Linear independence of linear forms in polylogarithms

Linear independence of linear forms in polylogarithms

RAFFAELEMARCOVECCHIO

Abstract. For x ∈ C,|x| < 1,s ∈ N, let Lis(x)be thes-th polylogarithm of x. We prove that for any non-zero algebraic number α such that|α|<1, theQ(α)-vector space spanned by 1,Li₁(α),Li₂(α), . . . has infinite dimension.

This result extends a previous one by Rivoal for rationalα. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Pad´e approximation problem.

Mathematics Subject Classification (2000): 11J72 (primary); 11J17, 11J91, 33C20 (secondary).

1.

Introduction

The aim of this paper is to prove the following statement: for x ∈

C

^{, |x| < 1,} s∈

N

^{, let Lis(x)be the}s-th polylogarithm ofx:

Lis(x)=

k≥1

x^k k^s;

then for any non-zero algebraic numberαsuch that|α|<1, in the sequence of com- plex numbers 1,Li₁(α),Li₂(α), . . . infinitely many terms are linearly independent over the number field

Q

^(α).

More precisely, for any non-zero algebraic numberαlet h(α)be the Weil ab- solute logarithmic height ofα(see (4.7) below). Let δ(α) = [

Q

^{(α) :}

Q

^{] if αis} real, and letδ(α) = [

Q

^{(α) :}

Q

^]/2 otherwise. Letτ_α(a)be the dimension of the

Q

^(α)-vector space spanned by 1,Li₁(α), . . . ,Li_a(α).

We prove the following:

Theorem 1.1. For anyε, ω,H such thatε,H >0and0< ω <1, there exists an integera=a(ε, ω,H)such that for every a≥a and for every non-zero algebraic Received August 29, 2005; accepted in revised form November 17, 2005.

numberαwith|α| ≤ωandh(α)≤H the dimensionτ_α(a)satisfies τ_α(a)≥ 1

δ(α)· 1−ε

1+log 2loga. (1.1)

In the special case of a non-zero rational numberαsuch that|α| < 1, Rivoal [4]

recently proved this result by applying the Nesterenko criterion to suitable linear forms in 1,Li₁(α), . . . ,Li_a(α)with rational coefficients. Rivoal’s method is based on the series

n!^a−r

k≥1

(k−1)(k−2)· · ·(k−r n)

k^a(k+1)^a· · ·(k+n)^a z^−k, (1.2) wherer = r(a)is an integer such that 1 ≤r < a,n ∈

N

^{andz = α−1. He men-} tioned that his result easily extends to real algebraic numbersα, but for technical reasons the case of complexαis delicate. Indeed, Nesterenko’s criterion needs both lower and upper bounds of the sequence of the linear forms asn → ∞. Now, to obtain an asymptotic estimate of (1.2) is straightforward ifαis real, but seems hard for non-realα.

The method of [4] is inspired by Nikishin’s series (see [3])

k≥1

(k−1)(k−2)· · ·(k−an−q+1)

k^a· · ·(k+n−1)^a(k+n)^q z^−k, q=0, . . . ,a. (1.3) Using this tool, Nikishin proved that, for suitable negative rational numbersα, all the real numbers 1,Li₁(α), . . . ,Li_a(α)are linearly independent over

Q

. Each series (1.3) is easily seen to be a linear form in 1,Li₁(1/z), . . . ,Li_a(1/z)with polynomial coefficients lying in

Q

[z]. The main feature of (1.3) is that, forq = 0, . . . ,a, the a+1 vectors of such polynomial coefficients are linearly independent. Thus, by a straightforward application of our Proposition 4.1 below, the use of the a+1 series (1.3) requires only upper bounds of the coefficients and of the linear forms, whereas the use of a single series such as (1.2) also requires a lower estimate of the corresponding linear form, and here is precisely where one would require a subtle application of the saddle point method. On the other hand, Nikishin’s method essentially corresponds to the special caser =ain Rivoal’s series (1.2), so that, for fixeda, it applies only to a class ofα’s smaller than Rivoal’s.

In the present paper we employ a system ofa+1 series that encompasses both Nikishin’s and Rivoal’s ideas:

n!^a−r

k≥1

(k−r n)r n

(k)^a_n(k+n)^q z^−k, q =0, . . . ,a, (1.4) where(β)mdenotes the Pochhammer symbol:

(β)0=1, (β)m=β(β+1)· · ·(β+m−1) (m=1,2, . . . ).

