Bruno Anglès

Federico Pellarin

Functional identities for L-series values in positive characteristic * †

Keywords: Anderson-Thakur function, L-functions in positive characteristic, Function fields of positive characteristic, AMS Classification 11F52, 14G25, 14L05

. Our results will be applied to obtain a new class of congruences for Bernoulli-Carlitz fractions, and an analytic conjecture is stated, implying an interesting behavior of such fractions modulo prime ideals of Fq[θ].

Introduction, results

Let F q be a finite field having q elements, let θ be an indeterminate over F q , A = F q [θ] and K = F q (θ). In all the following, we denote by v ∞ the θ -1 -adic valuation normalized by setting v ∞ (θ) = -1. Let K ∞ be the completion of K for v ∞ , and let us consider the completion C ∞ of an algebraic closure of K ∞ for the unique extension of this valuation, in which we embed an algebraic closure K alg of K. Carlitz's exponential function is the surjective, F q -linear, rigid analytic entire function exp

C : C ∞ → C ∞ defined by exp C (z) = n≥0 z q n d n ,
where d 0 = 1, d n = (θ q n -θ)(θ q n -θ q ) • • • (θ q n -θ q n-1 ), n > 0.

The kernel of this function, surjective, is equal to πA, where the period π, unique up to multiplication by an element of F × q , can be computed by using the following product expansion π := θ(-θ)

1 q-1 ∞ i=1 (1 -θ 1-q i ) -1 ∈ (-θ) 1 q-1 K ∞ , (1)
once a (q -1)-th root of -θ is chosen.

Anderson-Thakur function. This function, introduced in [2, Proof of Lemma 2.5.4 p. 177], is defined by the infinite product ω(t) = (-θ)

1 q-1 i≥0 1 - t θ q i -1 , (2)
with the same choice of the (q -1)-th root as in [START_REF] Anderson | Log-Algebraicity of Twisted A-Harmonic Series and Special Values of L-Series in Characteristic p[END_REF], converges for t ∈ C ∞ such that |t| ≤ 1 (where | • | is an absolute value associated to v ∞ ) and can be extended to a non-zero rigid analytic function over C ∞ \ {θ q k ; k ≥ 0} with simple poles at θ q k , k ≥ 0.

Let t be a variable in C ∞ and let us consider the F q -algebra homomorphism

χ t : A → F q [t]
defined by formal replacement of θ by t. In other words, χ t may be viewed as the unique F q -algebra homomorphism from A to the ring of rigid analytic functions C ∞ → C ∞ such that χ t (θ) = t. More generally, we shall consider s independent variables and the F q -algebra homomorphisms χ ti : A → F q [t 1 , . . . , t s ], i = 1, . . . , s defined respectively by χ ti (θ) = t i . To simplify our notations, we will write χ ξ (a) for the evaluation at t = ξ of the polynomial function χ t (a) at a given element ξ ∈ C ∞ . Let α be a positive integer and let β 1 , . . . β s be non-negative integers. The following formal series was introduced in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]:

L(χ β1 t1 • • • χ βs ts , α) = d≥0 a∈A + (d) χ t1 (a) β1 • • • χ ts (a) βs a -α ∈ K ∞ [[t 1 , . . . , t s ]]. (3)
Here and in all the following, A + (d) denotes the set of monic polynomials of A of degree d. It is easy to see that this series is well defined. As claimed in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]Remark 7], this series converges for all (t 1 , . . . , t s ) ∈ C s ∞ to a rigid analytic entire function of s variables t 1 , . . . , t s ; see Proposition 6. The residue of ω(t) at t = θ is -π: π = -lim t→θ (t -θ)ω(t).

In [13, Theorem 1], it is proved that

L(χ t , 1) = π (θ -t)ω(t)
.

Taking into account the functional equation ω(t) q = (t q -θ)ω(t q ) apparent in [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF], this implies that, for m ≥ 0 integer, V q m ,1 (t) := π -q m ω(t)L(χ t , q m ) = 1 (θ q m -t)(θ q m-1 -t) • • • (θ -t) .

This result provided an awaited connection between the function ω of Anderson and Thakur and the "even" values of the Goss zeta function (or Carlitz zeta values)

ζ(n) = BC n π n Π(n)
, n > 0, n ≡ 0 (mod q -1)

where BC n and Π(n) denote respectively the n-th Bernoulli-Carlitz fraction and Carlitz's factorial of n, see Goss' book [START_REF] Goss | Basic structures of function field arithmetic[END_REF]Section 9.1]. Indeed, evaluating at t = θ, we get

L(χ θ , q m ) = ζ(q m -1), m ≥ 1.
More generally, it is proved in [13, Theorem 2] that, if α ≡ 1 (mod q -1) and α ≥ 1, then

µ α = π -α L(χ t , α)ω(t)
is a rational function in F q (θ, t). In [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF], it is suggested that this result could be a source of information in the study of the arithmetic properties of the Bernoulli-Carlitz fractions. However, the methods of loc. cit. (based on deformations of vectorial modular forms and Galois descent) are only partially explicit. More recently, Perkins [START_REF] Perkins | On Pellarin's L-series[END_REF] investigated the properties of certain special polynomials associated to variants of the functions L(χ β t , α) with α ≤ 0 which turn out to be polynomial. He notably studied the growth of their degrees. Moreover, by using Wagner's interpolation theory for the map χ t , Perkins [START_REF] Perkins | Explicit Formulae for L-values in Finite Characteristic[END_REF] obtained explicit formulas for the series

L(χ t1 • • • χ ts , α), α > 0, 0 ≤ s ≤ q, α ≡ s (mod q -1).
We quote here a particular case of Perkins' formulas for the functions L(χ t , α) with α ≡ 1 (mod q -1)

L(χ t , α) = µ j=0 d -1 j (t -θ)(t -θ q ) • • • (t -θ q j-1 )ζ(α -q j )L(χ t , q j ), ( 4
)
where µ is the biggest integer such that q µ ≤ α. It seems difficult to overcome the threshold s ≤ q giving at once expressions for L(χ t1 • • • χ ts , α) with the effectiveness of Perkins' results.

In the next Theorem, we extend the previous results beyond the mentioned threshold, providing at once new quantitative information. Let n be a non-negative integer, let us consider the expansion n = r i=0 n i q i in base q, where n i ∈ {0, • • • , q -1}. We denote by ℓ q (n) the sum of the digits n i of this expansion: ℓ q (n) = r i=0 n i . We have: Theorem 1 Let α, s be positive integers, such that α ≡ s (mod q -1). Let δ be the smallest non-negative integer such that, simultaneously, q δ -α ≥ 0 and s + ℓ q (q δ -α) ≥ 2. The formal series:

V α,s (t 1 , . . . , t s ) = π -α L(χ t1 • • • χ ts , α)ω(t 1 ) • • • ω(t s ) s i=1 δ-1 j=0 1 - t i θ q j ∈ K ∞ [[t 1 , . . . , t s ]] (5)
is in fact a symmetric polynomial of K[t 1 , . . . , t s ] of total degree δ(α, s) such that

δ(α, s) ≤ s s + ℓ q (q δ -α) q -1 -s.
This statement holds if α = q m and s ≥ 2 (so that δ = m) assuming that empty products are equal to one by convention. In this case, since s ≡ α (mod q -1), we have s + ℓ q (q δ -α) ≡ 0 (mod q -1) so that in fact, s ≥ max{2, q -1}. The reader may have noticed that the choice α = q m and s = 1 is not allowed in Theorem 1. However, as mentioned above, the computation of V q m ,1 is completely settled in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]. This discrimination of the case α = q m , s = 1 should not be surprising neither; similarly, the Goss zeta function associated to A has value 1 at zero, but vanishes at all negative integers divisible by q -1.

In Section 3, we will be more specifically concerned with Bernoulli-Carlitz numbers. A careful investigation of the polynomials V 1,s and an application of the digit principle ( [START_REF] Conrad | The Digit Principle[END_REF]) to the function ω will allow us to show that, for s ≥ 2 congruent to one modulo q -1,

B s = Π(s) -1 V 1,s (θ, . . . , θ)
is a polynomial of F q [θ] (Proposition 24) (1 ). We don't know wether B s vanishes or not for general s. In all the following, a prime is by definition a monic irreducible polynomial in A. We shall then show the next Theorem, which highlights the interest of these polynomials in θ. Theorem 2 Let s ≥ 2, s ≡ 1 (mod q -1). Let us consider the expansion s = r i=0 s i q i of s in base q. Let d be an integer such that q d > s and let p be a prime of degree d. Then:

B s ≡ (-1) s BC q d -s r i=0 l siq i d-i-1 Π(q d -s) (mod p).
In this result, l d denotes the polynomial (-1) d d i=1 (θ q i -θ); we observe that the latter polynomial is invertible modulo p just as Π(q d -s). The non-vanishing of B s for fixed s signifies the existence of an explicit constant c > 0, depending on s and q, such that for all d ≥ c, BC q d -s ≡ 0 (mod p), for all p such that deg p = d.

However, the non-vanishing of B s is also equivalent to the fact that the function

L(χ t1 • • • χ ts , 1) s i=1 (t i -θ) -1 ,
entire of s variables as we will see, is a unit when identified to an element of C ∞ [[t 1 -θ, . . . , t s -θ]]; we presently do not know how to prove this property for all s. Therefore, the property ( 6) is linked with the following conjecture of nature analogue of classical results on the simplicity of the zeroes of Goss zeta functions and L-series, which should be, we believe, true.

Conjecture 3 Let s ≥ 2 be congruent to one modulo q -1. Then, locally at

t 1 = • • • = t s = θ, the set of the zeroes of the function L(χ t1 • • • χ ts , 1
) is equal to the set of zeroes of the polynomial

i (t i -θ).
Numerical computations on Bernoulli-Carlitz fractions made by Taelman provide some evidence to support this hypothesis. The Conjecture follows from Perkins results [START_REF] Perkins | Explicit Formulae for L-values in Finite Characteristic[END_REF] in the case s ≤ q and α = s. The conjecture is also verified if ℓ q (s) = q and α = 1, thanks to Corollary 26.

2 Functional identities for L-series Let d, s be non-negative integers. We begin with the study of the vanishing of the sums

S d,s = S d,s (t 1 , . . . , t s ) = a∈A + (d) χ t1 (a) • • • χ ts (a) ∈ F q [t 1 , . . . , t s ],
which are symmetric polynomials in t 1 , . . . , t s of total degree ≤ ds, with the standard conventions on empty products. We recall that, for n ≥ 0, a∈Fq a n equals -1 if n ≡ 0 (mod q -1) and n ≥ 1, and equals 0 otherwise. We owe the next Lemma to D. Simon [START_REF] Anglès | [END_REF]. We give the proof here for the sake of completeness.

Lemma 4 (Simon's Lemma) We have S d,s = 0 if and only if d(q -1) ≤ s.

Proof. Since

S d,s = a0∈Fq • • • a d-1 ∈Fq s i=1 (a 0 + a 1 t i + • • • + a d-1 t d-1 i + t d i ), the coefficient c v1,...,vs of t v1 1 • • • t vs s with v i ≤ ds (i = 1, . . . , s
) is given by the sum:

a0∈Fq • • • a d-1 ∈Fq a v1 • • • a vs .
The last sum can be rewritten as:

c v1,...,vs =   a0∈Fq a µ0 0   • • •   a d-1 ∈Fq a µ d-1 d-1   , (7)
where µ i is the cardinality of the set of the indices j such that v j = i, from which one notices that

d-1 i=0 µ i ≤ s
(notice also that si µ i is the cardinality of the set of indices j such that v j = d). For any choice of µ 0 , . . . , µ d-1 such that i µ i ≤ s, there exists (v 1 , . . . , v s ) such that (7) holds. If s < d(q -1), for all (v 1 , . . . , v s ) as above, there exists i such that, in [START_REF] Anglès | [END_REF], µ i < q -1 so that S d,s = 0. On the other hand, if s ≥ d(q -1), it is certainly possible to find (v 1 , . . . , v s ) such that, in [START_REF] Anglès | [END_REF],

µ 0 = • • • = µ d-1 = q -1
so that the sum does not vanish in this case.

