M Emilia Alonso

Henri Lombardi
email: henri.lombardi@univ-fcomte.fr.

Local Bézout Theorem for Henselian rings

Keywords: Local Bézout Theorem, Henselian rings, Roots continuity, Stable computations, Constructive Algebra. MSC 2010. primary 13J15; secondary 13P10, 13P15, 14Q20, 03F65

HAL is

Introduction

In this paper we use ideas from computer algebra to prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and m-adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities.

As far as we know the proofs of this classical result in the literature are essentially: by Arnold using powerfully the topological degree and Weierstrass theorems [4, section I-4.3], and another one that can be deduced from Griffiths-Harris [9, Residue Theorem, p. 656] using the theory of residues and its relationship with multiplicity.

Here we state and prove an algebraic version of this theorem in the setting of arbitrary Henselian rings and m-adic topology. We are somehow inspired by Arnold in [4, section I-4.3], exploiting an abstract version of Weierstrass division (in a Henselian ring) and introducing also an abstract version of which he called the "multilocal ring". Roughly speaking we consider a finitely presented A-algebra, where (A, m, k) is a local Henselian ring such that the special point is a k-algebra with an isolated zero of multiplicity r and we prove that the "multilocal ring" determined by this point is a free A-module of rank r.

Our main tools are, first the border bases [START_REF] Mourrain | A new criterion for normal form algorithms[END_REF], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [START_REF] Smit | Finite complete intersection algebras and the completeness radical[END_REF], for which there exists a constructive proof in [START_REF] Quitté | Le théorème de de Smit et Lenstra, démonstration élémentaire[END_REF].

In Section 2.2 we recall the definition and main properties of border bases.

We point out that, to obtain a fully algorithmic proof of our results, we rely on the constructive proof of the Multivariate Hensel Lemma (MHL for short) given in [START_REF] Emilia | Revisiting Zariski main theorem from a constructive point of view[END_REF]. Also, in order to get true algorithms, fields are assumed to be discrete and local rings to be residually discrete.

The pure abstract algebraic form of Local Bézout Theorem is given in Theorems 3.3 and 3.7.

Going futher into details let us explain the form of Local Bézout Theorem we are interested in.

Assume (A, m, k) is a local normal domain, f 1 , . . . , f n ∈ A[X 1 , . . . , X n ] and let B := A[X 1 , . . . , X n ]/ f 1 , . . . , f n and C = B m+ x = A[x 1 , . . . , x n ] m+ x ,
where x = x 1 , . . . , x n , x i is X i mod f 1 , . . . , f n , and f i (0) ∈ m for i = 1, . . . , n. We denote by K the quotient field of A. We assume K to be an algebraically closed field and therefore A to be a Henselian ring.

We assume that the k-algebra

C := (k[X 1 , . . . , X n ]/ f 1 , . . . , f n ) x1,...,xn = k[x 1 , . . . , x n ] x1,...,xn is zero-dimensional, where f i (resp. x i ) is the image of f i (resp. x i ) by ⊗ A k.
Since K is algebraically closed it is plausible to speak about the continuity of the roots. The algebraic form of Local Bézout Theorem (Theorem 3.7) says that there are finitely many zeroes of (f 1 , . . . , f n ) above the residual zero (0, . . . , 0) (i.e., with coordinates in m), and the sum of their multiplicities equals the dimension r of C as k-vector space, i.e., the multiplicity of the residual zero. This implies also that the K-algebra C ⊗ A K is finite free of rank r.

As application of the precedent sections, in section 4, we obtain, from the abstract theorem, the classical one. Here the characterization of border bases introduced by Bernard Mourrain in [START_REF] Mourrain | A new criterion for normal form algorithms[END_REF] reveals to be a crucial tool.

2 Useful tools 2.1 Theorem of de Smit and Lenstra Theorem 2.1. (Theorem of de Smit & Lenstra, flatness, [START_REF] Smit | Finite complete intersection algebras and the completeness radical[END_REF]) Let A be an arbitrary commutative ring. If an A-algebra

B = A[X 1 , . . . , X n ]/ f 1 , . . . , f n is finite, then it is flat (so, it is free if A is local).
A constructive proof due to Claude Quitté is explained in [START_REF] Quitté | Le théorème de de Smit et Lenstra, démonstration élémentaire[END_REF].

Border bases

In this subsection, A is an arbitrary commutative ring.

(2.2.1) In the sequel we shall identify the semi-groups

X N 1 • • • X N n and N n . A nonempty finite subset B ⊂ N n is called closed by division if for every X γ , X γ , if X γ ∈ B and X γ | X γ then X γ ∈ B.
In the sequel any finite subset N n denoted by B will be assumed nonempty and closed by division. We call border of B the following finite subset of N n ,

∂B := (X 1 B ∪ • • • ∪ X n B) \ B (2.2.2) Let A be a ring, I a finitely generated ideal of A[X 1 , . . . , X n ] = A[X] and B := A[X]/I = A [x 1 , . . . , x n ]
(x i the image of X i in B and A the image of A). Given a finite subset B ⊂ N n as above, we call rewriting rules for B w.r.t. B a set of equalities in B as follows,

x β = α∈B h β,α x α : β ∈ ∂B where h β,α ∈ A (1)
Formally, we define the rewriting rules as being the polynomials h β (X) = X β -α∈B h β,α x α . The equalities (1) mean precisely that the h β (X)'s belong to the ideal I.

If B defined as above has rewriting rules w.r.t. to B, then the set { x α : α ∈ B } generate B as A-module.

In particular B is a finite A-module.

(

: Span A (B) → Span A (B)
the linear map given by

µ Xi (X α ) = X i X α if X i X α ∈ B, α∈B h β,α X α if X i X α = X β ∈ ∂B.
Now let Λ i denote the matrix of µ Xi w.r.t. the basis B of the free A-module Span A (B), then, see [6] and [START_REF] Mourrain | A new criterion for normal form algorithms[END_REF] Theorem 2.2. B is a free A-module with basis {x α : α ∈ B} iff

Λ i Λ j = Λ j Λ i for i, j = 1, . . . , n.
(2.2.5) In the particular case in which A is a discrete field k and B is an Artinian k-algebra, the Gröbner basis algorithm provides a border basis of B. Indeed, it is well known that, w.r.t. to an admissible monomial ordering, the monomials "under the staircase" form a k-basis B of B, and the rewriting rules are given by computing the remainder of the division of X β by the Gröbner basis (for β ∈ ∂B). Mourrain in [START_REF] Mourrain | A new criterion for normal form algorithms[END_REF] introduced an algorithm to compute a border basis of an Artinian k-algebra without using the theory of Gröbner basis and requiring for B a weaker property than to be closed by division.

(2.2.6) One important fact with border bases is that they are more suitable for computational purposes specially when data is given by approximate values. Another one is that they can be useful in more general cases when Gröbner bases over rings are not available or not easy to manage.

The Bézout local theorem

Let us recall two theorems in [3].

Theorem 3.1. ([3, Theorem 1]) Let (A, m, k) be a local Henselian ring. Let f 1 , . . . , f m ∈ A[X 1 , . . . , X n ] = A[X], f 1 , . . . , f m their images in k[X], B := A[X]/ f 1 , . . . , f m = A[x 1 , . . . , x n ] and B := B/mB = k[X 1 , . . . , X n ]/ f 1 , . . . , f m = k[x 1 , . . . , x n ]. Let B 1 := B m+ x , B 1 := B x1,...,xn and assume 1 ≤ dim k (B 1 ) < ∞.
Then the A-algebra B 1 is a finitely generated A-module.

