Emmanuel Hallouin
email: hallouin@univ-tlse2.fr

Marc Perret
email: perret@univ-tlse2.fr

From Hodge Index Theorem to the number of points of curves over finite fields

Keywords: AMS classification : 11G20, 14G05, 14G15, 14H99 Curves over a finite field, rational point, Weil bound

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve X over a finite field in term of euclidean relationships between the Neron Severi classes in X × X of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a second order Weil upper bound, to establish a new third order Weil upper bound, and using magma to produce numerical tables for higher order Weil upper bounds. We also give some interpretation for the defect of exact recursive towers, and give several new bounds for points of curves in relative situation X → Y .

Introduction

Let X be an absolutely irreducible smooth projective curve defined over the finite field F q with q elements. The classical proof of Weil Theorem for the number of F q -rational points X(F q ) rests upon Castelnuovo identity [START_REF] Weil | Sur les courbes algébriques et les variétés qui s'en déduisent[END_REF], a corollary of Hodge index Theorem for the smooth algebraic surface X × X. The intent of this article is to push further this viewpoint by forgetting Castelnuovo Theorem. We come back to the consequence of Hodge index Theorem that the intersection pairing on the Neron Severi space NS(X × X) R is anti-euclidean on the orthogonal complement E X of the trivial plane generated by the horizontal and vertical classes. Thus, the opposite •, • of the intersection pairing endows E X with a structure of euclidean space. Section 1 is devoted to few useful scalar products computations.

In section 2, we begin by giving a proof of Weil inequality which is, although equivalent in principle, different in presentation than the usual one using Castelnuovo Theorem given for instance in [Har77, exercice 1.9, 1.10 p. 368] or in [Sha94, exercises 8, 9, 10 p. 251]). We prove that Weil bound follows from Cauchy-Schwartz inequality for the orthogonal projections onto E X of the diagonal class Γ 0 and the class Γ 1 of the graph of the Frobenius morphism on X.

The benefit of using Cauchy-Schwartz instead of Castelnuovo is the following. We do not know what can be a Castelnuovo identity for more than two Neron Severi classes, while we do know what is Cauchy-Schwartz for any number of vectors. It is wellknown that Cauchy-Schwartz between two vectors is the non-negativity of their 2 × 2 Gram determinant. Hence, we are cheered on investigating the consequences of the non-negativity of larger Gram determinants involving the Neron Severi classes Γ k of the graphs of the k-th iterations of the Frobenius morphism.

For the family Γ 0 , Γ 1 , Γ 2 , we recover the well known Ihara bound [START_REF] Ihara | Some remarks on the number of rational points of algebraic curves over finite fields[END_REF] which improves Weil bound for curves of genus greater than g 2 = √ q( √ q-1)

2

, a constant appearing very naturally with this viewpoint in section 2.5.2, especialy looking at figure 1 in section 2.5.2. It follows that the classical Weil bound can be seen as a first order Weil bound, in that it comes from the euclidean constraints between Γ 0 and Γ 1 , while the Ihara bound can be seen as a second order Weil bound, in that it comes from euclidean constraints between Γ 0 , Γ 1 and Γ 2 . Moreover this process can be pushed further: by considering the family Γ 0 , Γ 1 , Γ 2 and Γ 3 , we obtain a new third order Weil bound (Theorem 18), which improves the Ihara bound for curves of genus greater than another constant g 3 = √ q(q-1) √

2 . The more the genus increases, the more the Weil bound should be chosen of high order in order to be optimal. Therefore it is useful to compute higher order Weil bounds. Unfortunately, establishing them explicitely requires the resolution of high degree one variable polynomial equations over R. For instance, the usual Weil bound requires the resolution of a degree one equation, Ihara and the third order Weil bounds require the resolution of second order equations, while fourth and fifth order Weil bounds require the resolution of third degree equations, and so on. Moreover, a glance at Ihara second order Weil bound and at our explicit third order Weil bound will convince the reader that they become more and more ugly as the order increase.

