A new exceptional polynomial for the integer transfinite diameter of $[0,1]$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

Q

IANG

W

U

A new exceptional polynomial for the integer

transfinite diameter of [0, 1]

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 847-861.

<http://www.numdam.org/item?id=JTNB_2003__15_3_847_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

A

new

exceptional polynomial

for the

integer

transfinite diameter of

_[0,1]

par

QIANG

WU

RÉSUMÉ. En améliorant

_{l’algorithme}

utilisé par

Habsieger

et

Sal-vy pour obtenir des

polynômes à

coefficients entiers de

plus

petite

norme infinie sur

[0, 1],

nous étendons leur table de

_polynômes

jusqu’au degré

100. Au

_degré

95 nous trouvons un nouveau

poly-nôme

_exceptionnel

_qui

a des racines

complexes.

Notre méthode

fait

_appel

à des

_polynômes

de

_{Müntz-Legendre}

_{généralisés.}

Nous améliorons un _peu la

_majoration

du diamétre transfini entier de

[0,1]

et nous donnons une démonstration élémentaire de la

mino-ration des _exposants de certains

_polynômes

_critiques.

ABSTRACT.

_Using

refinement of an

_{algorithm given by Habsieger}

and

_Salvy

to find

_{integer polynomials}

with smallest _sup norm

on

[0, 1]

we extend their table of

_polynomials

_{up to}

_degree

100.

For the

_degree

95 we find a new

_{exceptionnal polynomial}

which

has

_complex

roots. Our method uses

_generalized

_{Muntz-Legendre}

polynomials.

We

_improve

_slightly

the _upper bound for the

_integer

transfinite diameter of

_{[0, 1]}

and

_{give elementary proofs}

of lower bounds for the _exponents of some critical

_polynomials.

1. Introduction

For a

_{positive integer k,}

let be the set of

_polynomials

P of

_degree

k with

_integer

coefficients.

_Following

Borwein and

_Erdelyi

_[2]

we define the

integer Chebyshev polynomials

on the interval

[0, 1]

as the

polynomials

Pk

such that

where The

_integer

transfinite diameter of

_{[0, 1]}

is defined

_by

The exact value of

_{tz([O, 1])}

is not known. The best known result has been obtained

_by

Pritsker who

_gives

E

_{(0.4213,0.4232).}

For a

general exposition

see

(3~, [2]

or

[7].

The

_integer

_Chebyshev

_polynomials

have been studied

_{extensively by}

many authors such as

_Aparicio

(1~,

_Flammang,

Rhin and

Smyth

[4],

Borwein

and

_Erdelyi

_[2],

_Habsieger

and

_Salvy

_[5].

In

_[5]

the _{authors gave} a

_complete

list of

_integer

_Chebyshev

_polynomials

for the

_degrees

1 to 75.

_They

found for the

_degree

k = 70 _a factor of

_Pk

which has not all its _zeros in

[0, 1].

This

_gives

a

_negative

answer to a

_question

of

_[2]:

Do the

_{integer Chebyshev polynomials}

of

_{[0, 1]}

have all their zeros in

[0,1]?

This

_exceptionnal

_polynomial

is

which has four non real zeros. The

_question

remained _open to know whether there are other such

exceptional polynomials.

As

_{suggested by}

_Habsieger

and

_Salvy

we use a new

algorithm

to extend

their table and we

_give

a list of

_integer

Chebyshev

_polynomials

_up to

_degree

100. Nevertheless it seems that it is not

_possible

to reach the

_degree

200 with our

_algorithm

as

_suggested

in

[2].

For the

_degree

95 we have found a new

exceptional polynomial

which is

We will

_give

in section 4 a

_{good polynomial}

in

Z(x~,

of

_degree

108,

which

proves the

following

Theorem 1.1.

’

Remark. This

_improves

_slightly

Pritsker’s result.

One of the

_important

tools used

_by

the

_previous

authors is the

M3ntz-Legendre polynomials.

We will

_generalize

these

_polynomials.