As an immediate corollary of Lemma 4, we see that the series

F s = F s (t 1 , . . . , t s ) = d≥0 S d,s = d≥0 a∈A + (d) χ t1 (a) • • • χ ts (a)
defines a symmetric polynomial of F q [t 1 , . . . , t s ] of total degree at most s 2 q-1 . In the next Lemma, we provide a necessary and sufficient condition for the vanishing of the polynomial F s . Lemma 5 If s ≥ 1, then, F s = 0 if and only if s ≡ 0 (mod q -1).

Proof. We will use several times the following elementary observation : let G ∈ C ∞ [t 1 , . . . , t s ] and let S 1 , . . . , S s ⊂ F alg q (F alg q denotes the algebraic closure of F q , embedded in C ∞ ) be infinite sets such that G vanishes on

S 1 × • • • × S s . Then G = 0.
Let us assume first that s ≡ 0 (mod q -1). The hypothesis on s implies that

a∈A,deg θ (a)=d χ t1 (a) • • • χ ts (a) = -S d,s .
We denote by A(≤ d) the set of polynomials of A of degree ≤ d and we write

G d,s = a∈A(≤d) χ t1 (a) • • • χ ts (a).
We then have:

G s q-1 ,s = -F s . Let us choose now distinct primes p 1 , . . . , p s of respective degrees d 1 , . . . , d s ≥ s/(q -1) and f = p 1 • • • p s . For all i = 1, . . . , s, we choose a root ζ i ∈ F alg q of p i . Let us then consider the Dirichlet character of the first kind χ = χ ζ1 • • • χ ζs . We have: F s (ζ 1 , . . . , ζ s ) = -G s q-1 ,s (ζ 1 , . . . , ζ s ) = - a∈A(≤s/(q-1)) χ(a) = - a∈A(≤d1+•••+ds) χ(a) = - a∈(A/fA) × χ(a)
= 0, by [START_REF] Rosen | Number Theory in Function Fields[END_REF]Proposition 15.3]. By the observation at the beginning of the proof, this implies the vanishing of F s . On the other hand, if s ≡ 0 (mod q -1), then F s (θ, . . . , θ) = ζ(-s) the s-th Goss' zeta value which is non zero, see [10, Remark 8.13.8 1].

Analyticity

The functions L(χ t1 • • • χ ts , α) are in fact rigid analytic entire functions of s variables. This property, mentioned in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF], can be deduced from the more general Proposition 6 that we give here for convenience of the reader.

Let a be a monic polynomial of A. we set:

a = a θ deg θ (a) ∈ 1 + θ -1 F q [θ -1 ]. Let y ∈ Z p ,
where p is the prime dividing q. Since a is a 1-unit of K ∞ , we can consider its exponentiation by y:

a y = j≥0 y j ( a -1) j ∈ F q [[θ -1 ]].
Here, the binomial y j is defined, for j a positive integer, by extending Lucas formula. Writing the p-adic expansion i≥0 y i p i of y (y i ∈ {0, . . . , p -1}) and the p-adic expansion r i=0 j i p i of j (j i ∈ {0, . . . , p -1}), we are explicitly setting:

y j = r i=0 y i j i .
Note that, for a ∈ A + , we have a continuous function: Z p → K ∞ , y → a y . We also recall, from [START_REF] Goss | Basic structures of function field arithmetic[END_REF]Chapter 8], the topological group S ∞ = C × ∞ × Z p . For (x, y) ∈ S ∞ and d, s non-negative integers, we define the sum

S d,s (x, y) = S d,s (x, y)(t 1 , . . . , t s ) = x -d a∈A + (d) χ t1 (a) • • • χ ts (a) a y ∈ x -d K ∞ [t 1 , . . . , t s ],
which is, for all x, y, a symmetric polynomial of total degree ≤ ds.

Let us further define, more generally, for variables t 1 , . . . , t s ∈ C ∞ and (x, y) ∈ S ∞ , the series:

L(χ t1 . . . χ ts ; x, y) = d≥0 S d,s (x, y)(t 1 , . . . , t s ).
For fixed choices of (x, y) ∈ S ∞ , it is easy to show that

L(χ t1 . . . χ ts ; x, y) ∈ C ∞ [[t 1 , . . . , t s ]],
and with a little additional work, one also verifies that this series defines an element of the standard Tate C ∞ -algebra T t1,...,ts in the variables t 1 , . . . , t s . Of course, if (x, y) = (θ α , -α) with α > 0 integer, we find L(χ t1 . . . χ ts ; θ α , -α) = L(χ t1 . . . χ ts , α).

The next Proposition holds, and improves results of Goss; see [START_REF] Goss | On the L-series of F. Pellarin[END_REF]Theorems 1,[START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF]).

Proposition 6

The series L(χ t1 , . . . , χ ts ; x, y) converges for all (t 1 , . . . , t s ) ∈ C s ∞ and for all (x, y) ∈ S ∞ , to an entire function on C s ∞ × S ∞ in the sense of Goss [START_REF] Goss | Basic structures of function field arithmetic[END_REF]Definition 8.5.1].

The proof of this result is a simple consequence of the Lemma below. The norm • used in the Lemma is the supremum norm of T t1,...,ts .

Lemma 7 Let (x, y) be in S ∞ and let us consider an integer d > (s -1)/(q -1), with s > 0. Then:

S d,s (x, y) ≤ |x| -d q -q [d-s q-1 ]-1
, where for x ∈ R, [x] denotes the integer part of x.

Proof. Let us write the p-adic expansion y = n≥0 c n p n , with c n ∈ {0, . . . , p -1} for all n. Collecting blocks of e consecutive terms (where q = p e ), this yields a "q-adic" expansion, from which we can extract partial sums:

y n = en-1 k=0 c k p k = n i=0 u i q i ∈ Z ≥0 ,
where

u i = e(i+1)-1 j=ei c j p j-ei ∈ {0, . . . , q -1}.
In particular, for n ≥ 0, we observe that ℓ q (y n ) ≤ (n + 1)(q -1). Since

S d,s (x, y n ) = 1 x d θ dyn a∈A + (d) χ t1 (a) • • • χ ts (a)a yn = 1 x d θ dyn S d,r (t 1 , . . . , t s , θ, . . . , θ u0 times
, θ q , . . . , θ q u1 times , . . . , θ q n , . . . , θ q n un times ) with r = s + ℓ q (y n ), if d(q -1) > s + ℓ q (y n ), we have by Simon's Lemma 4:

S d,s (x, y n ) = 0. This condition is ensured if d(q -1) > s + (n + 1)(q -1). Now, we claim that S d,s (x, y) -S d,s (x, y n ) ≤ |x| -d q -q n+1 .
Indeed,

S d,s (x, y) -S d,s (x, y n ) = x -d a∈A + (d) χ t1 (a) • • • χ ts (a) j≥0 y j - y n j ( a -1) j ,
and y j = yn j for j = 0, . . . , q n+1 -1 by Lucas' formula and the definition of the binomial, so that

j≥0 y j - y n j ( a -1) j ≤ q -q n+1 .
The Lemma follows by choosing n

+ 2 = [d -s q-1 ].
In particular, we have the following Corollary to Proposition 6 which generalizes [12, Theorem 1], the deduction of which, easy, is left to the reader.

Corollary 8 For any choice of an integer α > 0 and non-negative integers M 1 , . . . , M s , the function

L(χ M1 t1 • • • χ Ms ts , α) = d≥0 a∈A + (d) χ t1 (a) M1 • • • χ ts (a) Ms a -α defines a rigid analytic entire function C s ∞ → C ∞ .

Computation of polynomials with coefficients in K ∞

Lemma 9 For all d ≥ 0, we have:

S d (-α) = a∈A + (d) a -α = 0.
Proof. This follows from [11, proof of Lemma 8.24.13].

We introduce, for d, s, α non-negative integers, the sum:

S d,s (-α) = a∈A + (d) χ t1 (s) • • • χ ts (a)a -α ∈ K[t 1 , . . . , t s ],
representing a symmetric polynomial of K[t 1 , . . . , t s ] of exact total degree ds by Lemma 9. We have, with the notations of Section 2.1:

S d,s (-α) = S d,s (θ α , -α).
From the above results, we deduce the following Proposition.

Proposition 10 Let α be a fixed positive integer. Let l ≥ 0 be an integer such that q l -α ≥ 0 and ℓ q (q l -α) + s ≥ 2. If ℓ q (q l -α) + s ≤ d(q -1), then:

S d,s (-α) ≡ 0 (mod s j=1 (t j -θ q l )).
Furthermore, let us assume that s ≡ α (mod q -1). With l as above, let k be an integer such that k(q -1) ≥ ℓ q (q l -α) + s. Then:

k d=0 S d,s (-α) ≡ 0 (mod s j=1 (t j -θ q l )).
Proof. Let us write m = ℓ q (q l -α). We have s -1 + m < d(q -1) so that, by Simon's Lemma 4, S d,s-1+m = 0. Now, let us write the q-ary expansion q l -α = n 0 + n 1 q + • • • + n r q r with n i ∈ {0, . . . , q -1} and let us observe that, since q l -α ≥ 0,

S d,s (-α)(t 1 , . . . , t s-1 , θ q l ) = a∈A + (d) χ t1 (a) • • • χ ts-1 (a)a q l -α = a∈A + (d) χ t1 (a) • • • χ ts-1 (a)χ θ (a) n0 χ θ q (a) n1 • • • χ θ q r (a) nr = S d,s-1+m (t 1 , . . . , t s-1 , θ, . . . , θ n0 times
, θ q , . . . , θ q n1 times , . . . , θ q r , . . . , θ q r nr times ) = 0.

Therefore t s -θ q l divides S d,s (-α). The first part of the Proposition follows from the fact that this polynomial is symmetric. For the second part, we notice by the first part, that the condition on k is sufficient for the sum S d,s (-α)(t 1 , . . . , t s ) to be congruent to zero modulo s i=1 (t i -θ q l ) for all d ≥ k. It remains to apply Lemma 5 to conclude the proof. Proposition 11 Let s, α ≥ 1, s ≡ α (mod q -1). Let δ be the smallest non-negative integer such that q δ ≥ α and s + ℓ q (q δ -α) ≥ 2. Then, the function of Theorem 1

V α,s (t 1 , . . . , t s ) = L(χ t1 • • • χ ts , α)ω(t 1 ) • • • ω(t s ) π -α   s i=1 δ-1 j=0 1 - t i θ q j   is in fact a symmetric polynomial of K ∞ [t 1 , . . . , t s ].
Moreover, its total degree δ(α, s) is not bigger than s s+ℓq(q δ -α) q-1 -s.

Proof. Let δ be the smallest non-negative integer such that q δ -α ≥ 0 and s + ℓ q (q δ -α) ≥ 2. We fix an integer k such that k(q -1) ≥ s + ℓ q (q δ -α).

We also set:

N (k) = δ + k - s + ℓ q (q δ -α) q -1 .
Obviously, N (k) ≥ δ. Let l be an integer such that

δ ≤ l ≤ N (k).
We claim that we also have k(q -1) ≥ s + ℓ q (q l -α).

Indeed, let us write the q-ary expansion α = α 0 + α 1 q + • • • + α m q m with α m = 0. Then, δ = m if α = q m and s ≥ 2 and δ = m + 1 otherwise. If l is now an integer l ≥ δ, we have

q l -α = q l -q δ + q δ -α = q δ (q -1) l-δ-1 i=0 q i + q δ -α,
where the sum over i is zero if l = δ, and

ℓ q (q l -α) = (q -1)(l -δ) + ℓ q (q δ -α)
because there is no carry over in the above sum. Now, the claim follows from [START_REF] Conrad | The Digit Principle[END_REF].