In Section 2 we state the upper bounds and the arithmetic properties of the linear forms and of the coefficients of (1.4).

The main new point of the present construction is to resume Nikishin’s original argument, which does not need a precise asymptotic estimate of the sequence of lin- ear forms. This is possible because a crucial property of (1.4) is the non-vanishing of the determinant of ordera+1 of the coefficients of 1,Li₁(α), . . . ,Li_a(α), for q=0,1, . . . ,aandz=α⁻¹. To achieve our plan, in section 3 we resort to a recent method introduced by Fischler and Rivoal ([2], Theorem 1), which in particular shows that the vector(P₀(z), . . . ,P_a(z))of the coefficients of the polylogarithms in the series (1.2):

n!^a−r

k≥1

(k−r n)r n

(k)^a_n+1 z^−k =^a

h=1

P_h(z)Li_h(1/z)−P₀(z)

is the unique non-zero solution (up to a multiplicative constant) of the following Pad´e approximation problem:

a h=1

P_h(z)Li_h(1/z)−P₀(z)= O(z^{−r n−1}) (z→ ∞)

a h=1

P_h(z)(log(1/z))^h−1

(h−1)! = O((1−z)^{an+a−r n−1}) (z→1)

P₀(z),P₁(z), . . . ,P_a(z)∈

C

^{[z], deg P1(z), . . . ,deg Pa(z)≤n.}

(1.5)

This property extends to the series (1.4) in a natural way (see our Remark 3.1 at the end of Section 3).

In the last section we complete the proof by using a suitable generalization of Nikishin’s determinant argument.

We remark that the present extension from polylogarithms of rational numbers to polylogarithms of algebraic numbers is analogue to the extension made in [1], where Amoroso and Viola obtain good approximation measures for logarithms of algebraic numbers by generalizing a previous method of Viola [5] for logarithms of rational numbers.

ACKNOWLEDGEMENTS. I am grateful to Professor Amoroso for helpful discus- sions and interesting comments on this work.

2.

Upper bounds

Throughout this paper,a,n,q,r are integers such that 0 ≤q ≤a,r =r(a)with 1≤r <aandn∈

N

^{(we letn→ ∞}). Let alsoD_λ= _λ¹_!(_dt^d)^λ, and let

R_n(k)=n!^a−r (k−r n)r n

(k)^a_n(k+n)^q for chosena,q,r.

In this section, following closely the methods of [3] and [4], we write each series (1.4) as a linear form in 1,Li₁(1/z), . . . ,Li_a(1/z)with polynomial coeffi- cients, and we derive the required upper bounds and arithmetic properties of the linear forms and of the coefficients.

For anya,n,q,r as above, and for any complex numberz with|z| > 1, let N_n(a,q,r;z)be the series (1.4):

N_n(a,q,r;z)=

k≥1

R_n(k)z^−k. (2.1) It is worth noting thatN_n(a,q,r;z)is a generalized hypergeometric function ofz⁻¹:

N_n(a,q,r;z)= n!^a−r(r n)!^a+1

((r +1)n)!^a−q((r +1)n+1)!^q z^{−r n−1}×

×

s≥0

(r n+1)^a_s⁺¹

((r +1)n+1)^as^−q((r+1)n+2)^qs

z^−s s! . We denote (for fixedr anda)

c_h,q,j,n =







D_a−h(R_n(t)(t+ j)^a)_|t=−j ∈

Q

^{if j} =0, . . . ,n−1;h=1, . . . ,a; D_q−h(R_n(t)(t+n)^q)_|t=−n ∈

Q

^{if j =n;h=1, . . . ,q;}

0 if j =n;h=q+1, . . . ,a.

The rational function R_n(t)decomposes as follows:

R_n(t)=^a

h=1

n j=0

c_h,q,j,n

(t + j)^h. (2.2)

We now define the following polynomials inz:

A⁽_n^h)(a,q,r;z)= n

j=0

c_h,q,j,nz^j (h=1, . . . ,a) B_n(a,q,r;z)=

a h=1

n j=1

c_h,q,j,n j k=1

1 k^h z^j−k.