This will

provide

us lower bounds for the

_exponent

of critical

polynomials :

an

irre-ducible

_polynomial

T in is a critical

_polynomial

for the interval

[0, 1]

if there exists a

_positive

constant

_C(T)

such that _{no every} inte-ger

Chebyshev

polynomial

of

degree n

is divisible

by

the

polynomial

Tk

with k &#x3E;

_C(T)n.

For instance we _prove that if

_Tn

is an

_{integer Chebyshev}

with

for n

large

enough.

This bound is not as

good

as Pritsker’s bound k &#x3E;

0.31n,

but the method of

_proof

is more

_elementary.

This paper is

organised

as follows : section 2 will be devoted to the

generalized Miintz-Legendre

polynomials

(GML)

on

[0,

1].

As

already

re-marked in

_[5]

it is

_interesting

to deal with

_{integer Chebyshev polynomials}

on

_~0,1/4~

because

_{t z ([0, 1]) =}

(tZ([0,1/4~))1~2.

Then section 3 is devoted

to the

_computations

on the interval

~0,1/4~.

In section 4 we summarize

briefly

the

_algorithm

that we use

_(a

more extensive version is available in

[8]

or

[9]),

and we

_give

a list of

integer

Chebyshev

polynomials

on

~0,1~

from

degree

76 to 100.

2. GML

_polynomials

on

[0,

1]

We

consider,

for 0 a

_b,

the scalar

_product

of two real continuous

functions on the interval

[a, b]

defined

_by

_{( f ,}

_g)w

= where

p is a real

_positive

function on the interval

[a, b~.

We note

In the _papers

_by

_Flammang,

_Rhin,

_Smyth

_[4]

and

_{Borwein, Erdelyi}

_[2],

the authors

_study

the

_integer

transfinite diameter of

_[0,

_1]

_{by using}

the

M3ntz-Legendre polynomials

which

_belong

to the real vector _space

_Vnk

generated by

(xn,

xn‘1, ~ ~ ~ ,

I

xk)

(n &#x3E;

k &#x3E;

0).

They apply

the

Gram-Schmidt process to the basis

x’~-1, ~ ~ ~ ,

xk )

and the usual scalar

product

( f , g)

f (x)g(x)dx

and obtain the

_{orthogonal Miintz-Legendre}

polynomials

Let

_R",~x~

be the set of

_polynomials

of

_degree

~t with real coefficients.

For a

_polynomial

P(x)

=

xkR(x)

_E

belonging

to

Vn,k

and

R(x)

_E

we

_get

Let F be a fixed non zero

polynomial

of

R[x].

We now consider the

vec-tor _space

_Vn,k

_{generated by}

_{(x’~F, xn~lF, ~ ~ ~ ,}

_By

the Gram-Schmidt process with a scalar

_product

( , ) ~

on the interval

[a, b],

we

_get

the set

Suppose

then that P =

belongs

to

_V,,,k

then we

_get

the

inequality

[8]

We consider now the case of the interval

[0,1]

with F =

(1 - x)9

and

= 1. Then we

have,

if _q &#x3E;

_0,

k n are

_integers,

Lemma 2.1. On the interval

_~0,1~

_for

F =

(1 - x)q,

the

orthogonal

GML

polynomials

are

Proof.

It is sufficient to _prove that for We know that

We consider

_Si

as a rational function in the

indeterminate q

where _n, i are

rational

_parameters.

The

_degree

of the denominator

_(2q

+ n -

_{i) ...}

_{(2q + 1)}

is

_equal

to n-i. The

_degree

of each numerator is

1)

= n-i. So if _we

_keep

the denominator in the form

(2q+~-i) ~ ~ ~ (2q+1),

then the numerator is of

_degree

less or

equal

to n - i. We will show that

Si

is a constant in

(~(q).

For

_that,

we show that the denominator divides

the numerator. Let _q =

-l/2

which is _{a zero} of the denominator with

J -"

The numerator of this sum is

_equal

to zero because it is

equal

to

so

(2q

+ n -

_{i) ...}

_(2q

_{+ 1)}

divides the numerator of

Si,

i.e.

