessler, Computing a reduced lattice basis _from a generating _system, to

appear.

[3]

Dobrowolski, On a _{question of} Lehmer and the number _of irreducible _{factors of} a

polynomial, Acta arithmetica 34

_(1979),

391-401.

[4]

Schinzel, Zassenhaus, A _{refinement of} two theorems _{of Kronecker,} Mich. Math. J. 12

_(1965),

81-84.

[5]

Smyth, On the _{product of} the _conjugates outside the unit circle of an _algebraic integer, Bull. London Math. Soc. 3

_(1971),

169-175.

Volker Kessler Siemens AG ZFE ST SN 5

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