For the forthcoming analysis it is useful to remark that the degree ofA⁽_n^h)(a,q,r;z) does not exceednfor 1≤ h ≤q, and does not exceedn−1 forq+1≤h ≤a.

Moreover, for anyh=1, . . . ,athe degree of A⁽_n^h)(a,h,r;z)is exactlyn, since 0=c_h,h,n,n=(R_n(t)(t+n)^h)_|t=−n = ±((r+1)n)!

n!^r+1 . (2.3) We also notice that the coefficient of the powerz^−t in N_n(a,q,r;z) is zero for t = 1, . . . ,r n, and non-zero for all t ≥ r n+1. In addition, the coefficient of z^{−r n−1}in N_n(a,0,r;z)is

n!^a−r(r n)!^a+1

((r+1)n)!^a . (2.4)

Using the decomposition (2.2) in the formula (2.1), one readily sees that N_n(a,q,r;z)=

a h=1

A⁽_n^h)(a,q,r;z)Li_h(1/z)−B_n(a,q,r;z). (2.5) The next three Lemmas 2.1, 2.2 and 2.3 are very similar to Lemmas 3, 4 and 5 of [4], and to Lemmas 1, 2 and 3 of [3], so we omit the proofs.

Lemma 2.1. For all z∈

C

^with|z|>1 lim sup

n→∞ |N_n(a,q,r;z)|^1/n ≤ 1

|z|^rr^a−r. Lemma 2.2. For all z∈

C

^{we have}

lim sup

n→∞ |A⁽_n^h)(a,q,r;z)|^1/n ≤r^r2^a+r+1max{1,|z|} (h=1, . . . ,a), lim sup

n→∞ |B_n(a,q,r;z)|^1/n ≤r^r2^a+r+1max{1,|z|}.

Lemma 2.3. Let d_n be the least common multiple of the integers1, . . . ,n.

Then

d_n^a−hA⁽_n^h)(a,q,r;z)∈

Z

^{[z] (h=1, . . . ,a),} d_n^aB_n(a,q,r;z)∈

Z

^[z].

3.

A non-vanishing determinant

To shorten our notation, from now on we put A⁽_n^h)(q)= A⁽_n^h)(a,q,r;z), B_n(q)= B_n(a,q,r;z),N_n(q)= N_n(a,q,r;z).

LetM_nbe the following square matrix of ordera+1:

M_n=







−B_n(0) A⁽_n¹⁾(0) · · · A⁽_n^a)(0)

−B_n(1) A⁽_n¹⁾(1) · · · A⁽_n^a)(1) . . . .

−B_n(a) A⁽_n¹⁾(a) · · · A⁽_n^a)(a)





. (3.1)

We claim that for any complex numberz=1 the matrixM_nis non-singular.

Clearly, det(M_n) is a polynomial inz. On the other hand, for any complex numberzsuch that|z|>1, we can add to the first column ofM_na linear combina- tion of the other columns, namely we can replace−B_n(0)with the sum−B_n(0)+ A⁽_n¹⁾(0)Li₁(1/z)+ · · · + A⁽_n^a)(0)Li_a(1/z) in the first row of (3.1), −B_n(1) with

−B_n(1)+A⁽_n¹⁾(1)Li₁(1/z)+ · · · +A⁽_n^a)(1)Li_a(1/z)in the second row of (3.1), and so on, and by applying (2.5) we immediately see that

det(M_n)=det







N_n(0) A⁽_n¹⁾(0) · · · A⁽_n^a)(0) N_n(1) A⁽_n¹⁾(1) · · · A⁽_n^a)(1) . . . . N_n(a) A⁽_n¹⁾(a) · · · A⁽_n^a)(a)





. (3.2)

We now recall that for t = 1, . . . ,r n the coefficient of z^−t is zero in each of N_n(0), . . . ,N_n(a), and degA⁽_n^h)(q) ≤ nfor h = 1, . . . ,a and forq = 0, . . . ,a.