_Si

is a constant.

So,

to

_compute

_Si,

we can

take q

= -(n

+ i +

_2)/2,

and then

Then

and

LJ

Then we show how we can deduce a result of

_type

(3)

_using

the

Propo-large

enough.

In

_fact,

let k the minimum of the

_exponents

of x and

_{1 - x,}

then

_Pn

is the form

_Pn

=

xk(1 -

X)kSn-2k(X)

and

and We remark that o i.e. If we

_put

a

= ~ ,

and then

By

the

_inequality

0.423164171 we have

Since

it is

_enough

to take a such that

0.423164171 ( f (a) ) 2

&#x3E;

1,

so a &#x3E;

_0.2907588,

0.2907588n for n

_large

_enough.

Remark.

_Using

_Proposition

2.2 for

_{tz([O, 1/4])}

we will _prove the better

3. GML

_polynomials

on

[0,

1]

We also know that

A2 = 1 - 2x

[5]

is a critical

polynomial

on

[0,

1],

with the

_change

of variable u =

~),

we obtain

A2

= 4u -1 which is a

critical

_polynomial

for the

_integer

_{Chebyshev polynomial}

on

[0, 1/4].

We

consider here the _{vector space}

_generated

_by

the basis

_((1-4x)9,

1

4)9, ,

7

4x)9)

and the scalar

product

( f, g)2

As for the lemma

_2.1,

we have

Lemma 3.1. We

_put

then is an

_orthogonal

GML

_family.

Proof.

We

_put

_y = 4x and it is _easy to

verify

that

3.1. The factor x.

If we want to determine a lower bound for the

_exponent

of the factor x

when the

_exponent

of the factor 1 - 4x is

_equal

to _q, we take a

polynomial

of

_degree

_nd-q,

_Q

=

+anxn(I-4x)Q,

then

_Q

=

AkMk

_{+ + ... +} with

, - - ,

then

We have

Proposition

3.2.

Proof. By taking y

=

4x,

we check that

Then

This _proves the

_proposition.

Proof.

We have

If we

_put

a

= ~

and

{3

= -1,

we have

Then

o

i.e.

Since

If we take

{3

=

0,

i.e. the

exponent

of 1 - 4x is

equal

to zero, we have

-

~ -

B I V ,v _/

and we obtain 1~ &#x3E; 2 x 0.2907588n which

_corresponds

to the result on the

interval

_[0,

_1]

in the section 2. 3.2. The factor 1 - 4x.

To find a lower bound for the

_exponent

of the factor 1- 4~ if the

exponent

of x is

_fixed,

we consider the vector space

generated

by

the basis

4x) n,

Xq (1 -

... ,

4x)k

and the scalar

product

(f,

2.

We

put

We have so and

and

Proposition

3.4. then

We can

estimate the exponent k of the factor 1 - 4x, because it

is clear

that is an

integer.

As in lemma

3.3,

we can _prove

Lemma 3.5. We

_put

_Pn

and _ai as above.

If {3

= * is

given,

then k &#x3E; an +

_o(n),

where cx is the smallest

_positive

root

_{o, f}

where

_{f (a, /3)}

is

_defined

in l emma 3.3.

So,

with lemma

_3.3,

_by

_taking

the

_exponent

of 1 - 4x

_equal

to zero,

we obtain that the

_exponent

of x is at least

_equal

to

_{0.581517fi99fin,}

and with this result we obtain that the

_exponent

of 1 - 4x is at least

_equal

to

0.0949fi70642n

_by

lemma 3.5. We

_put

this last result in lemma

_3.3,

we thus

get

that the

_exponent

of x is at least

_equal

to

_{0.5947009759n,}

we continue

and obtain the

_following

results :

TABLE 1

By

these

_results,

we obtain the

_proof

of the relation

(3),

i.e. let

_Pn

E

0.5952253n,

q &#x3E; 0.0966218n.

4.