A domain A is called a DVR with uniformizing parameter p when every nonzero element of A is uniquely expressed in the form up m for some m > 0 and u ∈ A × . In this case A is local with radical pA.

Theorem 3.2. ([3, Theorem 5]) Let (A, m, k) be a Henselian DVR. Let f 1 , . . . , f n ∈ A[X 1 , . . . , X n ] = A[X], f 1 , . . . , f n their images in k[X], B := A[X]/ f 1 , . . . , f n = A[x 1 , . . . , x n ] and B := B/mB = k[X 1 , . . . , X n ]/ f 1 , . . . , f n = k[x 1 , . . . , x n ]. Let B 1 := B m+ x , B 1 := B x1,...,xn and assume dim k (B 1 ) = r ≥ 1.
Then the A-algebra B 1 is a free A-module of rank r, whose basis is given by lifting any k-basis of B 1 .

We give first a result that implies a generalization and an important precision in Theorems 3.1 and 3.2 in case of a residually finite global complete intersection (m = n and the residual algebra is finite): see corollary 3.5.

In the sequel we use the notation m[X] for the ideal of A[X] generated by m (the polynomials with coefficients in m) and m+ X for the ideal m[X] + X (the polynomials ∈ A[X] with constant coefficient in m). (b) In case we know a border basis of B as k-algebra, its rewriting rules can be lifted, to provide, for a suitable s ∈ 1 + mB, a system of generators of B[1/s] as an A-module.

2. Assume in addition that the A-algebra B is presented with an equal number of generators and relations, that is Note. In order to be able to compute a k-basis of B, we need a priori that B is finitely presented as an A-algebra. So, in item 1 (a) we do not assume that B has a k-basis, and in item 2 the hypothesis implies that we are able to compute a border basis of B.

B := A[X 1 , . . . , X n ]/ f 1 , . . . , f n = A[x 1 , . . . , x n ]. (
An intuitive meaning of the first part of the statement of the theorem is that, when A is a domain with quotient field K and the polynomial system is residually zero-dimensional, inverting s "maps at infinity" all zeroes of I in K which are not integral over A. We explain this intuition through Corollary 3.4. An A-algebra is a commutative unital ring E with a ring morphism ρ : A → E, in the sequel we use in this case the terminology "an A-algebra ρ : A → E".

Corollary 3.4. Same hypotheses and notations as in Theorem 3.3 1. Assume moreover that B := A[X 1 , . . . , X n ]/ f 1 , . . . , f m = A[x 1 , . . . , x n ] and we are given an A-algebra ρ : A → E and a zero (ξ 1 , . . . , ξ n , ζ) of {f 1 , . . . , f m , ZS(X) -1} in E. Then each ξ i is integral over ρ(A).

Proof. Since B[1/s] is a finite A-module, each x i is integral over A in B[1/s]: in fact x i annihilates the characteristic polynomial of the matrix of multiplication by x i with respect to a system of generators of B[1/s] as A-module. The morphism ρ gives by factorization a morphism of A-algebras B[1/s] → E mapping (x 1 , . . . , x n , z) to (ξ 1 , . . . , ξ n , ζ). So each ξ i annihilates a monic polynomial with coefficients in ρ(A).

Corollary 3.5. Same hypotheses and notations as in Theorem 3.3 1, and in addition assume that the kalgebra B is local, more precisely that B = B x1,...,xn (this means that each x i is nilpotent, or also that (0, . . . , 0) is the unique zero of f 1 , . . . , f m in an algebraic closure of k).

1. There exists an s ∈ 1 + mB such that B[1/s] = B m+ x is a finitely generated A-module.

If in addition B

:= A[X 1 , . . . , X n ]/ f 1 , . . . , f n = A[x 1 , . . . , x n ],
then the A-algebra B m+ x is a free A-module of rank r, whose basis is given by lifting any k-basis of B with its rewriting rules.

Proof (as in Corollary 3.5 1). Then each ξ i is "integral over m" (see the precise meaning in the proof).

Proof. Since x i is nilpotent, the multiplication µ xi by x i in B is given by a lower triangular matrix M i with zeroes on the diagonal (after a suitable change of basis as a k-vector space). Since we have a generator system B of B[1/s] as an A-module which is lifted from a border basis B of B, and since the rewriting rules are also lifted, the multiplication by x i in B[1/s] can be expressed w.r.t. a generator system B (obtained from B by lifting the above change of vector space basis) by a lifting of the above matrix M i and the characteristic polynomial of this lifted matrix has the form T r + r-1 j=0 µ i,j T r-j with µ i,j ∈ m j . So, we get an equality in E: ξ r i + r-1 j=0 ρ(µ i,j )ξ r-j i = 0 and µ i,j ∈ m j .

Proof of Theorem 3.3. 1. We set B = A[X 1 , . . . , X n ]/I for some ideal I ⊆ A[X]. As B is a finitely generated nonzero k-vector space, we can represent it as a quotient of some finitely presented nonzero k-algebra

k[X 1 , . . . , X n ]/ f 1 , . . . , f m for some f i ∈ I. So B is a quotient of the finitely presented A-algebra B 0 = k[X 1 , . . . , X n ]/ f 1 , . . . , f m
whose residual k-algebra B 0 is a k-vector space with a finite k-basis. Hence w.l.o.g. we can assume that B is finitely presented. Hence B has a border basis. There exists a finite set of monomials B ⊂ N n containing 1 and closed by division, such that {x α : α ∈ B} is a k-basis of B, and for

β ∈ ∂B h 0 β (X) := X β - α∈B u 0 β,α X α (with u 0 β,α ∈ k)
are the corresponding rewriting rules.

Now we follow the constructions in the proof of [3, Theorem 1] (see Claim 4 in [3]). As

f 1 , . . . , f m = h 0 β : β ∈ ∂B ⊆ k[X 1 , . . . , X n ], we have h 0 β = m i=1 p i,β f i , with some p i,β ∈ A[X]. For β ∈ ∂B we put H β (X) := m i=1 p i,β f i (so H β = h 0 β ).
Next we reduce these polynomials H β with the following "formal rewriting rules"

h β := X β - α∈B u β,α X α ,
where u β,α are variables, whose suitable values in A we are looking for. Following [3] and [START_REF] Emilia | Stable Deformation of Zero-Dimensional Quotient Algebras[END_REF] we get

H β = β ∈∂B Q β,β h β + R β (2) for polynomials Q β,β (( u β,α ), X) ∈ A[( u β,α ) β∈∂B,α∈B , X] and R β = α∈B R β,α X α with R β,α (( u β,α )) ∈ A[( u β,α )]. Moreover ∆(( u β,α ), X) := det(( Q β,β ) β,β ∈∂B ) ∈ 1 + m[( u β,α ), X].
In [3] and [START_REF] Emilia | Stable Deformation of Zero-Dimensional Quotient Algebras[END_REF] we proved that (u 0 β,α ) is an isolated simple zero of the system R β,α = 0 . Since A is Henselian, by MHL, there exists a unique solution (u β,α ) ∈ A ∂B×B of the system R β,α = 0 lifting the solution (u 0 β,α ) ∈ k ∂B×B . For β ∈ ∂B, we define

h β := X β -α∈B u β,α X α , Q β,β (X) := Q β,β ((u β,α ), X) ∈ A[X], S(X) := ∆((u β,α ), X) = det((Q β,β ) β,β ∈∂B ) ∈ 1 + m[X].
Equalities [START_REF] Emilia | Revisiting Zariski main theorem from a constructive point of view[END_REF] give

H β = β ∈∂B Q β,β h β . ( 3
) Let s = S(x) ∈ A[x] = B. The ideal f 1 , . . . , f m , SZ -1 ⊆ I ⊆ A[X, Z] contains polynomi- als h β 's for β ∈ B since h β ≡ ZSh β
and Sh β is expressed from the H β 's using the cotransposed matrix of (Q β,β ) β,β ∈∂B . As f i belong to the ideal I, we get a well defined A-algebra morphism

Φ : C = A[X]/ (h β ) β∈∂B -→ A[X, Z]/ I, SZ -1 = B[1/s] X i -→ X i As an A-module, C is generated by the classes X α mod J = (h β ) β∈∂B (for α ∈ B).
Moreover Φ is surjective because S has an inverse in Φ(C). Indeed Φ(C) is a finitely generated A module and the multiplication by S in Φ(C) is residually onto (it is the identity), therefore by Nakayama lemma the multiplication by S is itself onto on Φ(C).