Therefore, expanding the determinant on the right hand side of (3.2) along the first column, we obtain

det(M_n)=z^{an−r n−1}

k≥0

u_kz^−k

for suitable rational numbersu_k. Moreover, since the elements above the principal diagonal of the matrix in (3.2) satisfy degA⁽_n^h)(q)≤n−1 (h=q+1, . . . ,a) and for the elements of that diagonal degA⁽_n^h)(h)=n(h=1, . . . ,a), the coefficientu₀ is the product of the coefficient ofz^{−r n−1}in N_n(0)and of the leading coefficients of the polynomialsA⁽_n¹⁾(1), . . . ,A⁽_n^a)(a). Thus

0=u₀= ± (r n)!

n!^r a+1

(3.3) by (2.3) and (2.4). Sincez is arbitrary in the domain|z| > 1, from the previous analysis we infer that the degree of the polynomial det(M_n)is exactlyan−r n−1, and that its leading coefficient isu₀.

Following Fischler and Rivoal [2], we now introduce, for z ∈/]− ∞,0], the function

J_n(a,q,r;z)= 1 2πi

|t|=µn

R_n(t)z^−t dt

(i =√

−1) whereµ>1 is arbitrary. As above, we abbreviateJ_n(q)=J_n(a,q,r;z). Using the decomposition (2.2) and Cauchy’s integral formula we obtain

J_n(q)= A⁽_n¹⁾(q)+^a

h=2

A⁽_n^h)(q)(log¹_z)^h−1

(h−1)! . (3.4) Furthermore,

d^k

dz^k J_n(a,q,r;z)

|z=1

= (−1)^k 2πi

|t|=µn

n!^a−r (t −r n)r n

(t)^an(t+n)^q (t)k dt, and lettingµ → ∞we see that J_n(q) vanishes atz = 1 to the orderan +q − r n−1. As before, we can add to the second column of (3.1) a linear combination of the subsequent columns, namely we can replace A⁽_n¹⁾(0)with the sumA⁽_n¹⁾(0)+ A⁽_n²⁾(0)^log_1!^1z+· · ·+A⁽_n^a)(0)^(log_(a−^1z₁^)a−1_)! in the first row of (3.1),A⁽_n¹⁾(1)withA⁽_n¹⁾(1)+ A⁽_n²⁾(1)^log_1!^1z + · · · +A⁽_n^a)(1)^(log_(a−^1z)₁^a−1_)! in the second row of (3.1), and so on, and by using (3.4) we obtain

det(M_n)=det







−B_n(0) J_n(0) A⁽_n²⁾(0) · · · A⁽_n^a)(0)

−B_n(1) J_n(1) A⁽_n²⁾(1) · · · A⁽_n^a)(1) . . . .

−B_n(a) J_n(a) A⁽_n²⁾(a) · · · A⁽_n^a)(a)





. (3.5)

Thus, expanding the determinant on the right hand side of (3.5) along the second column, we see that det(M_n)vanishes atz=1 with multiplicity at leastan−r n−1.

Since deg(M_n)=an−r n−1, we must have

det(M_n)= ±(r n)!^a+1(z−1)^{an−r n−1} n!^r(a+1) . In particular, for anyz=1 this implies that

det(M_n)=0, (3.6)

as we claimed.

Remark 3.1. As is noted in [2, page 1383], by generalizing (1.5) one can prove that the so-called non-diagonal Pad´e approximation problem

a h=1

p_h(z)Li_h(1/z)− p₀(z)=O(z^{−r n−1}) (z→ ∞) a

h=1

p_h(z)(log(1/z))^h−1

(h−1)! = O((1−z)^{an+q−r n−1}) (z→1) p₀(z),p₁(z), . . . ,p_a(z)∈

C

^[z]

deg p₁(z), . . . ,deg p_q(z)≤n, deg p_q+1(z), . . . ,deg p_a(z)≤n−1

(3.7)

has a unique solution (up to a multiplicative constant), namely p₀(z) = B_n(q), p_h(z)= A⁽_n^h)(q) (h=1, . . . ,a).

4.

Generalization of the Nikishin determinant method

Let

K

^⊂

C

be a number field. Let

M

Kbe the set of places of

K

, and let Id∈

M

K

be the archimedean place associated with the usual absolute value in

C

^{. For any} v ∈

M

K, let

K

v be the completion of

K

with respect tov, and let η_v = [

K

v :

Q

v]. Thus,ηId = 1 if

K

⊂

R

^{, and}ηId = 2 otherwise. Let| · |v be the absolute value associated withv, normalized as follows: ifv|∞andvis associated with an embeddingσ :

K

^→

C

, we denote|β|_v = |σ(β)| for β ∈

K

; if, instead, v|p, where pis a rational prime, we let|p|_v=1/p. We also putδ=[

K

^:

Q

^]/ηId.