By Proposition 10 we have, with k as above, that the following expression

W k,s,α :=   s i=1 N (k) j=δ 1 - t i θ q j -1   k d=0 S d,s (-α)
is in fact a symmetric polynomial in K[t 1 , . . . , t s ]. By Lemma 9, S d,s (-α) ∈ K[t 1 , . . . , t s ] is a symmetric polynomial of total degree ds; indeed, the coefficient of α). Hence, the total degree of k d=0 S d,s (-α) is exactly ks. The total degree of the product

t d 1 • • • t d s is exactly S d (-
s i=1 N (k) j=δ 1 - t i θ q j is equal to s(1 + N (k) -δ) so that, by the definition of N (k): deg(W k,s,α ) = sk -s -sN (k) + sδ = sk -sk -sδ + sδ -s + s s + ℓ q (q δ -α) q -1 = s s + ℓ q (q δ -α) q -1 -s,
independent on k. We now let k tend to infinity. The Proposition follows directly from the definition (2) of ω as an infinite product, the fact that, in (2), πθ -1/(q-1) ∈ K ∞ , and the definition of L(χ t1 • • • χ ts , α).

Preliminaries on Gauss-Thakur sums

We review quickly the theory of Gauss-Thakur sums, introduced by Thakur in [START_REF] Thakur | Gauss sums for F q [T ][END_REF].

Let p be an irreducible monic polynomial of A of degree d, let ∆ p be the Galois group of the p-cyclotomic function field extension K p = K(λ p ) of K, where λ p is a non zero p-torsion element of K alg (the algebraic closure of K in C ∞ ). Gauss-Thakur sums can be associated to the elements of the dual character group ∆ p via the Artin symbol (see [11, Sections 7.5.5 and 9.8]). If χ is in ∆ p , we denote by g(χ) the associated Gauss-Thakur sum. In particular, we have the element ϑ p ∈ ∆ p obtained by reduction of the Teichmüller character [11, Definition 8.11.2], uniquely determined by a choice of a root ζ of p, and the Gauss-Thakur sums g(ϑ q j p ) associated to its q j -th powers, with j = 0, . . . , d -1, which can be considered as the building blocks of the Gauss-Thakur sums g(χ) for general χ ∈ ∆ p .

Definition 12 With p, d, ϑ p as above, the basic Gauss-Thakur sum g(ϑ q j p ) associated to this data is the element of K alg defined by:

g(ϑ q j p ) = δ∈∆p ϑ p (δ -1 ) q j δ(λ p ) ∈ F p [λ p ].
The same sum is denoted by g j in [START_REF] Goss | Basic structures of function field arithmetic[END_REF][START_REF] Thakur | Gauss sums for F q [T ][END_REF]. The basic Gauss-Thakur sums are used to define general Gauss-Thakur sums associated to arbitrary elements of ∆ p . For instance, if χ = χ 0 is the trivial character, then g(χ) = 1. The group ∆ p being isomorphic to ∆ p , it is cyclic; it is in fact generated by ϑ p . Let χ be an element of ∆ p . There exists a unique integer i with 0 < i < q d , such that χ = ϑ i p . Let us expand i in base q, that is, let us write i = i 0 + i 1 q + • • • + i d-1 q d-1 with i j ∈ {0, . . . , d -1}. Then, χ = d-1 j=0 (ϑ q j p ) ij .

Definition 13

The general Gauss-Thakur sum g(χ) associated to χ ∈ ∆ p as above, is defined by:

g(χ) = d-1 j=0 g(ϑ q j p ) ij .
More generally, let us now consider a non-constant monic polynomial a ∈ A. We denote by ∆ a the dual character group Hom(∆ a , (F alg q ) × ), where ∆ a is the Galois group of the extension K a of K generated by the a-torsion of the Carlitz module.

If χ is in ∆ a , we set: F q (χ) = F q (χ(δ); δ ∈ ∆ a ) ⊂ F alg q . We also write

F a = F q (χ; χ ∈ ∆ a )
and we recall that Gal(K a (F a )/K(F a )) ≃ ∆ a . We observe that ∆ a is isomorphic to ∆ a if and only if a is squarefree.

If a = p 1 • • • p n with p 1 , . . . , p n distinct primes, then ∆ a ≃ n i=1 ∆ pi .
Let us then assume that a is non-constant and square-free. We want to extend the definition of the Gauss-Thakur sums to characters in ∆ a . For χ ∈ ∆ a , χ = χ 0 , there exist r distinct primes p 1 , • • • , p r and characters χ 1 , . . . , χ r with χ j ∈ ∆ pj for all j, with

χ = χ 1 • • • χ r .
Definition 14 The Gauss-Thakur sum associated to χ is the product:

g(χ) = g(χ 1 ) • • • g(χ r ).
The polynomial

f χ = p 1 • • • p r is called the conductor of χ; it is a divisor of a.
The degree of f χ will be denoted by d χ . If a itself is a prime p of degree d, then F χ = p and d χ = d.

The following result collects the basic properties of the sums g(χ) that we need, and can be easily deduced from Thakur's results in [START_REF] Thakur | Gauss sums for F q [T ][END_REF]Theorems I and II].

Proposition 15 Let a ∈ A be monic, square-free of degree d. The following properties hold.

1. For all δ ∈ ∆ a , we have δ(g(χ)) = χ(δ)g(χ).

If

χ = χ 0 , then g(χ)g(χ -1 ) = (-1) dχ f χ .
By the normal basis theorem, K a is a free K[∆ a ]-module of rank one. Gauss-Thakur's sums allow to determine explicitly generators of this module:

Lemma 16 Let us write η a = χ∈ ∆a g(χ) ∈ K a . Then :

K a = K[∆ a ]η a ,
and

A a = A[∆ a ]η a ,
where A a is the integral closure of A in K a .

Moreover, let χ be in ∆ a . Then, the following identity holds:

K a (F a )g(χ) = {x ∈ K a (F a ) such that for all δ ∈ ∆ a , δ(x) = χ(δ)x}. (9)
Proof. Let us expand a in product

p 1 • • • p n of distinct primes p i . To show that A a = A[∆ a ]η a (this yields the identity K a = K[∆ a ]η a ) one sees that A a ≃ A p1 ⊗ A • • • ⊗ A A pn ,
because the discriminants of the extensions A pi /A are pairwise relatively prime and the fields K pi are pairwise linearly disjoint (see [9, Property (2.13)]). One then uses [3, Théorème 2.5] to conclude with the second identity. We now prove the identity [START_REF] Frohlich | Algebraic Number Theory[END_REF]. We recall that if we set, for χ ∈ ∆ a ,

e χ = 1 |∆ a | δ∈∆a χ(δ)δ -1 ∈ F q (χ)[∆ a ]
(well defined because p, the rational prime dividing q, does not divide |∆ a |), then the following identities hold:

• e χ e ψ = δ χ,ψ e χ (where δ χ,ψ denotes Kronecker's symbol),

• for all δ ∈ ∆ a , δe χ = χ(δ)e χ ,

• χ∈ ∆a = 1.
This yields e χ η a = g(χ). Now, by K a (F a ) = K a (F a )η a , we get e χ K a (F a ) = K a (F a )g(χ). The second part of the Lemma then follows by observing that if M is an F a [∆ a ]-module, then e χ M = {m ∈ M such that for all δ ∈ ∆ a , δm = χ(δ)m}.

An intermediate result on special values of Goss L-functions

Let χ be a Dirichlet character of the first kind, that is a character

χ : (A/aA) × → (F alg q ) × ,
where a is a non-constant squarefree monic element of A which we identify, by abuse of notation, to a character of ∆ a still denoted by χ, of conductor f = f χ , and degree d = deg θ f . Let s(χ) be the type of χ, that is, the unique integer s(χ) ∈ {0, • • • , q -2} such that:

χ(ζ) = ζ s(χ) for all ζ ∈ F × q .
We now consider the special value of Goss' abelian L-function [START_REF] Goss | Basic structures of function field arithmetic[END_REF]Section 8]:

L(α, χ) = a∈A + χ(a)a -α , α ≥ 1.
The following result is inspired by the proofs of [1, Proposition 8, VII] and [4, Proposition 8.2]:

Proposition 17 Let α ≥ 1, α ≡ s(χ) (mod q-1). Then there exists an element B α,χ -1 in F q (χ)(θ) such that :

L(α, χ)g(χ) π α = B α,χ -1 Π(α) ,
where Π(α) is the Carlitz factorial of α (see [11, chapter 9, section 1]).

Proof. The proposition is known to be true for the trivial character (see [START_REF] Goss | Basic structures of function field arithmetic[END_REF]Section 9.2]); in this case, we notice that:

B α,χ -1 0 = BC α , α ≥ 1, α ≡ 0 (mod q -1),
where we recall that BC α is the α-th Bernoulli-Carlitz number (see [START_REF] Goss | Basic structures of function field arithmetic[END_REF]Definition 9.2.1]. We now assume that χ = χ 0 . Since:

exp C (z) = z a∈A\{0} 1 - z πa ,
We have:

1 exp C (z) = a∈A 1 z -πa .
Let b ∈ A be relatively prime with f and let σ b ∈ Gal(K f /K) be the element such that σ b (λ f ) = φ b (λ f ). We have:

1 exp C (z) -σ b (λ f ) = - n≥0 f n+1 π n+1 a∈A 1 (b + af) n+1 z n .
Therefore, we obtain:

b∈(A/fA) × χ(b) exp C (z) -σ b (λ f ) = - n≥0 f n+1 π n+1   a∈A\{0} χ(a) a n+1   z n .
If n + 1 ≡ s(χ) (mod q -1), we get: a∈A\{0} χ(a) a n+1 = 0, and if n + 1 ≡ s(χ) (mod q -1), we have:

a∈A\{0} χ(a) a n+1 = -L(n + 1, χ).
Thus:

b∈(A/fA) × χ(b) exp C (z) -σ b (λ f ) = i≥1, i≡s(χ) (mod (q-1)) f i L(i, χ) π i z i-1 . (10)
But note that by the second part of Lemma 16:

b∈(A/fA) × χ(b) exp C (z) -σ b (λ f ) ∈ g(χ -1 )F q (χ)(θ)[[z]].
Since by Proposition 15, g(χ)g(χ -1 ) = (-1) d f,

where d = deg θ f χ , we get the result by comparison of the coefficients of the series expansion of both sides of [START_REF] Goss | v-adic zeta functions, L-series and measures for function fields[END_REF].

Remark 18 In the above proof of Proposition 17, if we set α = 1 and we assume s(χ) ≡ 1 (mod q -1), we have, by comparison of the constant terms in the series expansions in powers of z in (10):

π -1 fL(1, χ) = - b∈(A/fA) × χ(b) σ b (λ f ) ∈ g(χ -1 )F q (χ)(θ).
Assuming that f is not a prime, by [START_REF] Rosen | Number Theory in Function Fields[END_REF]Proposition 12.6],

λ f is a unit in the integral closure A f of A in K f . Therefore, b∈(A/fA) × χ(b) σ b (λ f ) ∈ g(χ -1 )F q (χ)[θ]
and we deduce that

π -1 L(1, χ)g(χ) ∈ F q (χ)[θ].
This remark will be crucial in the proof of Corollary 21.

Proof of Theorem 1

In [5, Theorem 3] (see also [START_REF] Anglès | [END_REF]), we noticed that the function ω can be viewed as a universal Gauss-Thakur sum. We review this result, which will be used a little later.

Theorem 19 Let p be a prime element of A of degree d and ζ a root of p as above. We have:

g(ϑ q j p ) = p ′ (ζ) -q j ω(ζ q j ), j = 0, . . . , d -1.
In this theorem, p ′ denotes the derivative of p with respect to θ.

The next Lemma provides a rationality criterion for a polynomial a priori with coefficients in K ∞ , again based on evaluation at roots of unity.