_Application

for

tZ

((0,1~)

and a new

_{exceptional polynomial}

In the _paper

_by

_Habsieger

and

_Salvy

_[5],

we find a table of

_integer

Chebyshev

polynomials

of

_degree

less than or

equal

to 75 on the

In this

_section,

we first consider the interval

[0,

.1].

For the search of

the factors of

_{integer Chebyshev}

_polynomials,

we use the method which we

detail in

_[8]

and

_[9]

with the

_following

_steps:

1. Find a

good

_upper bound for 1 In this

_step

we use the LLL

algorithm

[6]

which furnishes a LLL-reduced basis of H where

H is a lattice with basis

(ei),

i.e. such that the norm

(euclidean norm)

of

vectors

_fi

is small and in

_particular

such that the vector

_fl

is not so far

from

_being

the smallest non zero vector in H. Then we can find a

good

upper bound.

2. Use this bound to deduce

_polynomials

that are _necessary factors of

Now we use the

generalized Muntz-Legendre

method with this bound

to

_give

an _upper bound for the

_exponents

of critical

_polynomials

as in

section 2 and section 3. More

_precisely

we

_comput

_explicitely

the bound of

the coefficients _ak when F is an

explicit polynomial having

even more than one irreducible factor

(such

as

_A3,

A4).

3. Perform an exhaustive search for the

_missing

factors : We have so a

system

of

_inequalities

cn where F is determined

by

the

step

2,

c,~, is the

good

upper bound in

step

1,

Q(x)

is a

_polynomial

of

_degree

k = _{n -}

deg F

whose unknown coefficients _are to be determined and the

xi are control

_points

in the interval

[a, b]

which are different from the roots

of

_F(x).

This

_system

defines a

_polyhedron

of which we must determine the

_{integer points.}

We solve this

_system

with a method

adapted

from the

simplex

method and the LLL

_algorithm.

We thus obtain a

polynomial

Pn

which is

_appropriate.

We thus find a new factor

and we extend the table _up to

_degree

50

_(i.e.

on the interval

~0,1~

we can

obtain the

_{integer Chebyshev polynomials}

of

_degree

up to 100 if we consider

only

the

_{polynomials Q}

of even

_degree

such that

Q(x)

=

Q(1 - x)).

Otherwise, by

lemma 1 of

_Habsieger

and

_Salvy

_[5],

we can also search

the factors of

_{integer Chebyshev polynomials}

of odd

_degree

on the interval

[0, 1]

by

transferring

the search on the interval

[0,

1],

i.e. we

_compute

Pn(u)

such that

and we

replace u by

x(1 - x),

the table 2 is thus obtained. In this

table,

for the

_{integer Chebyshev polynomial}

of

_degree

_95,

we have

which is a new

exceptional polynomial,

i.e. it has four non real zeros. For

the

_polynomials

of

_degree

less or

_equal

to

_75,

we find of course the same ones as those of the table of

_Habsieger

and

_Salvy.

We will now

_explain

how our

_computation

let us obtain a

polynomial

of

degree

108

which will

_imply

the Theorem. Let

_Bl

=

969581u8-1441511u?+

928579u 6- 338252u5

+

7fi143u4 -10836u3

+

951u2 -

47u

_{+ 1, B2}

=

49u2

-20u + 2,

B3 = 34u2 -12u + 1,

B4 = 193u3 -104u2

+

_{18u -1, B5}

=

199u3

-105u2 + 18u -1, B6

=

182113u7 233968us + 127434u5 38125u4 +6763u3

-711 u2

+

_{41 u -1,}

where the

_polynomials

_Bi

_appear

_during

the

_computation

of the table 2.

_Using

a classical semi-infinite linear

_programming

_[8],

we

get

the smallest bound for max

_[

with

0u1/4 2 10 11 1 6

the condition . Then we obtain the

TABLE 2

This proves the theorem.

Surprisingly

we see that the

exceptional

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57045 METZ Cedex 1, France E-mail : _{uuoponcelet .} univ-metz . fr