As we get Φ(C) = B[1/s], B[1/s] is generated as A-module by the lifting {x α ; α ∈ B} of B and the h β 's are rewriting rules for this generator system.

We have

B = A[X 1 , . . . , X n ]/ f 1 , . . . , f n . By item 1 we know that B[1/s] is a finitely generated A-module for some S(X) ∈ 1 + m[X].
We can apply the theorem of de Smit & Lenstra 2.1 to conclude that B[1/s] is a finite free A-module. Clearly its rank is the same after ⊗ A k, so it is equal to the dimension r = #B of the k-vector space B[1/s] = B. Since B[1/s] is generated by a lifting of B, we conclude by saying that in a free module of rank r, any generator system of r elements is a basis.

On the other hand, since the generator system {x α : α ∈ B} of the A-module C is mapped by Φ to a basis of the A-module B[1/s] with the same number of elements, we have that Φ is injective and therefore it is an isomorphism. This fact will be used in Section 4. In consequence we have finished the proof of the theorem.

2

Our main concern is the following result in which item 3 generalizes [3, Theorem 11]. We point out that in [3, Theorem 11] we forgot to give as an hypothesis the fact that A is integrally closed in K: the proof was given only for the case of a valuation domain and used implicitely the normality hypothesis in the general case. So C is a finitely presented A-algebra to which we can apply Corollary 3.5 1. There exists s ∈ 1+mC such that C[1/s] = C m+ x . Since e ∈ 1+mB+ x we see that s is written as v(x)/e m for a v ∈ 1 + mB + x and a suitable exponent m. Finally we see that

C[1/s] = B[1/u] = B m+ x
where u = sv ∈ 1 + mB + x . This finishes the proof of item 1.

2. Now we give some details for describing C and C[1/s] when C is presented with an equal number of generators and relations. Let E(X) ∈ A[X] s.t. e = E(x), we have

1 -E ∈ m + X + f 1 , . . . , f n and E 2 -E ∈ m[X] + f 1 , . . . , f n (inside A[X]) Therefore C := B[1/e] = A[x, 1/e] = A[X, T ]/ f 1 , . . . , f n , E(X)(T + 1) -1 = A[x, t]
Now we write s = S(x, t) where S(X, T ) ∈ 1 + m[X, T ] and Corollary 3.5 2 says us that

D := C[1/s] = A[X, T, Z]/ f 1 , . . . , f n , E(X)(T + 1) -1, S(X, T )(Z + 1) -1
is a free A-module of rank r, whose basis is given by lifting any k-border basis of B[1/e] = B x1,...,xn with its rewriting rules. This finishes the proof of item 2.

3. First of all notice that under the hypothesis of 3, A is a Henselian local ring and item 2 applies. Let

:= {f 1 , . . . , f n , E(X)(T + 1) -1, S(X, T )(Z + 1) -1}
Since D is a free A-module of rank r, D ⊗ A K is a finite K-vector space of dimension r.

Since K is an algebraically closed field, by Stickelberger's theorem (see [START_REF] Lombardi | Commutative Algebra. Constructive Methods[END_REF].17]), we can decompose D ⊗ A K as (ξ1,...,ξn+2)∈Z K (Σ)

K[X, T, Z] (X-ξ,T -ξn+1,Z-ξn+2) / Σ

In each (ξ 1 , . . . , ξ n+2 ) ∈ Z K (Σ) the corresponding local K-algebra is zero-dimensional and the sum of multiplicities is equal to r.

It remains to prove that the points of Z K (Σ) correspond exactly to the zeroes of (f 1 , . . . , f n ) in K having their coordinates in m, and that the corresponding local K-algebras are isomorphic. First let (ξ 1 , . . . , ξ n+2 ) ∈ Z K (Σ) where ξ 1 , . . . , ξ n ∈ m. Since E(x) ∈ 1 + m + x , we get ε := E(ξ 1 , . . . , ξ n ) ∈ 1 + m. This ε has a unique inverse in K, and this inverse is written 1 + µ with µ ∈ m. This forces ξ n+1 = µ. Morevover the two local algebras at (ξ 1 , . . . , ξ n ) and (ξ 1 , . . . , ξ n+1 ) are "equal" It remains to see that any zero (ξ 1 , . . . , ξ n+2 ) ∈ Z K (Σ) has its first n coordinates in m. Corollary 3.4 (with E = K and ρ : A → K the inclusion morphism) shows that ξ 1 , . . . , ξ n+1 are in A (recall that A is assumed to be integrally closed). Then Corollary 3.6 shows that ξ 1 , . . . , ξ n+1 are in m.

  Theorem 3.3. Let (A, m, k) be a local Henselian ring, B := A[x 1 , . . . , x n ] be a finitely generated Aalgebra and set B := B/mB = B ⊗ A k the residual algebra. Assume that B is a nonzero finitely generated k-vector space. Then it holds: 1. (a) There exists s ∈ 1 + mB s.t. the A-algebra B[1/s] is a finitely generated A-module.

  a) If dim k (B) = r ≥ 1, there exists S ∈ 1+m[X] s.t. letting s = S(x), B[1/s] is a free A-module of rank r. (b) More precisely, any border basis of the residual algebra B = B[1/s] can be lifted with its rewriting rules to a border basis of B[1/s] with its rewriting rules.

Theorem 3. 7 .Proof. 1 .

 71 (Bézout local) Let (A, m, k) be a Henselian local ring and B := A[x 1 , . . . , x n ] a finitely generated A-algebra. Let B := B/mB = k[x 1 , . . . , x n ] be the residual algebra. Assume that B is finitely presented 1 as a k-algebra and that dim k B x1,...,xn = r ≥ 1 (this means that (0) is an isolated residual zero of multiplicity r). Then mB + x 1 , . . . , x n (denoted as m + x ) is a maximal ideal of B and we have:1. (Pure algebraic form, without zeroes)The A-algebra B m+ x is a finitely generated A-module: there exists anu ∈ 1 + mB + x such that B[1/u] = B m+ x , B[1/u] = B x1,...,xn ,and we get a generator system of the A-module B m+ x by lifting a k-border basis of B[1/u].2. (Local complete intersection case, without zeroes) If in addition B is finitely presented asB := A[X 1 , . . . , X n ]/ f 1 , . . . , f n = A[x 1 , . . . , x n ],then B m+ x is a free A-module of rank r, whose basis is given by lifting any k-border basis of B x1,...,xn with its rewriting rules.3. (Local complete intersection case, usual form, with zeroes)Assume that (A, m, k) is a local normal domain with algebraically closed quotient field K and let B as in item 2. Then, there are finitely many zeroes of (f 1 , . . . , f n ) above the residual zero (0) (i.e., zeroes with coordinates in m) and the sum of their multiplicities equals r. Let B 1 := B m+ x , B 1 := B x1,...,xn . The hypothesis dim k (B 1 ) = r ≥ 1 means that (0) is residually an isolated zero of multiplicity r. So, we can construct an e ∈ B with e ∈ B such that • e is an idempotent in B, • for a suitable integer N , 1 -e = x 1 , . . . , x n N in B, • B[1/e] = B 1 . (See [10, Theorem IX-4.7].) In particular e ∈ 1 + mB + x and e 2 -e ∈ mB Let C := B[1/e]. Since e ∈ 1 + mB + x , we have C m+ x = B m+ x , and C = B[1/e] = B x1,...,xn = C x1,...,xn .