Lemma 20 Let F (t 1 , . . . , t s ) ∈ K ∞ [t 1 , . . . , t s ] such that for all ζ 1 , . . . , ζ s ∈ F alg q , pairwise not con- jugate over F q , F (ζ 1 , . . . , ζ s ) ∈ K(ζ 1 , . . . , ζ s ).
Then

F (t 1 , . . . , t s ) ∈ K[t 1 , . . . , t s ].
Proof. We begin by pointing out that if elements a 1 , . . . , a s ∈ K ∞ are K ⊗ Fq F alg q -linearly dependent, then they also are K-linearly dependent. The proof proceeds by induction on s ≥ 1. For s = 1, this is obvious. Now, let

s i=1 λ s a s = 0 (11)
be a non-trivial relation of linear dependence with the λ i ∈ K ⊗ F alg q \ {0}. We may assume that λ s = 1 and that there exists i ∈ {1, . . . , s -1} such that λ i ∈ K. Then, there exists

σ ∈ Gal(K ∞ ⊗ F alg q /K ∞ ) ≃ Gal(K ⊗ F alg q /K) ≃ Gal(F alg q /F q )
such that σ(λ i ) = λ i . Applying σ on both left-and right-hand sides of [START_REF] Goss | Basic structures of function field arithmetic[END_REF] and subtracting, yields a non-trivial relation involving at most s -1 elements of K ∞ on which we can apply the induction hypothesis.

We can now complete the proof of the Lemma. Let F be a polynomial in K ∞ [t 1 , . . . , t s ] not in K[t 1 , . . . , t s ]. It is easy to show that there exist a 1 , . . . , a m ∈ K ∞ , linearly independent over K, such that

F = a 1 P 1 + • • • + a m P m ,
where P 1 , . . . , P m are non-zero polynomials of K[t 1 , . . . , t s ]. Let us suppose by contradiction that there exists

F ∈ K ∞ [t 1 , . . . , t s ] \ K[t 1 , .
. . , t s ] satisfying the hypotheses of the Lemma. By the observation at the beginning of the proof of Lemma 5, there exists a choice of such roots of unity ζ 1 , . . . , ζ s and i ∈ {1, . . . , m} such that P i (ζ 1 , . . . , ζ m ) = 0. This means that a 1 , . . . , a m are K ⊗ F alg qlinearly dependent, thus K-linearly dependent by the previous observations; a contradiction.

Proof of Theorem 1. In view of Lemma 20, we want to show that the polynomial

V α,s = π -α L(χ t1 • • • χ ts , α)ω(t 1 ) • • • ω(t s )   s i=1 δ-1 j=0 1 - t i θ q j   ∈ K ∞ [t 1 , • • • , t s ]
of Proposition 11 takes values in K(ζ 1 , . . . , ζ s ) for all ζ 1 , . . . , ζ s ∈ F alg q pairwise non conjugate over F q . Let (ζ 1 , . . . , ζ s ) be one of such s-tuples of roots of unity and, for i = 1, . . . , s, let p i ∈ A be the minimal polynomial of ζ i , so that p 1 , . . . , p s are pairwise relatively prime. We choose the characters ϑ pi so that ϑ pi (σ θ ) = ζ i for all i. We construct the Dirichlet character of the first kind χ defined, for a ∈ A, by

χ(a) = χ ζ1 (a) • • • χ ζs (a).
By Proposition 17, we have

L(α, χ)g(χ) π α = (-1) dχ B α,χ -1 f α-1 χ ∈ F q (χ)(θ). Since L(α, χ) = L(χ ζ1 • • • χ ζs , α),
we get:

V α,s (ζ 1 , . . . , ζ s ) = L(α, χ)ω(ζ 1 ) • • • ω(ζ s ) π -α = L(α, χ)g(χ) π α ω(ζ 1 ) • • • ω(ζ s ) g(χ) = (-1) dχ B α,χ -1 f α-1 χ χ ζ1 (p ′ 1 ) • • • χ ζs (p ′ s ) ∈ K(ζ 1 , . . . , ζ s ),
where in the next to last step, we have used Theorem 19. The proof of Theorem 1 now follows from Lemma 20.

Congruences for Bernoulli-Carlitz numbers

In this Section, we shall prove Theorem 2. This is possible because in Theorem 1, more can be said when α = 1. In this case, one sees that the integer δ of Theorem 1 is equal to zero and s ≥ q, so that, with the notations of that result,

V 1,s = π -1 L(χ t1 • • • χ ts , 1)ω(t 1 ) • • • ω(t s ).
In the next Subsection we will show that the above is a polynomial of A[t 1 , . . . , t s ].

Functional identities with α = 1

We begin with the following Corollary of Theorem 1. The main result of this subsection is Proposition 24.

Corollary 21 Let s ≥ 2 be such that s ≡ 1 (mod q -1). Then the symmetric polynomial V 1,s ∈ K[t 1 , . . . , t s ] of Theorem 1 is in fact a polynomial of A[t 1 , . . . , t s ] of total degree ≤ s 2 /(q -1) -s in the variables t 1 , . . . , t s .

Proof. Let p 1 , . . . , p s be distinct primes in A, let us write a = p 1 • • • p s and let us consider the Dirichlet character of the first kind χ associated to ϑ p1 • • • ϑ ps that we also loosely identify with the corresponding element of ∆ a . Since a is not a prime power, Remark 18 implies that

π -1 L(1, χ)g(χ) ∈ F q (χ)[θ]. (12)
Now, specializing at t i = ζ i the root of p i associated to the choice of characters ϑ pi for all i = 1, . . . , s,

we obtain V 1,s (ζ 1 , . . . , ζ s ) = π -1 L(1, χ)g(χ) ∈ F q (ζ 1 , . . . , ζ s )[θ],
and the result follows from an idea similar to that of Lemma 20, the bound on the degree agreeing with that of Theorem 1.

Digit principle for the function ω and the L-series

We denote by T t the standard Tate C ∞ -algebra in the variable t. Let ϕ : T t → T t be the C ∞ -linear map defined by

ϕ   n≥0 c n t n   = n≥0 c n t qn , c n ∈ C ∞ .
We also set, for N a non-negative integer with its expansion in base q, N = N 0 + N 1 q + • • • + N r q r , N i ∈ {0, . . . , q -1}:

ω N (X) = r i=0 ϕ i (ω(X)) Ni .
We then have the next Lemma.

Lemma 22 Let p be a prime of A of degree d and let N be an integer such that 1 ≤ N ≤ q d -1.

The following identity holds:

ω N (ζ) = ϑ p (σ p ′ ) N g(ϑ N p )
, where ζ is the root of p that determines the character ϑ p .

Proof. This is a direct application of Theorem 19. Indeed,

ω N (ζ) = d-1 i=0 ω(ζ q i ) Ni = d-1 i=0 ϑ p (σ p ′ ) q i Ni g(ϑ q i p ) Ni .
Let X, Y be two indeterminates over K. We introduce a family of polynomials (G d ) d≥0 in F q [X, Y ] as follows. We set G 0 (X, Y ) = 1 and

G d (X, Y ) = d-1 i=0 (X -Y q i ), d ≥ 1.
This sequence is closely related to the sequence of polynomials G n (y) of [2, Section 3.6]: indeed, the latter can be rewritten in terms of the former:

G d (y) = G d (T q d , y q ), d ≥ 1,
in both notations of loc. cit. and ours ( 2 ). The polynomial G d is monic of degree d in the variable X, and (-1) d G d is monic in the variable Y of degree (q d -1)/(q -1). We now define, for N = N 0 + N 1 q + • • • + N r q r a non-negative integer expanded in base q, the polynomial

H N (t) = r i=0 G i (t q i , θ) Ni = r i=0 i-1 j=0 (t q i -θ q j ) Ni .
We also define the quantities associated to N and q:

µ q (N ) = r i=0 N i iq i , µ * q (N ) = N q -1 - ℓ q (N ) q -1 , ℓ ′ q (N ) = r i=0 N i i.
Lemma 23 Let N be a non-negative integer. The following properties hold.

1. The polynomial H N (t), as a polynomial of the indeterminate t, is monic of degree µ q (N ).

2. As a polynomial of the indeterminate θ, H N (t) has degree µ * q (N ) and the leading coefficient is (-1) ℓ ′ q (N ) .

We have H

N (θ) = Π(N ) and v ∞ (H N (θ)) = -µ q (N ), where v ∞ is the ∞-adic valuation of C ∞ .
4. We also have, for all ζ ∈ F alg q , v ∞ (H N (ζ)) = -µ * q (N ). Proof. Easy and left to the reader.

We observe that:

ϕ d ω(t) = 1 G d (t q d , θ) ω(t) q d = ω q d N (t), d ≥ 0 so that, with N as above, ω N (t) = ω(t) N r i=0 G i (t q i , θ) Ni = ω(t) N H N (t) . (13)
The following Proposition was inspired by a discussion with D. Goss.

2 As an aside remark, we also notice that we recover in this way the coefficients of the formal series in K[[τ ]] associated to Carlitz's exponential and logarithm

e = i≥0 d -1 i τ i , l = i≥0 l -1 i τ i , because d i = G i (θ q i , θ) and l i = G i (θ, θ q ). Moreover, if p is a prime of A of degree d, we observe that p = d i=1 (θ -ζ i ) = d-1 j=0 (θ -ϑp(σ θ q j )) = G d (θ, ϑp(σ θ )).

The polynomials W s

By Proposition 24, the function

W s (t) = π -1 L(χ s t , 1)ω s (t) = L(χ s t , 1)ω(t) s πH s (t)
is a polynomial of F q [t, θ]. Furthermore, we have:

Proposition 25 Assuming that s ≥ 2 is an integer congruent to 1 modulo q -1 and is not a power of q, the following properties hold.

1. The degree in t of W s does not exceed s(s -1)/(q -1) -s -µ q (s),

2. the degree in θ of W s is equal to (ℓ q (s) -q)/(q -1).

By the remarks in the introduction, we know how to handle the case of s = q i ; we then have

W q i (t) = 1 θ -t q i .
Proof of Proposition 25. The bound for the degree in t is a simple consequence of Proposition 24 and Lemma 23. To show the property of the degree in θ, we first notice that, by Lemma 23, for all

ζ ∈ F alg q , v ∞ (W s (ζ)) = - ℓ q (s) -q q -1 . (14)
The computation of W s (ζ) is even explicit if ζ ∈ F q . Indeed, with the appropriate choice of a (q -1)-th root of (ζ -θ), the fact that χ ζ = χ s ζ , [5, Lemma 12] and [13, Theorem 1],

W s (ζ) = L(χ s ζ , 1)ω(ζ) s πH s (ζ) = L(χ ζ , 1)ω(ζ) s πH s (ζ) = L(χ ζ , 1)ω(ζ) s π(ζ -θ) s-ℓq (s) q-1 = (ζ -θ) -1 q-1 (θ -ζ) -1 (ζ -θ) s q-1 (ζ -θ) ℓq (s)-s q-1 and W s (ζ) = -(ζ -θ) ℓq (s)-q q-1 . (15)
Let us write:

W s (t) = g i=0 a i t i , a i ∈ A.
By [START_REF] Perkins | Explicit Formulae for L-values in Finite Characteristic[END_REF], we have

a 0 = W s (0) = -(-θ) ℓq (s)-q q-1 (16)
and for all ζ ∈ F alg q we have, by [START_REF] Perkins | On Pellarin's L-series[END_REF],

|W s (ζ)| = |a 0 |.
This means that for i = 1, . . . , g, |a i | < |a 0 |, and the identity on the degree in θ follows as well.

Corollary 26 If ℓ q (s) = q, then W s = -1.

Proof. It follows from ( 16) and the fact that |a i | < |a 0 | for i = 1, . . . , g.

By Corollary 21, the function V 1,s (t 1 , . . . , t s ) = π -1 L(χ t1

-θ) i1 • • • (t s -θ) is , c i1,...,is ∈ C ∞ , (17)
where on both sides, we have entire analytic functions (see Corollary 8). This can also be seen, alternatively, by considering the function F s-1 of Lemma 5, which vanishes, and observing that L(χ t1 • • • χ ts , 1)| ti=θ = F s-1 (t 1 , . . . , t i-1 , t i+1 , . . . , t s ).