For_β=(β1, . . . , βm)∈

K

^{mwe define} h₀(β)= 1

[

K

^:

Q

^]

v∈MK

v=Id

η_vlog|β|_v,

where|β|_v =max{|β1|_v, . . . ,|βm|_v}. We remark that h₀(β)depends only on_β, and is independent of the field

K

^.

We need the following:

Proposition 4.1. Let_θ=(θ1, . . . , θm)∈(

C

^{×)m, let}

L

⁽_iⁿ⁾(x)=l_i⁽_,ⁿ₁⁾x₁+ · · · +l⁽_i,ⁿ_m⁾x_m (i =1, . . . ,m;n∈

N

^{) (4.1)} be linear forms with coefficients in a number field

K

⊂

C

, and suppose that for any n∈

N

the linear forms

L

⁽1ⁿ⁾, . . . ,

L

^(mn)are linearly independent.

Assume also that(for i =1, . . . ,m) lim sup

n→∞

1

nlog|

L

i⁽ⁿ⁾(θ)| ≤ −ρ lim sup

n→∞

1

nlog max{|l_i⁽_,ⁿ₁⁾|, . . . ,|l_i⁽_,ⁿ_m⁾|} ≤c lim sup

n→∞

1

nh₀(L⁽_iⁿ⁾)≤c,

(4.2)

whereL⁽_iⁿ⁾ =(l⁽_i,ⁿ₁⁾, . . . ,l_i⁽_,ⁿ_m⁾), andρ,c,c are real numbers satisfying c,c, ρ+c>0.

Then the dimensionτof the

K

-vector space spanned byθ1, . . . , θmsatisfies τ ≥ c+ρ

c+δc . (4.3)

Proof. Let

A=



a_τ+1,1 · · · a_τ+1,m . . . . a_m,1 · · · a_m,m





be a matrix with entries in

K

^{, havingm−τ} linearly independent rows such that a_i,1θ1+ · · · +a_i,mθm=0 (i =τ+1, . . . ,m). (4.4) Then for eachn, up to renumberingL⁽₁ⁿ⁾, . . . ,L⁽_mⁿ⁾, we can assume that the square matrix

⁽ⁿ⁾ =











l₁⁽ⁿ_,1⁾ · · · l₁⁽ⁿ_,m⁾ . . . .

l_τ,⁽ⁿ₁⁾ · · · l_τ,⁽ⁿ_m⁾ a_τ+1,1 · · · a_τ+1,m . . . . a_m,1 · · · a_m,m











(4.5)

is non-singular. On multiplying byθ1the first column of⁽ⁿ⁾and using (4.1) and (4.4) we get

0=θ1det⁽ⁿ⁾=det











L

⁽₁ⁿ⁾(θ) l₁⁽ⁿ_,2⁾ · · · l₁⁽ⁿ_,m⁾ . . . .

L

⁽τⁿ⁾(θ) l_τ,⁽ⁿ₂⁾ · · · l_τ,⁽ⁿ_m⁾

0 a_τ+1,2 · · · a_τ+1,m . . . . 0 a_m,2 · · · a_m,m











. (4.6)

We now apply the product formula toθ1det⁽ⁿ⁾, and for each placev ∈

M

K we use either (4.5) or (4.6), according asv=Id orv=Id, respectively. We have

0 =ηIdlog|θ1det⁽ⁿ⁾| +

v∈MK

v=Id

η_vlog|θ1det⁽ⁿ⁾|_v

≤ηId

log max

1≤i≤τ|

L

⁽_iⁿ⁾(θ)| +(τ−1) log max

1≤i≤τ 2≤j≤m

|l_i⁽_,ⁿ_j⁾|

+τ

v∈MK

v=Id

η_vlog max

1≤i≤τ 1≤j≤m

|l_i⁽_,ⁿ_j⁾|_v+γ,

for some real numberγ which is independent ofn. Dividing byηId×nand letting n → ∞, from (4.2) we obtain 0 ≤ −ρ+(τ − 1)c+τδc, which is the same as (4.3).