K[x 1

 1 , . . . , x n ] x1-ξ1,...,xn-ξn K[x 1 , . . . , , x n+1 ] x1-ξ1,...,xn+1-ξn+1 .This follows from the facts that K[x 1 , . . . , ,x n+1 ] = K[x 1 , . . . , x n ][1/E(x)] and that E(x) is invertible in K[x 1 , . . . , x n ] (x1-ξ1,...,xn-ξn) . Indeed E(x 1 , . . . , x n ) ≡ E(ξ 1 , . . . , ξ n ) = ε mod x 1 -ξ 1 , . . . , x n -ξ n in K[x 1 , . . . , x n ], so E(x 1 , . . . , x n ), x 1 -ξ 1 , . . . , x n -ξ n = 1 in K[x 1 , . . . , x n ].Similarly S(ξ 1 , . . . , ξ n , µ) ∈ 1 + m, it has a unique inverse in K, and this inverse is written 1 + ν with ν ∈ m. This forces ξ n+2 = ν and the equality K[x 1 , . . . , x n ] (x1-ξ1,...,xn+1-ξn+1) K[x 1 , . . . , , x n+1 ] (x1-ξ1,...,xn+2-ξn+2) .

  . 1. First we note that B m+ x = B x1,...,xn = B. We apply Theorem 3.3. We get an s ∈ 1 + mB

such that B[1/s] is a finitely generated A-module. Since s is invertible in B m+ x , we get an A-morphism ϕ : B[1/s] → B m+ x . We show that "ϕ is a canonical isomorphism": the two localizations of B are the same one. This means that any element c ∈ 1+mB+ x is invertible in B[1/s]. Since c = 1 in B x1,...,xn and B = B x1,...,xn , we get c = 1 in B[1/s] and the result follows from Nakayama: the multiplication by c in B[1/s] is an A-endomorphism which is residually onto, hence itself is onto and c is invertible in B[1/s]. 2. We have the hypothesis of Theorem 3.3 2 and B[1/s] = B m+ x . Corollary 3.6. Same hypotheses and notation as in Corollary 3.4. Assume moreover that B = B x1,...,xn

From a constructive point of view we cannot deduce that B is finitely presented from the fact it is finitely generated.

by Spanish GR MTM-2011-22435 and MTM-2014-

Remark 3.1. Theorem 3.7 1. can be seen as a generalization of the Mather-Weierstrass division theorem. In fact let k be a field,

. . , X n , Y 1 , . . . , Y ]] alg , I = F 1 , . . . , F q an ideal of A m , and assume that the morphism A n → C = A m /I is quasi-finite (i.e. C/m n C is a finite dimensional k-vector space). We will see that C is a finite A-module. As A m is the henselization of (A n [Y 1 , . . . , Y ]) mn+ Y1,...,Y , we can assume that I is contained in some

where

t ∈ A m and f (Y , t) = 0.

W.l.o.g. we assume that F i = P i (Y , t) where P i (Y , T ) ∈ A n [Y , T ]. We apply the theorem with A = A n , B = (A n [Y , T ]/ P 1 , . . . , P q ) m+ Y . Finally we get that B is a finite A-module. So, it is Henselian and

And C is a finite A-module. This is the Mather Theorem.

In the case of q = 1, we get the Weierstrass division theorem applying 3.7 2.

Application

This section is written in classical mathematics, allowing us to deal with C as if it were a discrete field (i.e. assuming we have an equality test for complex numbers). E.g., Theorem 4.1 is not written in a constructive form because, since there is no zero test for elements of C, it is impossible to know in the general situation what are exactly the distinct zeroes and their multiplicities for the perturbed polynomial. E.g., it is a priori impossible to know if a monic univariate polynomial of degree two has one double root or two distinct roots. A constructive and continuous form of the FTA has a slightly different formulation, the best one being [START_REF] Richman | The fundamental theorem of algebra: a constructive development without choice[END_REF], where the zeroes of a complex polynomial are seen as forming a multiset that varies continuously in a suitable complete metric space. See also [START_REF] Bishop | Foundations of constructive analysis[END_REF]Chapter 5] for another constructive formulation, and [12, Appendix A, page 276] for an optimal modulus of continuity. We think that Theorem 4.2 would need a subtle constructive reformulation, with a more precise proof than that we give here.

Finally we would like to point out that, in the spirit of the proof of [START_REF] Eisermann | The fundamental theorem of algebra made effective: an elementary real-algebraic proof via Sturm chains[END_REF]Eisermann] which is an analogous to Theorem 4.1 for the algebraic closure R[i] of a real closed field R, one could prove the classical multivariate version of Theorem 4.2 for R[i], taking into account the usual manipulation of semialgebraic continuous functions w.r.t. the topology in C n ∼ = R 2n defined by the order of the real closed field R.

We start with a classical result of complex analysis, about the continuity of roots of a univariate polynomial defined over C (the field of complex numbers), with respect to its coefficients. By Ω(ξ, ) we denote the polydisc of C n centered at ξ ∈ C n : that is

The statement is the following easy consequence of Rouché's Theorem.

exactly m j roots counting with multiplicities in Ω(ξ j , ), for every j ∈ 1..s . This section consists in giving a clear proof of the following theorem, which is a classical result about the continuity of the points in a 0-dimensional complete intersection C-algebra, as in [START_REF] Arnold | Classification of critical points, caustics and wave fronts[END_REF]9]. Theorem 4.2. Let g 1 , . . . , g n ∈ C[X 1 , . . . , X n ] be polynomials of degrees d 1 , . . . , d n respectively and let p ∈ Z C (g 1 , . . . , g n ) be an isolated zero of multiplicity r. Let N 1 , . . . , N n be the number of monomials of degree respectively d 1 , . . . , d n in the variables in X 1 , . . . , X n , and let N = i N di . We can see (g 1 , . . . , g n ) as an element (a) ∈ C N . Let us consider a "slightly perturbed system" ( g 1 , . . . , g n ) corresponding to an element ( a) ∈ C N . Then for all > 0 there exists 1 with 0 < 1 < and δ 1 > 0 such that for all ( a) ∈ Ω(a, δ 1 ), the perturbed system has only finitely many zeroes in Ω(p, 1 ); moreover the sum of multiplicities of these zeroes equals r.