Let us focus on the coefficient c 1,...,1 in the expansion [START_REF] Thakur | Gauss sums for F q [T ][END_REF]. We then have

c 1,...,1 = d dt 1 • • • d dt s L(χ t1 • • • χ ts , 1) t1=•••=ts=θ so that V 1,s (θ, . . . , θ) = (-1) s π s-1 d≥0 a∈A + (d)
a ′s a = (-1) s π s-1 c 1,...,1 ∈ F q [θ]

(by Corollary 8, the series on the right-hand side is convergent). Now, by Proposition 24, Π(s) divides the polynomial V 1,s (θ, . . . , θ) in A. We then set, as in the introduction: Let τ be the unique F q [t]-linear automorphism of T t which extends the automorphism of C ∞ defined, for c ∈ C ∞ , by c → c q . If B is a polynomial of A[t] and if p is a prime of degree d > 0, then

B s = V 1
τ d B ≡ B (mod p).
The reason for this is that p divides the polynomial θ q d -θ. In particular, (τ d B)(θ) ≡ B(θ) (mod p).

Recalling the C ∞ -linear operator ϕ of subsection 3.1.1, we have

τ ϕ = ϕτ = ρ,
where ρ is the operator defined by ρ(x(t)) = x(t) q for all x ∈ C ∞ ((t)). In particular, if s = r i=0 s i q i is expanded in base q and if d ≥ r ≥ i, from

τ d ϕ i = τ d-i τ i ϕ i = τ d-i ρ i we deduce (τ d ω s )(t) = r i=0 ((τ d-i ω)(t)) siq i , so that (τ d ω s )(t) = ω(t) s r i=0 G d-i (t, θ) si q i . (19)
We can finish the proof of Theorem 2. By (18), B s ≡ (τ d W s )(θ) (mod p).

We shall now compute (τ d W s )(θ). If d > r, then for i = 0, . . . , r we can write G d-i (t, θ) siq i = (t -θ) siq i d-i-1 j=1 (t -θ q j ) siq i , and

r i=0 G d-i (t, θ) siq i = (t -θ) s F (t),
where F (t) is a polynomial such that

F (θ) = r i=0 l siq i d-i-1 .
Since (τ d W s )(t) = π -q d L(χ s t , q d )(t -θ) s ω(t) s F (t)

and lim t→θ (t -θ)ω(t) = -π, we get lim t→θ (τ d W s )(t) = (-1) s π -q d ζ(q d -s) π s r i=0 l siq i d-i-1

= (-1) s BC q d -s Π(q d -s) r i=0 l siq i d-i-1 .

Our Theorem 2 follows at once.

3 . 2

 32 ,s (θ, . . . , θ) Π(s) = G s (θ) ∈ A.21 Proof of Theorem 2

  • • • χ ts , 1)ω(t 1 ) • • • ω(t s ) is, for s ≡ 1 (mod q -1)and s ≥ 2, a polynomial of A[t 1 , . . . , t s ]. Since where o(1) represents a function locally analytic at t = θ, the function L(χ t1 • • • χ ts , 1) vanishes on the set

	ω(t) =	π θ -t	+ o(1),
		s
	D =	D i ,
		i=1

where

D i = {(t 1 , . . . , t i-1 , θ, t i+1 , . . . , t s ) ∈ C ∞ }. In other words, in C ∞ [[t 1 -θ, . . . , t s -θ]], we have L(χ t1 • • • χ ts , 1) = i1,...,is≥1 c i1,...,is (t 1

Note that B 1 is not well defined

Acknlowledgements. The authors would like to warmly thank Vincent Bosser, David Goss, Matthew Papanikolas, Rudolph Perkins, Denis Simon, Lenny Taelman, Dinesh Thakur for fruitful discussions about the themes developed in this work. In addition, we are indebted with: David Goss who suggested the appropriate direction of investigation in order to obtain Proposition 24, Denis Simon who discovered Lemma 4, and Lenny Taelman who performed numerical computations on Bernoulli-Carlitz fractions providing additional evidence to our Conjecture 3. Moreover, we would like to thank the Istituto de Giorgi of Pisa, Italy, for hospitality during part of the editing of this work. Finally, we would like to express our deep gratitude to the referee for useful remarks which helped to improve the paper.

Proposition 24 Let s ≥ 2 be an integer. Let M 1 , . . . , M s be positive integers such that M 1 + • • • + M s ≡ 1 (mod q -1). Then:

For all i, the degree in t i of W satisfies

Proof. We shall write

We know from Lemma 23 that deg ti (H) = µ q (M i ). Let us consider the function

so that by ( 13), V = W H.

Corollary 21 implies that:

Now, H is a polynomial in θ with leading coefficient in F × q (see Lemma 23). Dividing V by H as polynomials in θ we find

where Q, R are polynomials in F q [θ, t 1 , . . . , t s ], and This implies R = 0 and thus W = Q ∈ F q [θ, t 1 , . . . , t s ].


Kirill Borissov
email: kirill@eu.spb.ru

Education, Wage Inequality Stephane Lambrecht
email: stephane.lambrecht@univ-lille1.fr

Growth ✩ Kirill Borissov

Hgh Document de travail

Keywords: Growth, Inequality, Human capital JEL classification: O41, O15, D31, D91

We model a successive-generation economy in which parents, motivated by family altruism, decide to finance or not their offspring's capital accumulation on the basis of their altruistic motive, their own income and the equilibrium ratio between skilled-labor and unskilled-labor wages. The question we ask is how the growth process in this economy shapes the wage inequality and the split of the population in two classes. We study the transitional dynamics of human capital accumulation and of wage inequality. First, we prove the existence of equilibrium paths. Then we show that there exists a continuum of steady-state equilibria and prove the convergence of each equilibrium path to one of the steady-state equilibrium. Also we look at the relationship between inequality and output on the set of steady states and find that this relationship is ambiguous. Finally, we deal with an endogenous-growth version of our model, which displays the ambiguous relationship between inequality and the rate of growth.

Introduction

The understanding of the role of human capital in economic activity has been decisively spurred by the work of G. [START_REF] Becker | Human capital. A theoretical and empirical analysis, with special reference to education[END_REF]. Human capital is a major feature of economic relations inside the family and, as such, is one of the key concepts for understanding of the individual decision-making over the life-cycle and the functioning of labor markets. On their side macroeconomists, and especially economic growth theorists, investigated the role of human capital in determining the growth rate of economies in the short and the long run 1 .

Modern growth theory has long been relying on the fiction of the representative agent. However, societies are patently not homogeneous, whether in incomes, wealth, or many other dimensions. In a sense, the question of inequality and its link with the growth process is an old one. The classical argument is that inequality is good for growth because the wealthy are more patient and accumulate more assets than the poor. Over the last 20 or 30 years, the nexus between inequality and growth has attracted a great deal of interest. We can distinguish two types of questions about this nexus. The first one is: "How does inequality affect growth?", namely do unequal economies perform better than those more equal? The policy implications of this first type of question are relevant for policies aiming at redistributing income and wealth among households.

The role played by capital market imperfection in discouraging human capital accumulation has been stressed by several important contributions (see, e.g., [START_REF] Loury | Intergenerational transfers and the distribution of earnings[END_REF], Galor and Zeira (1993)) 2 .

The second type of question about the inequality-growth nexus is then the following: "How does the growth process affect in turn inequality?", namely 1 To mention but one major contribution, see, e.g., [START_REF] Lucas | On the Mechanisms of Economic Development[END_REF]. 2 Other arguments have been put forward to emphasize the negative relationship between inequality and growth: political economy arguments [START_REF] Alesina | Distributive politics and economic growth[END_REF], [START_REF] Bertola | Factor shares and savings in endogenous growth[END_REF], [START_REF] Persson | Is inequality harmful for growth? Theory and evidence[END_REF]), or social conflict arguments [START_REF] Alesina | Income distribution, political instability, and investment[END_REF] and [START_REF] Borissov | Growth and Distribution in an AKmodel with Endogenous Impatience[END_REF]).

2 what is the feedback of growth into the evolution of inequality across time.

This second question is relevant also for the first one. Indeed, as [START_REF] Aghion | Inequality and economic growth: The perspective of new growth theories[END_REF] argue, if redistribution creates a virtuous circle by alleviating credit constraints to human capital accumulation, these policy efforts might be vain if growth in turn worsen inequalities. A virtuous circle would be more or less offset by a vicious circle. This paper is mainly about the second type of question.

It deals with it by introducing heterogeneity coming from the functioning of the labor markets, namely the occupational heterogeneity. According to this heterogneneity, workers in the economy need to occupy positions of different skill levels and get different endogenously determined wages. As a consequence, educational decisions determine not only the individual stock of human capital but also influences the choice of occupation and the wages structure. [START_REF] Ray | On the Dynamics of Inequality[END_REF] examines equilibrium paths of an economy in which skilled and unskilled labor are necessary to produce. Each generation decides whether to finance the offspring's acquisition of human capital out of a dynastic [START_REF] Barro | Are government bonds net wealth?[END_REF]) altruistic motive to finance educational expenses. Since both skilled and unskilled categories of labor are necessary to production, equilibrium wages adjust to insure that each category of labor has positive supply, i.e. that one share of the population occupies low-skill jobs and the other high-skill jobs.

As a result, inequality inside each generation must emerge. [START_REF] Ray | On the Dynamics of Inequality[END_REF] shows that this intragenerational inequality is persistent in the long run. Moreover a continuum of steady states is possible. Our article is close to [START_REF] Ray | On the Dynamics of Inequality[END_REF] and Mookherjee and Ray's (forthcom-ing) articles in the sense that it looks at the persistence of inequality in an economy populated with altruists facing occupational heterogeneity. Its main contribution is that it explicitly deploys and analyses the dynamics of human capital accumulation, and hence allows for different levels of human capital for the skilled workers.

We assume that individuals care about their offspring's net disposable income and that there is an accumulative education function. [START_REF] Becker | An Equilibrium Theory of the Distribution of Income and Intergenerational Mobility[END_REF] used this set of hypotheses to analyse the equilibrium distribution of income and intergenerational mobility. They labeled the approach based on the offspring's wealth by the term "quality of the children". They claimed that the implications in terms of income distribution of this approach are similar to those of [START_REF] Barro | Are government bonds net wealth?[END_REF] "dynastic altruism" approach, in which altruists care about their offsring's utility. [START_REF] Lambrecht | Public pensions and growth[END_REF] and [START_REF] Lambrecht | Capital accumulation and fiscal policy in an OLG model with family altruism[END_REF] studied the properties of fiscal policies under this approach, which they label "family altruism". They find less clear cut conclusions: pay-as-you-go policies are neutral but public debt is not. The family altruism approach enables to study the transitional dynamics of physical and/or human capital 4 . As an altruistic bequest motive, it is also preferable to [START_REF] Andreoni | Giving with impure altruism: applications to charity and Ricardian equivalence[END_REF] joy-of-giving or warm-glow motive because the latter is insensitive to the economic situation of the beneficiaries of transfers.

To summarize, this paper is based on the threefold assumption of (i) family altruism (ii) accumulative human capital and (iii) the existence of two distinct occupational choices (high-skil and low-skill jobs). Moreover, in the high-skill occupations, there is room for heterogeneity in human capital, and hence in income. In that sense it combines (i) the neoclassical approach which sees human capital as efficiency units of labor whose individual endowments vary across the skilled workers and (ii) the approach which emphasizes the role of indivisibilities in occupational choice 5 .

We study the equilibrium paths along which human capital is accumulated differently across each generation's family. The main results are the following. First, we prove that there exists a unique intertemporal equilibrium path starting from any initial distribution of human capital. Secondly, we establish necessary and sufficient conditions for the existence of stationary equilibrium paths compatible with inequality in income among families and show that there exist a multiplicity of steady states. At these steady states, one share of the population permanently supply unskilled labor while the remaining share maintain a unique and constant human capital across generations and supplies skilled labor. Then we show that any equilibrium path converges to a steady state equilibrium with inequality. Finally, we propose an endogenous version of the model by assuming that the productivity of unskilled labor benefits from the accumulation of human capital. In this version of the model, it is shown that the relationship between inequality and growth is ambiguous.