We are ready to prove Theorem 1.1. Before doing this we recall the definition of the Weil absolute logarithmic height:

h(β)= 1 [

K

^:

Q

^]

v∈MK

η_vlog⁺|β|_v (β∈

K

^{). (4.7)}

Here and in the sequel log⁺x = log max{x,1}for x ≥ 0. It is well known that h(β)depends only onβ, and is independent of the field

K

^.

Theorem 1.1 now follows from a straightforward application of the following analogue of Proposition 1 of [4]:

Proposition 4.2. Let a,r be integers such that1 ≤ r < a, and letα ∈

Q

^{× with}

|α|<1. Then

τ_α(a)≥ 1

δ(α)· alogr+(a+r+1)log 2−(r+1)log|α|

a+(a+r+1)log 2+rlogr+h(α) . (4.8) Proof. Letq∈[0,a] be an integer. We apply Proposition 4.1 to the linear forms

d_n^aN_n(q)=d_n^aA⁽_n¹⁾(q)Li₁(α)+ · · · +d_n^aA⁽_n^a)(q)Li_a(α)−d_n^aB_n(q), with

K

⁼

Q

^{(α),δ=δ(α)andz=α−1}. The Prime Number Theorem implies that

nlim→∞

logdn

n =1. Then by Lemma 2.1 we have lim sup

n→∞

1

nlog|d_n^aN_n(q)| ≤a−(a−r)logr+rlog|α| = −ρ.

Moreover, for all placesv∈

M

Ksuch thatv|∞, by Lemma 2 we get lim sup

n→∞

1

nlog max{|d_n^aB_n(q)|_v,|d_n^aA⁽_n¹⁾(q)|_v, . . . ,|d_n^aA⁽_n^a)(q)|_v}

≤a+rlogr+(a+r+1)log 2+log⁺|z|_v=c_v. Letc=c_Id, and note thatρ+c>0. Finally, Lemma 2.3 gives

max{|d_n^aB_n(q)|_v,|d_n^aA⁽_n¹⁾(q)|_v, . . . ,|d_n^aA⁽_n^a)(q)|_v} ≤(max{1,|z|_v})ⁿ for all placesv∈

M

_K^{such that}v

^{∞. Hence}

lim sup

n→∞

1 nh₀

d_n^aB_n(q),d_n^aA⁽_n¹⁾(q), . . . ,d_n^aA⁽_n¹⁾(q)

≤ 1

[

K

^:

Q

^] _v|∞

v=Id

η_vc_v+

v∞

η_vlog⁺|z|_v

=a+rlogr+(a+r +1)log 2+h(z)− c δ(α) =c.

As for c, we remark that h(z) = h(z⁻¹) = h(α). By (3.6) the linear forms d_n^aN_n(0), . . . ,d_n^aN_n(a)are linearly independent.

From Proposition 4.1 we get (4.8).

Proof of Theorem 1.1. Just as in [4], letr be the integer nearest toa(loga)⁻². From Proposition 4.2 we have

alogr+(a+r +1)log 2−(r+1)log|α| =aloga+o(aloga)

a+(a+r+1)log 2+rlogr+h(α)=(1+log 2)a+o(a), (a→ ∞), which implies (1.1) for alla≥a, whereadepends only onε,|α|and h(α).

References

[1] F. AMOROSOand C. VIOLA,Approximation measures for logarithms of algebraic numbers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)30(2001), 225–249.

[2] S. FISCHLERand T. RIVOAL, Approximants de Pad´e et s´eries hyperg´eom´etriques ´equi- libr´ees, J. Math. Pures Appl. (9)82(2003), 1369–1394.

[3] E. M. NIKISHIN, On the irrationality of the values of the functions F(x,s), Math. Sb.

(N.S.)109(151) (1979), 410–417 (in Russian); English translation in Math. URSS-Sb.37 (1980), 381–388.

[4] T. RIVOAL, Indep´endance lin´eaire de valeurs des polylogarithmes, J. Th´eor. Nombres Bordeaux15(2003), 551–559.

[5] C. VIOLA,Hypergeometric functions and irrationality measures, In: “Analytic Number Theory”, Y. Motohashi (ed.), London Math. Soc. Lecture Note Series 247, Cambridge Univ.

Press, 1997, 353–360.

Dipartimento di Matematica Universit`a di Pisa

Largo Bruno Pontecorvo, 5 56127 Pisa, Italy

marcovec@mail.dm.unipi.it