Proof. For sake of simplicity we assume p = (0, . . . , 0), that is

. . , g n . We consider a family of new indeterminates:

Let us denote by G 1 ( a (1) , X), . . . , G n ( a (n) , X) the generic polynomials of degrees d 1 , . . . , d n respectively. So, g i = G i (a (i) , X) for some a (i) ∈ C Ni . We set now

We can see the v (i) j 's as indeterminate perturbations of the a

j ) i=1,...,n;j=1,...,Ni ]] alg the ring of algebraic formal power series (or the ring

j ) i=1,...,n;j=1,...,Ni } of analytic power series) with coefficients in C in the variables v (i) j . In the second case this ring can be identified with the ring of germs of analytic functions in a neighborhood of a = (a (i) j ) i,j . In both cases the ring is Henselian with residue field C. Then, we apply Theorem 3.7 2 taking as

Let us recall some elements in the proof of 3. In this situation a C-border basis B of B 1 := B x1,...,xn (with r = #∂B) lifts with its rewriting rules to an A-basis of D. So we get in D for every β ∈ ∂B an equation

These U β,α are algebraic power series in the formal coefficients v's and they give analytic functions in a neighborhood of (a) = (a (i) j ) i,j . Let us denote by Λ i the multiplication matrix by x i in the free A-module D = Span A (B), w.r.t. the basis B using [START_REF] Bishop | Foundations of constructive analysis[END_REF]. Notice that the entries of Λ i are either 0 or 1 or one of the U β,α 's and, since B is a border basis, one has Λ i Λ j -Λ j Λ i = 0 (6) in A r×r for i = j. Now we remark that in the proof of 3.7 2 we have seen that the following ideals of A[X 1 , ..., X n ] coincide.

Hence one gets equalities (β ∈ ∂B):

for some W i , M, R ∈ A[X, T, Z], and on the other side (i = 1, . . . , n)

for some

. We know that identities ( 8) and (9) hold in the smallest sub-C-algebra of A containing the U β,α 's. We recall that m = v (i)

To stress the (analytic) dependence of E and S on the v (i) j 's we shall write E( v, X) and S( v, X, T ). Now let P i ∈ A[Y ] denote the characteristic polynomial of the multiplication by the image of x i in D (i.e. the characteristic polynomial of Λ i ). Its coefficients are Z-polynomials in the U β,α 's. By change of ring,

From , by usual arguments of Taylor calculus for convergent series we can find δ and < such that the following properties hold: i) All the U β,α are convergent and also the coefficients in ( 8) and (9) are convergent for any a (i) ∈ Ω(a (i) , δ). Therefore the same will hold for the coefficients of P i .

ii) The polynomial E( v, X) as a polynomial with coefficients which are analytic functions in a neighborhood of a (i) 's verifies that for every ( a (i) , x) ∈ Ω(a (i) ; δ) × Ω(0; ), |E( a, x) -1| < .

In particular for those points ( a, x) we have E( a, x) = 0 and we may also ask that 1 -1 E( a,x) < . iii) In the same way for ( a (i) , x, t) ∈ Ω(a (i) ; δ) × Ω(0; ) × Ω(0; ), S( a, x, t) does not vanish and 1 -1 S( a,x,t) < .

We apply Theorem 4.1 with and

is a monic polynomial of degree r, whose coefficients belong to a neighborhood Ω(0; δ ), then, the distance of each of the r complex roots of Q to 0 is less than . Notice that Q can be considered as a perturbation of P i . Then, for every a (i) ∈ Ω(a (i) ; δ ); we consider g i := G i ( a (i) , X), which is "a perturbation" of the g j , and we denote by U β,α the result of specializing U β,α in the values of a (i) 's. Then the following facts hold true and alltogether prove the theorem taking the value δ 1 as δ and 1 as :

1. {x β : β ∈ B} is a basis of the C-vector space, because the relations of commutations (6) still hold under specialization at a (i) (see 2.2.4 in section 2 on border bases). Consequently the ideal of C[X] I = I( a (i) ) := X α -β∈B U β,α X β has r zeroes in C n counted with multiplicity.

2. The characteristic polynomial of the multiplication by X i in the ring C[X]/ I is P i . For every (ξ) ∈ C n which is a zero of I its coordinates ξ i 's are roots of the polynomials P i (T ). We know that these roots belong to Ω(0; ). On the other hand these zeroes are zeroes of the ideal J( a (i) ) := ( g i ) i=1,...,n C[X], because of the specialization of (9).

3. Conversely given (ξ) ∈ C n which is a zero of the ideal J( a (i) ) and such that (ξ) ∈ Ω(0; ) we are going to prove that (ξ) is a zero of I( a (i) ). As a (i) ∈ and (ξ) ∈ Ω(0; ), by ii) above we have that t := 1 -1 E( a (i) ,ξ) ∈ Ω(0, ). Consequently it makes sense to substitute in [START_REF] Eisermann | The fundamental theorem of algebra made effective: an elementary real-algebraic proof via Sturm chains[END_REF], ( a (i) , ξ, t). since the right hand side vanishes, the same happen for the left hand side. [START_REF] Arnold | Classification of critical points, caustics and wave fronts[END_REF]. Given (ξ) ∈ Ω(0; ), zero of J( a (i) ) and hence of I( a (i) ) the local rings C[X]/I( a (i) ) (X-ξ) and C[X]/J( a (i) ) (X-ξ) coincide. In fact this comes from (9), and by substituting in (8) T by 1 -1 E( a (i) ,X) and Z by 1 -1 S( a (i) ,X,T ) , which is allowed since both expressions are rational regular at the point (ξ).


Gilles De Rapper

DEALING WITH THE GREEK ALBANIAN BORDER AMONG THE ALBANIAN-SPEAKING CHRISTIANS OF SOUTHERN ALBANIA

Following the publication of Peter Sahlins' Boundaries: the Making of France and Spain in the Pyrénées in 1989 1 , the 1990s have witnessed the development of an anthropology of international borders, specially thanks to the work of Hastings Donnan and Thomas M. Wilson 2 . The anthropological approach, with its focus on everyday life of border communities as well as on the symbolic dimension of boundaries, contributed to the understanding of borders and border areas, at a time when many international borders, particularly in Europe, were rapidly and radically changing. In this context, it is striking that the Balkan borders have been the centre of relatively limited attention.

The purpose of this article is to present the way Albanians from a specific area in Southwest Albania cross the border with Greece, and to question to what extent the ethnic or ethnonational boundary between Greeks and Albanians plays a role in those border crossings. The idea is that the specific position of those people in the Albanian context -regarding religious affiliation and alleged cultural characteristics -gives a special significance to their crossing of the border and influence their practice of the border and border area.

[163] I am thus making a distinction between the state border on the one hand, understood as a line delimitating two political territories, Greece and Albania, and the ethnic boundary on the other hand, understood as the more or less organised juxtaposition of attributes that makes one being Albanian or Greek. The former is a political or legal boundary, the latter a symbolic boundary.

Implicit is also the idea that crossing an international border (or state border) means more than a move through space, but has to do with the self and with the social existence of the person who is crossing the border. As is shown by Henk Driessen3 , there is a similarity between crossing an international border and what is known in social anthropology as rite de passage. This is specially true in the case of the new migration as it occurs today between Albania and Greece: due to the imposition of rules to cross the border and work in Greece (or just visit, as more and more Albanians do), most of the people who cross the border have to go through a change in their being. Border crossings have to be understood here as state border crossings as well as ethnic boundary crossings.

The area I am focusing on is a group of around twelve villages on the slopes of the mountain facing the town of Gjirokastër, and called Lunxhëri. In opposition to the surrounding areas, those twelve villages are presented by their inhabitants as being at the same time Orthodox Christian (as opposed to the Muslim area of Labëri), and Albanian (as opposed to the Greek minority officially recognised by the Albanian state).

The first part of the article is dedicated to the specific position of the Albanian-speaking Christians in the border area, which is a position marked by ambiguity. The second part looks at the way people from Lunxhëri cross the border -the international border between Greece and Albania as well as the ethnic boundary between Greeks and Albanians. In other words, the focus of this study is not the border per se, nor its creation in 1913-1919, but rather the way it is experienced today by the local inhabitants of the border zone. It also means that the people actually living in that area, rather than the migrants originating from it, are the centre of my attention.