The paper starts with the presentation of the model in Section 2. Then the competitive equilibrium is analyzed in Section 3. Section 4 studies steady state equilibria and Section 5 establishes the convergence to equilibrium paths to steady state equilibrium. In Section 6 we shortly describe an endogenousgrowth version of our model. Section 7 concludes.

The model

The firms

At each time t, the output of the representative firm, Y t , is determined by the Cobb-Douglas production function:

Y t = (H t ) α (L t ) 1-α ,
where H t is the supply of human capital and L t is the supply of unskilled labor. The wage rates of unskilled labor and human capital, w L t and w H t , are determined by their marginal products:

w H t = α L t H t 1-α , w L t = (1 -α) H t L t α .
Output is is either consumed or spent on education.

The households

Our model is one of successive generations. Each agent lives for one time period and has one offspring. The set of dynasties is the interval [0, 1]. Each dynasty is denoted by the variable i. Each agent is endowed with one unit of unskilled labor force that requires no higher education. Some agents (not necessarily all) are also endowed with some amount of human capital. We will call agents with positive endowment of human capital educated agents and those with zero endowment of human capital uneducated agents.

The human capital of the agent of dynasty i living at time t, h t (i), depends on the human capital of his parent, h t-1 (i) and the amount of money the parent spent for his higher education, e t-1 (i). We assume that this dependence is as follows:

h t (i) = e t-1 (i) κ (h t-1 (i) + 1) 1-κ , 0 < κ < 1, ∀t ≥ 0. (1)
It follows that if the parent of an individual spent nothing on his education, the human capital of this individual is nil.

Consider the individual that belongs to dynasty i and lives in period t. and w H t+1 are the expected time t + 1 wage rates and h t+1 (i) is the offspring's human capital. According to (1) at time t + 1, spending e t (i) on the offspring's education determines the latter's human capital, and thus his total income. The individuals' preferences are defined over consumption c t (i) and the offspring's expected total income ω t+1 (i) = max{w L t+1 , w H t+1 h t (i)} They are represented by the following log-linear utility function: ln c t (i) + ln ω t+1 (i). Hence the individual maximizes his utility function under his budget constraints considering current wages and expectations on next period wages as given. We state this problem as follows max

ct(i)≥0,et(i)≥0 ln c t (i) + ln ω t+1 (i)
under the following constraints:

e t (i) + c t (i) = ω t (i), ω t+1 (i) = max{w L t+1 , w H t+1 h t+1 (i)}, h t+1 (i) = e t (i) κ (h t (i) + 1) 1-κ .
This problem can be rewritten as follows:

max 0≤et(i)≤ωt(i) ln(ω t (i) -e t (i)) + ln(max{w L t+1 , w H t+1 e t (i) κ (h t (i) + 1) 1-κ }). (2)
Problem ( 2) can be solved in two steps. First we solve the two sub-problems defined by the two alternatives of the max function, max

0≤et(i)≤ωt(i) ln(ω t (i) -e t (i)) + ln(w L t+1 ), (3)
and max

0≤et(i)≤ωt(i) ln(ω t (i) -e t (i)) + ln(w H t+1 e t (i) κ (h t (i) + 1) 1-κ ), (4)
and then we select the solution leading to the highest utility.

The solution to problem ( 3) is e t (i) = 0. If at time t + 1 dynasty i is going to supply unskilled labor on the labor market, then at time t its expenditure on education is nil and all income ω t (i) is spent on consumption. Hence the human capital of this dynasty at time t + 1, h t+1 (i), is also nil. The value of problem

(3) is V L ω t (i), h t (i) := ln ω t (i) + ln w L t+1 .
The solution to problem ( 4) is

e t (i) = êH ω t (i), h t (i) := κ 1 + κ ω t (i).
If at time t + 1 dynasty i is going to supply human capital on the labor market, then at time t its expenditure on education is equal to the fraction κ 1+κ of its income ω t (i). The rest of the income, 1 1+κ ω t (i), is spent on consumption. Thus, the endowment of human capital of agent that belongs to dynast i and lives at time t + 1 is equal to

ĥH ω t (i), h t (i) := γ ω t (i) κ 1 + h t (i) 1-κ , (5)
where

γ = κ 1 + κ κ .
The value of problem ( 4) is

V H ω t (i), h t (i) := ln ω t (i) -ln(1 + κ) + ln w H t+1 + ln ĥH ω t (i), h t (i) . It is clear that if V H ω t (i), h t (i) < V L ω t (i), h t (i)
, then the unique solution to problem (2) coincides with the solution to problem (3). If

V H ω t (i), h t (i) > V L ω t (i), h t (i)
, then the unique solution to problem

(2) coincides with the solution to problem (4), and if

V H ω t (i), h t (i) = V L ω t (i), h t (i)
, then the solutions of both ( 3) and ( 2) are solutions to [START_REF] Alesina | Income distribution, political instability, and investment[END_REF].

It is easily checked that

V H ω t (i), h t (i) V L ω t (i), h t (i) ⇔ ĥH ω t (i), h t (i) 1 + κ w L t+1 w H t+1 .
Thus we can formulate the following proposition.

Proposition 1. 1) If ĥH ω t (i), h t (i) 1 + κ > w L t+1 w H t+1 , ( 6
)
then e t (i) = êH ω t (i), h t (i) is the unique solution to (2); 2) if ĥH ω t (i), h t (i) 1 + κ < w L t+1 w H t+1 , (7)
then e t (i) = 0 is the unique solution to [START_REF] Alesina | Income distribution, political instability, and investment[END_REF];

3) if ĥH ω t (i), h t (i) 1 + κ = w L t+1 w H t+1 , (8)
then there are two solutions to ( 2): e t (i) = 0 and e t (i) = êH ω t (i), h t (i) .

At each time t it is convenient to order the set of dynasties in a way such that the function h t (•) defined on the interval [0, 1] and describing the distribution of human capital across dynasties is non-decreasing. At the same time it follows from (5) that if for some i and j, h t (i) ≥ h t (j) and ω t (i) ≥ ω t (j), then ĥH ω t (i), h t (i) ≥ ĥH ω t (j), h t (j) . This give us the opportunity to order the set of dynasties at time 0 and restrict our consideration to paths of the economy such that all functions h t (•) are non-decreasing at the initial order.

Competitive equilibrium

To study the general equilibrium of this economy we proceed in two steps 6 .

We first study the time t temporary equilibrium in subsection 3.1 given past variables and expectations of the future. In subsection 3.2, we then describe the intertemporal equilibrium with perfect foresight as a sequence of temporary equilibria with some adequate initial conditions and rule for expectations.

Time t temporary equilibrium

To define the time t temporary equilibrium, we consider all past variables and expectations of the future as given. The latter are expectations of the next period wages w L t+1 and w H t+1 and the former are the time t -

1 human capital levels, h t-1 (i) ∀i ∈ [0, 1], total incomes ω t-1 (i) = max{w L t-1 , w H t-1 h t-1 (i)} ∀i ∈ [0, 1],
and educational spendings e t-1 (i) ∀i ∈ [0, 1]. Since these given past variables determine time t human capital levels h t (i) ∀i ∈ [0, 1], we can say that these levels are completely pre-determined by time t -1 decisions. To be more precise, all we need to know to construct the time t temporary equilibrium is the function h t (•).

Let us assume that the function h t (•) is non-decreasing and upper semicontinuous and that

1 0 h t (i)di > 0.
A time t temporary equilibrium is defined by a quadruple of functions • all agents, households and firms, are at their optima;

{ω t (•), c t (•), e t (•), h t+1 (•)}, defined on [0, 1],
• the set of dynasties supplying unskilled labor at time

t is [0, i H t ) = {i | 0 ≤ i < i H t } and the set of dynasties supplying human capital is [i H t , 1] = {i | i H t ≤ i ≤ 1};
• all markets clear.

It should be noticed that the pivotal dynasty i H t shows the fraction of unskilled labor suppliers in the population. The fraction of human capital suppliers is respectively 1i H t . To make our presentation simple we also impose the following requirement on temporary equilibrium, which will not lead to any loss of generality:

• h t+1 (i) is a non-decreasing upper semi-continuous function defined on [0, 1].
We determine this equilibrium at time t by writing the variables of the above-mentioned tuples as functions of past variables and expectations. To find a temporary equilibrium at time t it is sufficient to determine the pivotal dynasty i H t . Knowing it, one can easily determine the equilibrium values of all other variables.

In equilibrium, the supply of unskilled labor is equal to

L t = i H t 0 di = i H t , (9)
the supply of human capital is equal to

H t = 1 i H t h t (i)di (10)
and human capital and unskilled labor are paid at their marginal products:

w H t = α   i H t 1 i H t h t (i)di   1-α , w L t = (1 -α)   1 i H t h t (i)di i H t   α . (11)
Therefore,

w L t w H t = (1 -α) α   1 i H t h t (i)di i H t   .
Also, in equilibrium, we need to have

w H t h t (i) ≤ w L t ⇔ h t (i) ≤ w L t w H t , 0 ≤ i < i H t ,
and

w H t h t (i) ≥ w L t ⇔ h t (i) ≥ w L t w H t , i H t ≤ i ≤ 1.
Therefore, given the time t human capital levels h t (i)∀ i ∈ [0, 1], the time t equilibrium pivotal dynasty, i H t , is determined by the following conditions:

h t (i) ≤ (1 -α) α   1 i H t h t (i)di i H t   , 0 ≤ i < i H t , (12)
h t (i) ≥ (1 -α) α   1 i H t h t (i)di i H t   , i H t ≤ i ≤ 1. ( 13
)
It is clear that (1-α)

α 1 i H ht(i)di i H is a continuous decreasing function of i H .
At the same time h t (i) is a non-decreasing function of i. Therefore, to find the pivotal dynasty it is sufficient to "solve" the following equation in i H :

h t (i H ) = (1 -α) α 1 i H h t (i)di i H . ( 14
)
If the solution to equation ( 14) exists, the time t equilibrium pivotal dynasty coincides with this solution. In this case this dynasty is the one with human capital just equal to the ratio between unskilled labor wage rate and human capital wage rate and hence for this dynasty supplying unskilled labor will result in the same income as supplying human capital:

w L t = w H t h t (i H t ).
A solution to [START_REF] Lambrecht | Public pensions and growth[END_REF] may not exist because h t (i) is not necessarily continuous.

But even in the case of non-existence there is a unique i H t satisfying ( 12)-( 13). Knowing i H t , we get H t , L t , w H t and w L t from ( 9)- [START_REF] Galor | Income distribution and macroeconomics[END_REF]. Also we are able to determine the total income of all households:

ω t (i) =    w L t , 0 ≤ i < i H t , w H t h t (i), i H t ≤ i ≤ 1.
With the pairs {h t (i), ω t (i)} ∀i, we can now determine the time t equilibrium educational expenditures, e t (i), i.e. the optimal educational expenditures at equilibrium prices given expectations on next period wage rates w L t+1 and w H t+1 . Once this variable is determined, it will give us the next period distribution of human capital, the h t+1 (i)'s. Here we should notice that the functions e t (•) and h t+1 (•) are not necessarily uniquely determined, because for i satisfying (8), e t (i) is equal to either 0 or êH ω t (i), h t (i) . However, this non-uniqueness plays no role in our model, because, as will be shown in the next subsection, in intertemporal equilibrium non-uniqueness does not appear.

For short, in what follows we identify any temporary equilibrium at time t with the couple {i H t , h t+1 (•)}.

The intertemporal equilibrium with perfect foresight

Suppose we are given an initial state of the economy represented by a nondecreasing upper semi-continuous function h 0 (•) showing the distribution of human capital across dynasties at the initial time. We assume that 1 0 h 0 (i)di > 0 and define an intertemporal equilibrium path {i H t , h t+1 (•)} ∞ t=0 starting from h 0 (•) as a sequence of temporary equilibria, such that at each time t each dynasty has perfect foresight, that is, correctly anticipate time t + 1 wage rates.

Theorem 1. For any initial state h 0 (•) there is a unique intertemporal equilibrium path starting from this initial state.