Albanian-speaking Christians in Southern Albania: context and significance

To illustrate the specificity of the Albanian-speaking Christians as a border community and in relation to border crossing, it is worth looking first at another Albanian border area, inhabited mostly by Muslims. In the Devoll, in South-east Albania, things seem to be quite clear: the state [164] border between Albania and Greece is also an ethno-national boundary between Albanians and Greeks, as it separates Muslims from Christians, and appears in collective memory as a place of confrontation and conflict. People recall stories of Greeks burning Muslim villages at the time of the First World War, of Muslims being expelled from their villages on the Greek side at the time of the Lausanne Treaty (1923), and of the last battles of the Greek Civil War (1949), in the Grammoz mountains, known on the Albanian side as 'the Greek provocations'. In the 1990s, however, people from the Devoll started to cross the border and migrate to Greece. The result of this migration, as far as identity or ethnicity are concerned, is the appearance of a double identity phenomenon. People choose or are given a second, Christian name, that they call their 'Greek name', they use false documents stating that they are of Greek descent, present themselves as Northern Epirotes, and sometimes go as far as converting to Christianity -that they practice in Greece rather than in Albania. In consequence, one can hear in Muslim villages people joking about their Northern Epirote identity, calling their friends by their Greek names, and using more Greek words than necessary in conversations4 . I am not pretending that things are easy for those people; actually, it is not easy at all and generally comes along with great suffering. But at least things are in a way quite clear: they are Albanian Muslims and, in order to cross the border, pretend to be Christians of Greek descent. At least until 1997, this double identity process was facilitated by the seasonal character of their migration: people would live one half of the year in Greece, among friends and male relatives, and the other half in Albania, with their families. Such changes of name and religion also appear in the Muslim area of Labëri, and it is a source of amusement among the Christians of Lunxhëri and Dropull, who laugh at 'those people with a moustache who bear Christian names and pretend to be Christian'5 .

In the case of the Albanian-speaking Christians on the other hand, no such things are possible, as their position always appears, from the start, ambiguous. In his book on Nationalism, Elie Kedourie reports the story of a villager from Southern Albania, asked by the international commission in charge of delineating the border, if he was Greek or Albanian, and answering -in Albanian -'I am Greek'6 . One can guess that what the man actually meant was that he was a Christian. As it is well known, in that part of the Balkans, the ethnonym 'Greek' was widely used with a religious meaning, and all the Orthodox Christians were called 'Greeks'. Still today, this ambiguity creates space for the discussion over the size of the Greek minority in Albania: are all Orthodox Christians of Greek descent, or only a part of them?

[165] If the ethnonym 'Greek' is not used today by the Lunxhotes, there is evidence that they did use it in the past, at least in its religious dimension. Moreover, from the Muslim point of view, all the Christians are kaur (infidels), either Greek or Albanian, and this religious community is sometimes seen as more important than the national community that makes Albanian Muslims and Christians alike.

In fact, religion, or more accurately religious affiliation, is too important in the construction of collective identity in Albania to be sidelined in favour of the sense of national belonging7 . It can be shown that collective identity in Lunxhëri is to a large extent constructed in opposition to the neighbouring area of Labëri, and one of the main dimensions of this opposition is religion. Moreover, in the social and political context of the 1990s, Christianity means Europe and European integration, and in that respect, it cannot be given up. Being Christian is thus one of the most important features of Lunxhote identity.

At the same time, nationalism has had a strong impact on Lunxhëri (as it usually has in border areas), and being Christian does not, in Lunxhëri, mean being Greek. Albanian nationalism developed in Southern Albania in reaction to the territorial gains made by Greece since its independence. One of the main tasks of the nationalists was to convince the world, and the Albanians themselves, that there was such a thing as an Albanian nation, and that either Christian or Muslim, the population of Epirus was Albanian. As we will see later on, not all the Lunxhotes agreed on that, and it took time to convince the others, but it can be said that the aim was reached during communism. Today, not only are the Lunxhotes aware (and sometimes proud) of being Albanian, they also know how to consider Greece as an enemy of the nation. The image of Greece also depends on the new emigration and the feeling that Albanians are not rewarded as they should for their work in Greece. 'Greeks are not fair, says a man from the village of Stegopul, the Albanians have rebuilt all their villages for a mouthful of bread, they could treat them better'. Lunxhot identity can thus be summarised as one Lunxhot from Gjirokastër does, by two negative statements: 'neither Islamised, nor Hellenised' (as islamizuar, as greqizuar), one related to religion, the other to national belonging. It can also be heard however that the non-islamisation of Lunxhëri is a result of the strong influence of Greece in the area, and that, without Greece, Lunxhëri would have become Muslim.

Particularly illustrative of the ambiguity of the position of the Lunxhotes is the case of the Greek language. Although Greek is acknowledged as the only written language in former times, the question [166] is: do the Lunxhotes speak Greek in addition to Albanian, or not? To what extent can they admit that they speak Greek without being Greek? Here is for instance what a retired school teacher from the village of Dhoksat says about the links between Lunxhëri and Ioannina, before the Second World War:

The relations with Ioannina where intense, people where trading, were going to the market there. Songs confirm that people from Lunxhëri could go there without any problem (...). Very few Lunxhotes had settled in Ioannina, because Lunxhëri is autochthonous (autokton) and does not speak Greek. Ioannina is Greece, and we, we are Albania. Relations with Ioannina are somehow superficial. The Albanian is Albanian, the Greek is Greek. In 1913, someone from Qestorat [in Lunxhëri] was leading the government of Northern Epirus (...) they wanted Korçë and Gjirokastër to be part of Greece. But Korçë and Gjirokastër remained Albanian. That man was not representative of all Lunxhëri (...). Greeks came during the first world war, in 1920, in 1941, but there was no problem: we call them dajko [mother's brother], because they were protecting us against the Turks. But no villager knows Greek. Even if until 1915 there was a Greek school in every village, there was no transmission. Old people know Greek.

Thus the use of Greek is acknowledged, in trade and education, but at the same time, the knowledge of Greek has to remain superficial, because Lunxhëri is 'autochthonous', as he says, that is Albanian, and the Albanians do not speak Greek, because they are Albanian. Though definitely Albanian, the language spoken by the Lunxhotes is nonetheless rich of numerous Greek borrowings, much more than, for instance, the language spoken by the Devoll Orthodox Christians.

Crossing the ethnic boundary and the state border

Obviously, the ambiguity of their position does not prevent the Lunxhotes from crossing the border. Although there are no figures on migration from Lunxhëri, personal observations reveal that the villages have lost a large part of their population, who moved to town or abroad, Greece being the main country of destination. The demographic changes, however, are not limited to the 1990s, and the new emigration is only a moment among others in the history of population movements and border crossings in the area. To keep things short, I will consider briefly the main three periods of this history.

[167] The first period goes up to the time between the creation of the border in 1913 and its closing in 1944. Up to that point, even though the ethnic boundary between Greek and Albanian Christians existed (as it appears in the relations of 19 th century travellers, for instance), national consciousness was generally too weak to prevent people from moving from one side of this boundary to the other. Moreover, this was the period of the kurbet, of the migration of both Greek-and Albanian-speaking Christians from Epirus towards Istanbul, Egypt and America.

The second period goes approximately from the creation of the border up to its opening in 1991. It is a time of crystallisation of national feelings (people have to choose their side) and of 'national enculturation', as Anastasia Karakasidou calls it for the Macedonian border area8 , meaning the process through which people come to identify with a national culture, rather than with their ethnic or local culture. This period culminated during the communist rule in Albania, when the border was closed and, at least from the Albanian point of view, separating two enemies, but already at the turn of the 20 th century, people started to split in filogrek (pro-Greek) and filoshqiptar (pro-Albanian). Both groups were in power at various times, but the former seem to have been powerful until the Second World War, such as in this story, from the village of Selckë:

Among the men of the village, there were filogrek [pro-Greek], and M. P. made a speech to the men and said that this village would be part of Albania. One of his cousins, M. N., told him to leave the village, the lineage T. was filogrek, and they would kill him. He took his wife along and went away, through the village of Sopik, for America. The T. later on went to Greece, abandoning their house, which is today inhabited by Vlachs.'

The third and last period started with the re-opening of the border in 1991. People started again to cross the border, but the national or nationalist context is still dominant, people do not perceive themselves outside national categories such as 'Albanian' and 'Greek', and tend to apply those same national categories to past events or periods, when they were not as relevant as they became later on. If it is a period of 'soft' border compared to the former, crossing is not as easy, and one must either be on the 'right' side (i.e. member of the Greek minority, or at least Orthodox Christian) or cheat and cross illegally.

It is no surprise then, that observation reveals that the Lunxhotes rely on existing networks to cross the border in the most legal way, rather than doing it clandestinely, as recent research has shown 9 . Most of those networks date back from the period of the kurbet, [168] and people try to reactivate them whenever possible to facilitate their entry in Greece and to rapidly find better job opportunities. Those networks are not, however, the only thing to be called back from the past and used in the present. The crossing of the ethnic boundary, and sometimes the crossing of the state border as well, are embedded in a wide range of memories and images of the past.

First of all comes the feeling of close contact and shared attributes between Lunxhotes and Greeks. By Greek are meant here both the Greeks from Greece and those of Albania, mainly inhabiting the Dropull and Pogon areas. All those areas are said to share the same high level of culture and development, as opposed to the Muslim areas. Intermarriage, even if it only occurs to a limited extent, is felt to be possible between those communities. It is also interesting to hear people telling stories about the Second World War, and especially of its first phase in Epirus, the Greek-Italian war of winter 1940-41.

The Greeks were not afraid here, says an old man from the village of Selckë, it is a Christian area, with a lot of minoritarë [members of the Greek minority]. They were much welcome. When they reached Gjirokastër, that was different; in Kardhiq and Golem, they had to fight against the Muslims.

Greek soldiers and officers were hosted in Lunxhëri (just as Italians had been a little earlier), where they found no difficulty to communicate in Greek and sometimes happened to meet people they already knew from Greece. There are cases of people originating from Lunxhëri, who had left Albania for Greece before the war, and came back at that time as Greek soldiers. It also happened that, after 1991, Albanians met the families of those soldiers and officers their parents or grand-parents had hosted in Lunxhëri.

School and church also seem responsible for this feeling of community, or at least of close ties, between Albanians and Greeks. As already mentioned, in most villages, the only school had been Greek up to the end of the First World War, and even later in some villages. Some people also say that when the Greeks occupied the area during the Italian-Greek war, men had to go to school and learn Greek. Greek was also the language of the church until 1966. Education and religion have therefore been the target of the Albanian 'nationalisation' or 'national enculturation' programmes. Already at the end of the 19 th century there was a dispute between the supporters of Greek education and those of Albanian education. Two villagers from Qestorat, one of Greek national consciousness, the other of Albanian national consciousness, fought over the school opened in 1874 by the former, Kristaq Zografi (or Christakis [169] Zographos). The school was eventually closed in 1891, but the Albanian Koto Hoxhi was beaten to death by his opponents in 1895. His statue has since been erected in front of the former Greek school.

The spread of Greek schools and religion also means that many school teachers and priests were of Greek background, coming either from Dropull and Pogon or from Greece. Several families of Lunxhëri thus trace back their origin to a Greek priest or school teacher who had arrived to work in the village and married there. Today, they nonetheless perceive themselves as Albanians. Crossings also happened in the other direction: quite a few people from Lunxhëri went to Greece to study (in Athens, Corfou and Voshtinë-Pogoniani) and did not come back to the village, where employment was not easy to find. It also happened, however, that in some periods and for some people, this contact with Greece was the cause of anti-Greek feelings and pro-Albanian commitment, as was the case for instance for Spiro Konda, born in 1862, who later wrote studies on the ancient history and origin of Albanians.

Although not originating from Lunxhëri, his example is worth mentioning, as particularly illustrative of the role of language in both creating and crossing boundaries: My work on this issue [i.e. the origin of Albanians] started when I was a pupil in the high school of Messolonghi (Greece). During the lessons on Iliad and Odyssey, my teacher was impressed by the fact that I was able to understand many words through the help of Albanian, words that sounded and had a meaning in Albanian. Later on, as I was a student at the University of Athens, I started to study this issue more seriously. I thus became aware that the Albanian language was the key to many questions raised by ancient history and ancient languages10 .

For others, marriage of relatives was a source of interest for Greece and Greek language. A villager from Saraqinishtë, born in the 1920s, explains: I went to the Albanian school for five years. I wanted to learn Greek. I talked to my teacher, but he was filoshqiptar [pro-Albanian], and said I should not learn it. I wanted to learn Greek because my sister was married in Poliçan [in Pogon, Greek minority], and I was going there quite often.

As can be seen, kinship and marriage also played a great role in maintaining a feeling of community and in facilitating the crossing of the ethnic boundary. Genealogies and family histories reveal a great number of intermarriages in which, at least in today's recollections, husband and wife [170] are said to be of different ethnic background. What is more important perhaps is not the intermarriages themselves (one could expect them in that context) but the fact that they are still perceived as the sign or evidence of both the existence and permeability of the ethnic boundary. In fact, the social and ethnic origin of women was (and still is) under heavy consideration before marriage, as it was assumed that women, while not going away in kurbet, were the guardians of local tradition and excellency.

Another way of crossing the boundary that is still remembered today is emigration. At the time of the kurbet, emigration did not mean automatically crossing of a state border: until 1912, Lunxhëri and Istanbul, for instance, where part of the same political unit, the Ottoman empire. But kurbet seems to have played a role in the construction and crossing of the ethnic boundary that was to become a national boundary and state border after 1912. In Istanbul, as well as in Egypt and in the United-States, and furthermore in Athens, the Lunxhotes would live together with the 'Greeks', that is with other Orthodox Christians, but most of the time of Greek ethnic and/or national consciousness. They would therefore learn Greek (rather than Turkish, if we can rely on what is said today of the kurbet in Istanbul), work with Greeks, and, after the First World War and collapse of the Ottoman empire, go to Greece or follow the Greeks in their migration patterns (towards America and Australia). The crossing of the boundary was facilitated in the case of the migrants who, due to the closing of the border at the time of the Second World War, remained away from Albania and identified with the Greek migrants among whom they were living (and then, eventually, with the country they lived in).

Finally, crossings of the ethnic boundary are said to have occurred during the period of crystallisation of the national boundary and when the state border was closed. These have to do with kinship and intermarriages, as the number of the latter seems to have grown after 1944 (as members of the Greek minority could not marry in Greece any longer and had to turn to the Albanian-speaking Christians). They also had to do with the social and political environment in Albania at that time. It could thus be of interest for a member of the Greek minority to become Albanian, but crossings also happened the other way round. This was for instance the case of one village in Lunxhëri, as it was incorporated in one cooperative and one administrative unit with Greek-speaking villages in Pogon. As a local analyst states: Until 1945, people would rather declare themselves Greek, later Albanian, especially after 1967. In some families, cousins gone away [171] to Greece declare themselves Greek, while others who moved to Gjirokastër or Tiranë declare themselves Albanian. Today, people rather tend to declare themselves Greek.