Proof. Let h t (•) be given. Let further i H t , w L t , w H t and ω t (•) be found as described in the previous section. To find the function h t+1 (•), we start with using the time t + 1 human capital function associated with positive investment in education given by equation ( 5) in section 2.2, namely ĥH ω t (i), h t (i) . Since the analysis will focus on the pivotal dynasty and all the arguments of this function depend on i, we re-write it as a function ht+1 of i.

Namely, let the function ht+1 : [0, 1] → R + be defined by

ht+1 (i) = ĥH ω t (i), h t (i) (= γ [ω t (i)] κ 1 + h t (i) 1-κ ).
Clearly, ht+1 (•) is a non-decreasing upper semi-continuous function such that

ht+1 (i) = γ w L t κ , 0 ≤ i < i H t , (15) ht+1
(i) = γ w H t κ ψ(h t (i)), i H t ≤ i ≤ 1, (16)
where the function ψ : R + → R + is given by

ψ(h) = h κ (h + 1) 1-κ .
Let further the function H t+1 : [0, 1] → R + be defined by

H t+1 (i H ) = 1 i H ht+1 (i)di. ( 17
)
This function is continuous and decreasing. It shows the dependence of the aggregate supply of human capital on the pivotal dynasty. In equilibrium at time t + 1 the ratio of the wage rates of common labor and human capital is endogenously determined by the marginal productivities of these inputs, which in turn is determined by the relative masses of these inputs. So we have:

1 -α α H t+1 (i H t+1 ) i H t+1 = w L t+1 w H t+1
and, at the same time, from Proposition 1 and the definition of the function

ht+1 (•) ht+1 (i) 1 + κ ≤ w L t+1 w H t+1 , 0 ≤ i < i H t+1 , (18) ht+1
(i) 1 + κ ≥ w L t+1 w H t+1 , i H t+1 ≤ i ≤ 1. ( 19
)
Whereas

Ht+1(i H ) i H is a decreasing function of i H , ht+1 (i) is a non-decreasing function. Moreover, 1 -α α H t+1 (i H ) i H > ht+1 (i H )
1 + κ for sufficiently small i H > 0 and

1 -α α H t+1 (i H ) i H < ht+1 (i H ) 1 + κ
for i H sufficiently close to 1.

To find the time t + 1 equilibrium pivotal dynasty i H t+1 , it is sufficient to "solve" the following equation in i H :

1 -α α H t+1 (i H ) i H = ht+1 (i H ) 1 + κ (20)
If this equation has a solution, it is unique. Since ht+1 (•) may be discontinuous, the non-existence of a solution to equation ( 20) is possible. But even if [START_REF] Ray | On the Dynamics of Inequality[END_REF] has no solution, there exists a unique i H t+1 satisfying the following conditions:

1 -α α H t+1 (i) i > ht+1 (i) 1 + κ , 0 ≤ i < i H t+1 , 1 -α α H t+1 (i) i ≤ ht+1 (i) 1 + κ , i H t+1 ≤ i ≤ 1.
This i H t+1 is the required time t + 1 equilibrium pivotal dynasty As for h t+1 (•), it is determined as follows:

h t+1 (i) = 0, 0 ≤ i < i H t+1 , h t+1 (i) = ht+1 (i), i H t+1 ≤ i ≤ 1.
It should be noticed that, unlike temporary equilibrium, non-uniqueness of equilibria does no appear in intertemporal equilibrium. This is because in the definition of temporary equilibrium at time t agents take the wage rates at time t + 1 as given, whereas in intertemporal equilibrium they are determined endogenously. Also it is noteworthy that in intertemporal equilibrium an agent spend a positive fraction of his income on education if and only if his offspring will be human capital supplier on the labor market.

Steady-state equilibria

We now turn to the examination of steady-state equilibria. They are characterized by the feature that the wage rates and the fractions of educated agents supplying human capital on the labor market and uneducated agents supplying unskilled labor are constant over time and that inside each dynasty children find themselves in the same position as their parents.

At a steady-state equilibrium, the amount of human capital h * supplied by agents from an educated dynasty depends only on the wage paid to one unit of human capital, w H * , because h * is the solution to the following equation:

h = γ(w H * ) κ ψ(h).
Hence, at a steady-state equilibrium all educated dynasties supply the same amount of it. Therefore we can define steady-state equilibria as follows.

A couple (i

H * , h * ), i H * ∈ (0, 1), h * > 0, is called a steady-state equilibrium if the sequence {i t , h t+1 (•)} ∞ t=0 given by i t = i H * , t = 0, 1, 2, . . . , h t+1 (i) = 0, 0 ≤ i < i H * , t = 0, 1, 2, . . . , h t+1 (i) = h * , i H * ≤ i ≤ 1, t = 0, 1, 2, . . . ,
is an equilibrium path starting from h 0 (•) defined as follows:

h 0 (i) = 0, 0 ≤ i < i H * , h 0 (i) = h * , i H * ≤ i ≤ 1.
It follows from ( 10) and ( 11) that at any steady state equilibrium (i H * , h * ) the total supply of human capital, H * and the wage rates of unskilled labor and human capital, w L * and w H * , are given as follows:

H * = (1 -i H * )h * , w H * = α i H * H * 1-α = α i H * (1 -i H * )h * 1-α , ( 21
)
w L * = (1 -α) H * i H * α = (1 -α) (1 -i H * )h * i H * α . ( 22
)
We are more precisely interested in the ratio of unskilled labor wage to human capital wage:

w L * w H * = 1 -α α (1 -i H * )h * i H * , ( 23
)
which is a decreasing function of i H * and an increasing function of h * . Also we are interested in the skill premium P * , which is a reasonable measure of income inequality at a steady-state (i H * , h * ) equilibrium of our model. It is defined as the proportion of the wage earned by an educated individual to the wage of an unskilled individual:

P * := w H * h * w L * .
It follows from (23) that on the set of steady-state equilibria the skill premium can be considered as an increasing function of i H * :

P * = α 1 -α i H * 1 -i H * .
Let (i H * , h * ) be a steady-state equilibrium and {i t , h t+1 (•)} ∞ t=0 be the corresponding equilibrium path. It follows from ( 15)-( 16) that for this path,

ht+1 (i) =    γ(w L * ) κ , 0 ≤ i < i H * , γ(w H * ) κ ψ(h * ), i H * ≤ i ≤ 1.
Therefore ( 18) and ( 19) can be rewritten as respectively

γ(w L * ) κ 1 + κ ≤ w L * w H * (24) and γ(w H * ) κ ψ(h * ) 1 + κ ≥ w L * w H * . ( 25
)
The first of these inequalities means that the uneducated agents have no incentives to spend money on the education of their offsprings and the second that the educated agents have such incentives. It is also clear that

h * = γ(w H * ) κ ψ(h * ). ( 26
)
One can easily prove the following proposition.

Proposition 2. A couple (i H * , h * ), 0 < i H * < 1, h * > 0, is a steady-state equilibrium if and only if for w H * and w L * given by ( 21) and ( 22) respectively, ( 24)-( 26) hold true.

Let us now describe the relationship between the share of uneducated agents in the population, i H * , and the human capital accumulated by an educated agent in a steady-state equilibrium, h * . It is reasonable to conjecture that this relationship is increasing because a higher fraction of uneducated agents can lead to a larger skill premium and wages of educated individuals and hence to higher individual educational expenditures. The following lemma says that this conjecture is true.

Lemma 2. There is a smooth increasing function χ : (0, 1) → R + and numbers L 1 and L 2 , 0 < L 2 < L 1 ≤ 1, such that for any i * ∈ (0, 1) and for w H * and w L * given by ( 21) and ( 22) respectively, ( 26) is equivalent to

h * = χ(i H * ), (27)
(25) is equivalent to i H * ≥ L 2 , (24) is equivalent to i H * ≤ L 1 .
The proof of this lemma is relegated to Appendix.

The following theorem describing the the structure of steady-state equilibria follows directly from Proposition 2 and Lemma 2.

Theorem 3. There is a smooth increasing function χ : (0, 1) → R + and numbers L 1 and L 2 , 0 < L 2 < L 1 ≤ 1, such that a couple (i H * , h * ) is a steady-state equilibrium if and only if either

L 2 ≤ i H * ≤ L 1 (if L 1 < 1) or L 2 ≤ i H * < L 1 (if L 1 = 1) and h * = χ(i H * ).
Theorem 3 reads that, like in [START_REF] Ray | On the Dynamics of Inequality[END_REF], the set of steady-state equilibria is a continuum. More precisely, this set is essentially an interval which can be parameterized by the fraction of uneducated agents in the population.

Another interesting parametrization of the set of steady-state equilibria is that by the skill premium. This parametrization can help us to explain why all equilibrium values of the fraction of uneducated agents lies in the interval

[L 2 , L 1 ]. If i H * < L 2 , then the couple (i H * , χ(i H *
) is not a steady-state equilibrium because the wage rate of unskilled labor is so high and the skill premium is so small that even the educated parents have no incentives to spend a positive fraction of their incomes on the education of their children.

If L 2 ≤ i H * ≤ L 1 ,
then the skill premium is such that the educated individuals prefer to see their children educated while the uneducated agents find it too expensive to spend money on the education of their children. Finally, if i H * > L 1 , the couple (i H * , χ(i H * ) is not a steady-state equilibrium because the wage rate of unskilled labor is so small and the skill premium is so high that even the uneducated individuals are ready to spend money on the education of their children.

Let us now consider the question of what is the relationship between inequality, measured by the skill premium, and output on the set of steady-state equilibria. To sketch the broad outlines of this relationship, it is sufficient to look at the dependence of output on i H * , because the skill premium P * is an increasing function of i H * . The level of output, Y * , corresponding to a steadystate equilibrium (i H * , χ(i H * )) is

Y * = [(1 -i H * )χ(i H * )] α [i H * ] 1-α .
It would be difficult to derive an analytical form of the function [(1 -

i)χ(i)] α [i] 1-α .
The shape of the graph of this function depends on the parameters of the model, α and κ. However, it is clear that the dependence of Y * on i H * is quite ambiguous. Our computational experiments show that this dependence on the interval [L 2 , L 1 ] can be of an inverted U-shaped form or increasing.

On Fig. 1 we present L 1 , L 2 and the graph of

[(1 -i)χ(i)] α [i] 1-α on the segment [0, 1] at α = 0.4, κ = 0.5. On [L 2 , L 1 ] the function [(1 -i)χ(i)] α [i] 1-α
has an inverted U-shaped form. On Fig. 2 we present L 1 , L 2 and the graph of Γ(i) on the segment [0, 1] at

α = 0.3, κ = 0.96. On [L 2 , L 1 ] the function [(1 -i)χ(i)] α [i] 1-α increasing.

Convergence of equilibrium paths

The following theorem reads that any equilibrium path converge to a steadystate equilibrium and that the number of uneducated agents does not increase in time (except, perhaps, at time t = 1).

Theorem 4. For any equilibrium path {i

H t , h t+1 (•)} ∞ t=0 the sequence {i H t } ∞ t=1 is non-increasing (it may be that i H 1 > i H 0 )
and there is a steady-state equilibrium

(i H * , h * ) such that i H t -→ t→∞ i H * , h t (i) -→ t→∞ h * , i H t ≤ i ≤ 1.
Proof We have

ht+1 (i) = γ w H t κ ψ(h t (i)), i H t ≤ i ≤ 1.
Therefore,

H t+1 (i H t ) = γ w H t κ 1 i H t ψ(h t (i))di,
where H t+1 (•) is defined by [START_REF] Lucas | On the Mechanisms of Economic Development[END_REF].

It is clear that ψ(h) h decreases as h > 0 increases and hence

ψ[h t (i)] h t (i) ≤ ψ[h t (i H t )] h t (i H t ) , i H t ≤ i ≤ 1.
It follows that

ψ[h t (i)] ≤ ψ[h t (i H t )] h t (i H t ) h t (i), i H t ≤ i ≤ 1.
Therefore,

H t+1 (i H t ) ≤ γ w H t κ ψ[h t (i H t )] h t (i H t ) 1 i H t h t (i)di = ht+1 (i H t ) H t (i H t ) h t (i H t )
.