All this has an impact on the way people cross the state border today. First of all, as already mentioned, the Lunxhotes use their direct or indirect knowledge of the other side in their migration. But the past should not be seen only as a tool in the present migration. Some Lunxhotes crossed the border at the very beginning of the 1990s with no 'migration strategy' in mind, but rather to meet relatives gone at the time of the war or to search information on properties their families used to own on the Greek side (this is specially true in Corfou). Even today, some people are aware of the existence of possible remote relatives in Greece, but do not try to take advantage of it to migrate, if they consider their situation to be good enough in Albania. For others, relatives or members of former filogrek families who had moved earlier to Greece are considered as potential support in educational rather than migration strategy: many Lunxhotes want their children to go and study in Greece rather than in Albania, and this seems safer with somebody to look on them in Greece.

Others crossed the border to attend religious festivals on the Greek side, or to get married at the church (what was forbidden in communist Albania), or have their children baptised. Plans for work migration seem to have come only later on, either as a consequence of these renewed relationships with the Greek side, either because there came a time when everybody was leaving. The example of the Vlachs or Aromanians inhabiting Lunxhëri seem to have been of importance here: just as the Lunxhotes, the Vlachs were able to find relatives on the Greek side, as most of them have their origin in the village of Mexhide-Kefalovrisso, in Greek Pogoni. But contrary to the Lunxhotes, they had suffered from persecution in Albania during communism, and were eager to move away from Albania. Actually, they were also the first ones to come back to Albania, built new houses and start private business (often transborder business), when Albanians were still expecting everything from the state.

When communism collapsed, many Lunxhotes looked towards the towns and cities of central Albania, where some of their relatives or neighbours had already moved in the 1950s and 1960s, and started to leave the villages for the urban centres. Emigration, in a way, was for the poor, for the Muslims and Vlachs, not for them. And still today there is a kind of pride in saying that you can go to Greece whenever you want, to go shopping, to visit friends, or go to doctors and hospitals. This sense of distinction, or of 'aristocracy', as they say, might also explain that, [172] generally speaking, the Lunxhotes tend to cross legally and work legally in Greece. As one of them says, 'we, the Albanians from the borderland, we are considered close to them by the Greeks, because we are Orthodox. We don't need to cheat '. On the contrary, 'the Labes are using false documents, as if they were kaur (Christians), they are getting baptised.' On the other hand, assistance from the ones on the Greek side is not looked for only by those who migrate. People who remain in the villages are willing to apply to the 'retirement of the Greek' (pensioni i grekut), i.e. financial assistance offered by Greece to members of its minority in Albania, and to that aim they have to declare themselves Greek.

All this in turn has two main aspects: the first one is that while making efforts to obtain visas and documents, people tend to take advantage of relatives and kinship relations which might not be considered relevant in the local kinship system, such as relations on the mother's side, for instance. A villager of Saraqinishtë, who did not know Greek before the 1990s, has nonetheless got a certificate stating he is a member of the Greek minority, because his mother was from the village of Selckë, and though Albanian-speaking, Selckë is included in a single administrative unit with the Greek-speaking villages of Pogon. After telling how he became friends with an old man living in Athens, while he was looking for his mother's brother's wife, a villager from Selckë explains:

We Albanians, even the Northern Epirotes [i.e. Orthodox Christians], are not considered very well by the Greeks. This is why I hold on to this friendship with the old Greek man. Who knows? It can be of some help.

It also develops a growing interest for genealogies and onomastics.

The second aspect is that formalities and difficulties to which migrants are confronted are felt as something extremely unfair for people who consider themselves close to Greece and to the Greeks, who have or had relatives or properties in Greece, and whose collective identity lays to a large extent on another migratory experience, that of the kurbet, perceived today as a golden age, and as the origin and justification of the feeling of aristocracy that characterises the relations between the Lunxhotes and the others, Muslims and Vlachs.

It can thus be said that from the Lunxhotes' point of view (as probably for other border communities), the state border is more than a political line separating two countries, more than an obstacle to migration. It is part of local history and local construction of collective identity. Due to the ambiguity of the Albanian Christians in the national [173] landscape of the border area, and to the fluidity of the ethnic boundary that comes along with this ambiguity, crossing the state border since the 1990s has had an impact on both the self-perception of the Lunxhotes and their relations with the other communities inhabiting the area. Crossing the border does not happen in the same way for the different groups, and it does not have the same meaning.

Conclusion

I have considered here the impact of the state border on the definition of local communities and on their relations. I have also considered how this impact determines, in turn, the way the different communities perceive and cross the border. I believe that the dynamic, or diachronic, aspect of borders and border crossing is extremely important. We may say that the creation of the international border between Greece and Albania, and its changing status through different periods of time, has brought as a consequence the reorganizing, or renegotiation of ethnic relations and boundaries in the border area. This process is connected to various aspects of social relations, of which migration may be the most well known, or at least the most studied, but we would have much to learn by looking at kinship relations and family organisation, for instance, as these fields can also be crossed by ethnic boundaries. Particularly interesting would be to study the articulation between the rhetorics of blood descent and the perception of ethnic boundaries, just as when the Lunxhotes say 'we are not Greek, but...', meaning that they are not of Greek descent, or do not belong to the Greek nation as long as it is based on common descent, but that apart from that, they have too many things in common not to form a kind of transborder community.

In this respect, although it is obviously beyond the scope of this article, it would be necessary to have a closer look at the other side of the border in terms of narratives and representations from the Greek border zone, but also with regard to the role of shared institutions and ways of life on both sides, as they seem to have been important at least until the Second World War, and are, again involved in the relations and mutual representations of Greeks and Orthodox Christians from South Albania.

Peter Sahlins, Boundaries: the Making of France and Spain in the Pyrénées(Berkeley, 1989).

Hastings Donnan and Thomas M. Wilson, Borders: Frontiers of Identity,Nation and State (Oxford, 1999).

Henk Driessen, 'The 'new immigration' and the transformation of the European-African frontier', in Thomas M. Wilson and Hastings Donnan (eds.), Border Identities. Nation and state at international frontiers (Cambridge, 1998), p. 101.

For an example of this behavior, see Gilles de Rapper, 'Grenzen überschreiten: Migration in der albanischen Grenzregion Devoll', in Karl Kaser, Robert Pichler and Stephanie Schwandner-Sievers (eds.), Die weite Welt und das Dorf. Albanische Emigration am Ende des 20. Jahrhunderts(Wien, 2002), p. 105.

It is interesting to note that this is not the only forced change of names mentioned in the area. As a villager from Selckë says: 'The [communist] regime imposed Muslim names on Christians, such as Agron, Ilir, etc.' (which in fact are not Muslim at all, but have been common among the Muslims, specially after 1976 and the ban on religion-based names).

Elie Kedourie, Nationalism² (London, 1966), p. 124.

Gilles de Rapper, 'Espace et religion : chrétiens et musulmans en Albanie du Sud', Etudes Balkaniques, Cahiers PierreBelon, 9 (2002), pp. 17-39.

Anastasia N. Karakasidou, 'Women of the Family, Women of the Nation: National Enculturation among Slavspeakers in North-west Greece', in Peter Mackridge and Eleni Yannakakis (eds.), Ourselves and Others. The Development of a Greek Macedonian Identity Since 1912 (Oxford, 1997), p

. 91. 9 Pierre Sintès, 'Structure de la migration albanaise en Grèce : le rôle des réseaux préexistants', paper given at the conference 'Etudes balkaniques : état des savoirs et pistes de recherche', Paris, 2002.

Spiro N. Konda, 'Albanët (shqiptarët) dhe problemi pellazgjik', Buletin i universitetit shtetëror të Tiranës. Seria Shkencat shoqerore,XVI, 3 (1962), p. 182.