Thus, H t+1 (i H t ) ht+1 (i H t ) ≤ H t (i H t ) h t (i H t )
, which implies i H t+1 ≤ i H t . To complete the proof, it is sufficient to note that the sequence {i H t } ∞ t=1 is bounded from below and therefore converges to some i H * . It is no difficult to check that i H * > 0 and the required steady-state equilibrium is the couple (i H * , h * ), where h * = χ(i H * ).

Endogenous growth

In this section we propose an endogenous growth version of our model. To do this, we introduce an endogenously formed variable reflecting the state of technology at time t, A t . An increase in its value leads (i ) to a higher effectiveness of unskilled labor and (ii ) promote accumulation of human capital. In its turn, the accumulation of human capital contribute to the growth of the value of this variable through a macroeconomic externality.

More precisely, our assumptions are as follows. The output at time t, Y t , is given by

Y t = H α t (A t L t ) 1-α .
Therefore the wage earned by each agent supplying unskilled labor on the labor market is w L t = (1α)H α t (A t L t ) 1-α . The human capital of an agent of dynasty i at time t, h t (i), depends not only on the human capital of his parent, h t-1 (i) and the amount of money the parent spent for his higher education, e t-1 (i), but also on the state of technology at time t -1, A t-1 :

h t (i) = e t-1 (i) κ (h t-1 (i) + A t-1 ) 1-κ , 0 < κ < 1. ( 28
)
As for the formation of A t , we assume that

A t = Φ(H t-1 , A t-1 ),
where Φ : R 2 + → R is a continuous homogeneous of degree one concave function. Thus, the variable A t i) shows the efficiency of unskilled labor and ii) plays the role of an input in the educational production function. Its value can grow over time through the accumulation of human capital.

The behavior of individuals is the same as in the case of the exogenous growth model with the only difference that (1) is replaced by (28). The temporal and intertemporal equilibria are also defined in practically the same way as above.

The difference is that a temporal equilibrium at each time t is described not by a couple {i H t , h t+1 (•)}, but by a triple {i H t , A t+1 , h t+1 (•)} satisfying the requirements formulated in Subsection 3.1 and the equation

A t+1 = Φ( 1 0 h t (i)di, A t ).
Clearly, an initial state of an intertemporal equilibrium path is determined by a couple {A 0 , h 0 (•)}. The existence of an intertemporal equilibrium paths is also proved in the same way.

As for a steady-state equilibrium, it is defined as a triple {i H * , 1 + g * , h * }, i H * ∈ (0, 1), h * > 0, such that the sequence {i t , A t+1 , h t+1 (•)} ∞ t=0 given by

i t = i H * , t = 0, 1, 2, . . . , A t = (1 + g * ) t , t = 0, 1, 2, . . . , h t+1 (i) = 0, 0 ≤ i < i H * , t = 0, 1, 2, . . . , h t+1 (i) = A t+1 h * , i H * ≤ i ≤ 1, t = 0, 1, 2, . . . ,
is an equilibrium path starting from the initial state {A 0 , h 0 (•)} given by A 0 = 1,

h 0 (i) = 0, 0 ≤ i < i H * , h 0 (i) = h * , i H * ≤ i ≤ 1.
As in the case of exogenous growth, it is not difficult to show that any equilibrium path converges to a steady-state equilibrium. The structure of the set of steady-state equilibria is described in the following theorem.

Theorem 5. There is a smooth increasing function χ : (0, 1) → R + and numbers

L 1 and L 2 , 0 < L 2 < L 1 ≤ 1, such that a triple (i H * , 1 + g * , h * ) is a steady-state equilibrium if and only if either L 2 ≤ i H * ≤ L 1 (if L 1 < 1) or L 2 ≤ i H * < L 1 (if L 1 = 1) and h * = χ(i H * ), 1 + g * = Φ((1 -i H * )h * , 1)
What does the endogenous growth version of our model tell about the nexus of income inequality and the rate of economic growth? The debate on this issue is not settled. The classical approach suggests that inequality stimulates capital accumulation and thus promotes economic growth, whereas the modern approach argues in contrast that for sufficiently wealthy economies equality stimulates investment in human capital and hence may enhance economic growth.

In our model, the nature of the relationship between income inequality and the rate of growth on the set of steady-state equilibria can be illustrated by the graph of the dependence of (1i H * )χ(i H * ) on i H * because, on the one hand, the rate of growth is an increasing function of (1i H * )χ(i H * ) and, on the other hand, the skill gap P * increases with an increase in i H * . Our simulations show that this dependence of (1i H * )χ(i H * ) on i H * on the interval [L 2 , L 1 ] can increasing (α = 0.4,κ = 0.99) or decreasing (α = 0.2,κ = 0.3) or of an inverted U-shaped form (α = 0.8,κ = 0.9).

In our model, the nature of the relationship between income inequality and the rate of growth on the set of steady-state equilibria can be illustrated by the graph of the dependence of (1i H * )χ(i H * ) on i H * because, on the one hand, the rate of growth is an increasing function of (1i H * )χ(i H * ) and, on the other hand, the skill gap P * increases with an increase in i H * . Our computational experiments show that this dependence of (1-i H * )χ(i H * ) on i H * on the interval [L 2 , L 1 ] can be increasing, decreasing or of an inverted U-shaped form.

On Figs.

Conclusion

To answer the question of how the growth process affects inequality, namely the question of the feedback of growth into the evolution of inequality across time, we developed a model characterized both by (i) capital accumulation, out of a family altruism motive, and (ii) a twofold occupational choice between skilled and unskilled position.

We showed how the initial wage distribution evolves across time. The pivotal dynasty splitting the population in two classes evolves in the direction of including more and more families in the skilled-labor class. However in the long run, wage inequality and the division of population between skilled and unskilled remains.

The steady state equilibrium itself is actually a continuum and for each of these steady state a different wage inequality and splitting prevails. We show that the relationship between the location of the steady state pivotal dynasty, and hence the long run wage inequality, and the level of output (or the rate of growth of output) is ambiguous.

capital, h * , supplied by each educated agent. Second, we show that the set of equilibrium values of i H is an interval.

Let us first prove the following lemma.

Lemma 6. 1) There is a smooth increasing function χ : (0, 1) → R + such that for any i ∈ (0, 1),

h = γ α i (1 -i)h 1-α κ ψ(h) ⇔ h = χ(i). (A.1)
This function satisfies χ(i) → 0 as i → 0 and χ(i) → ∞ as i → 1. (A.2)

2) 1-i i χ(i) monotonically decreases from ∞ to (γα κ )

1

(1-α)κ as i increases from 0 to 1.

Proof. 1) Given that ψ(h) = h κ (h + 1) 1-κ , we can rewrite the first equation in (A.1) as

h = γα κ i 1 -i (1-α)κ h ακ (h + 1) 1-κ ,
or, after dividing both sides by h ακ (h + 1) 1-κ , as h 1-ακ (h + 1) 1-κ = γα κ i 1i (1-α)κ .

(A.

3)

The LHS of the last equation is continuous and increasing in h, tends to 0 as h → 0 and to +∞ as h tends to +∞ since 1ακ > 1κ. The RHS of this equation is continuous and increasing in i, tends to 0 as i → 0 and to +∞ as i → 1. It is then obvious that for any i there exists a solution to (A.3) in h.

To complete the proof, denote this solution by χ(i) and notice that the both properties in (A.2) hold true.

2) After some rearrangement of (A.3) we can get

1 -i i χ(i) = (γα κ ) 1 (1-α)κ 1 + 1 χ(i) (1-κ)/(1-α)κ .
Since χ(i) monotonically increases from 0 to ∞ , 1-i i χ(i) monotonically decreases from ∞ to (γα κ ) 1 (1-α)κ as i increases from 0 to 1.

Because of (21) and Lemma 6, equation (26) can be rewritten as (27), where χ(•) is the function introduced in Lemma 6. Thus, the amount of human capital supplied by each educated agent is an increasing function of the fraction of uneducated agents in the population. Also, by ( 23) and (26), we can rewrite (25) as

h * 1 + κ ≥ 1 -α α (1 -i H * )h * i H * ,
or, equivalently, as

i H * ≥ L 2 := (1 -α)(1 + κ) (1 -α)(1 + κ) + α .
Let us now rewrite (24) as

γ 1 + κ (1 -α) κ (1 -i H * )h * i H * ακ ≤ 1 -α α (1 -i H * )h * i H * ,
This inequality can be re-written as

γ 1 + κ (1 -α) κ ≤ 1 -α α 1 -i H * i H * h * 1-ακ = 1 -α α 1 -i H * i H * χ(i H * ) 1-ακ
or, after substituting (27) as

γ 1 + κ (1 -α) κ ≤ 1 -α α 1 -i H * i H * χ(i H * ) 1-ακ . (A.4)
By Lemma 6, 1-i i χ(i) monotonically decreases from ∞ to (γα κ ) 1

(1-α)κ as i increases from 0 to 1. Therefore, if

αγ 1 + κ (1 -α) κ-1 1 1-ακ ≥ (γα κ ) 1 (1-α)κ , then (A.4) is equivalent to i H * ≤ L 1 , (A.5)
where L 1 is the solution to the following equation in i:

γ 1 + κ (1 -α) κ = 1 -α α 1 -i i χ(i) 1-ακ . If αγ 1 + κ (1 -α) κ-1 1 1-ακ < (γα κ ) 1
(1-α)κ , then (A.4) holds for all i H * ∈ (0, 1).

  In a recent contribution, Mookherjee and Ray (forthcoming) extend and modify Ray's (2006) model. They provide a small-open economy model with physical capital, a continuum of occupations, training costs and a mix of utilitybased and wealth-based motivation for bequest. They derive conditions under which the steady state exhibits inequality. These conditions rely on the share of occupations with high training costs 3 being non degenerate.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

  [START_REF] Alesina | Distributive politics and economic growth[END_REF][START_REF] Andreoni | Giving with impure altruism: applications to charity and Ricardian equivalence[END_REF][START_REF] Barro | Are government bonds net wealth?[END_REF] we present graphs of (1i)χ(i) on the segment [0, 1] and indicate L 2 and L 1 for different values of parameters α and κ.

Figure 3 .

 3 Figure 3. α = 0.4, κ = 0.5.

Figure 4 .

 4 Figure 4. α = 0.7, κ = 0.8.

Figure 5 .

 5 Figure 5. α = 0.3, κ = 0.96.

  a pair of prices {w L t , w H t }, a triplet of aggregate variables {L t , H t , Y t } and a pivotal dynasty i H t satisfying the following requirements:

  . It is sufficient to prove that for any equilibrium path {i H t , h t+1 (•)} ∞

	t=0
	the sequence {i H t } ∞ t=1 is non-increasing (it may be that i H 1 > i H 0 ).
	Let {i H t , h t+1 (•)} ∞ t=0 be an equilibrium path. Show that for any t = 1, 2, . . .,
	i H t+1 ≤ i H t .

Higher than an endogenous threshold.

We confine our analysis to human capital only.

Indivisibilities may also come from the educational system like inChusseau and Hellier (2010).

See Hicks (1939) or, more recently[START_REF] Grandmont | Money and value: a reconsideration of classical and neoclassical monetary theories[END_REF] for the articulation of these two steps.

✩ The research developed in this paper is part of the project Mondialisation, restructurations et déclassement social et professionnel (MONDES), which is financed by the French National Agency of Research (ANR) through its programme Vulnérabilités : à l'articulation du sanitaire et du social. We thank the ANR for its financial support. The usual disclaimer applies.

Appendix A. Proof of Lemma 2

To describe the set of steady-state equilibria we verify two things. First we show that on the set of steady-state equilibria there is a monotonically decreasing relationship between the equilibrium pivotal dynasty i H * , which shows the fraction of uneducated agents in the population, and the amount of human To complete the the proof of Lemma 2, it is necessary to show that L 2 < L 1 .

To do this, let

We have

It follows that

It is clear that (24) fulfills as a strict inequality if and only if (A.5) fulfills as a strict inequality. Therefore the last chain on inequalities implies L 2 < L 1 .