Hence, we give up the hope to establish explicit formulae for the Weil bounds of order from 4. We then turn to a more algorithmic point of view. We use an algorithm which, for a given genus g and a given field size q, returns the best upper order Weil bound for the number of F q -rational points of a genus g curve, together with the corresponding best order n. The validity of this algorithm requires some results proved in the second section. In the figure below, we represent the successive Weil bounds (in logarithmic scales) of order from 1 to 5. Note that taking into account the logarithmic scale for the y-axis, higher order Weil bounds become significantly better than usual Weil's one.

where green third order weil bound becomes the best. From some genus g4, light blue fourth order Weil bound is the best up to some genus g5, where dark blue fifth order Weil bound becomes better, and so on. Here, we have chosen q = 2, for which the genera g2, g3, g4, g5 are particularly small, respectively about 0.3, 1, 2.35 and 4.67. This means for instance that the best bound for q = 2 and g = 3, 4 is the fourth order one!

In order to illustrate the efficiency of our algorithm, we display in section 2.6.3 a numerical table.

We acknowledge that for any pair (g, q) we have tested, a comparison of our numerical results with the table given on the website http://www.manypoints.org/ reveals that we always recover the very same numerical upper bound for N q (g) than those coming from Oesterlé bounds! We were not been able to understand this experimental observation.

Nevertheless, we think that the viewpoint introduced in this article is preferable to Osterlé's one. First, our viewpoint is more conceptual in nature. Any constraints we use to obtain our bounds come in a quite pleasant way either from algebraic-geometry or from arithmetic, as explained in the introduction of section 2 in which we outline our approach. We think moreover that the graph displayed just above is very satisfying.

Second, the viewpoint introduced in this article is perfectly adapted to the study of bounds for the numbers of rational points. As the reader can see, many1 known results can be understood with this viewpoint -even asymptotical ones, and new results can be proved. We are pretty sure that we have not extracted all potential outcomes of this viewpoint. Third, new questions can be raised from this viewpoint. We propose a few in Section 4.

To conclude Section 2, we push to the infinite-order Weil bound using the nonnegativity of the Gram determinants of any orders, which imply that some symmetric matrix is semi-definite positive. Applied to some very simple vector, this lead us to Theorem 22, a stronger form of [START_REF] Michael | Some remarks on the asymptotic number of points, Coding theory and algebraic geometry[END_REF] Tsfasman bound in that it gives a new interpretation for the defect of an exact tower as a limit of nice euclidean vectors in (E, •, • ).

In Section 3, we study the relative situation. Given a finite morphism f : X → Y between two absolutely irreducible smooth projective curves defined over F q , the pullback functor on divisors from the bottom algebraic surface Y × Y to the top one X × X induces a map from the Neron Severi space NS(Y ×Y ) R to NS(X ×X) R . Restricting this map to the subspaces F X and F Y generated by the classes of the graphs of iterations of the Frobenius morphisms, and after a suitable normalization (which differs from the normalization chosen in Section 2), we prove that it becomes an isometric embedding. We thus have an orthogonal decomposition

F X = F * Y ⊕ F X/Y , involving a relative subspace F X/Y of F X .
Now, Cauchy-Schwartz inequality applied to the orthogonal projections of Γ 0 X and Γ 1 X on the relative space F X/Y is equivalent to the well known relative Weil bound that

| X(F q ) -Y (F q )| ≤ 2(g X -g Y ) √ q.
With regard to higher orders relative Weil bounds, we encounter a difficulty since the arithmetical constraints X(F q r ) -Y (F q r ) ≥ X(F q ) -Y (F q ) for r ≥ 2, similar to that used in Section2 2 does not hold true! Hence, the only constraints we are able to use are the non-negativity of Gram determinants. This lead to an inequality relating the quantities X(F q r ) -Y (F q r ) (corollary 29). We also prove a bound involving four curves in a cartesian diagram under some smoothness assumption (Theorem 33).

As stated above, we end this article by a very short Section 4, in which we raise a few questions.

1 The euclidean space (E, •, • ) Let X be an absolutely irreducible smooth projective curve defined over the finite field F q with q elements. We consider the Neron-Severi space NS(X × X) R = NS(X × X) ⊗ Z R of the smooth surface X × X. It is well known that the intersection product of two divisors Γ and Γ on X × X induces a symmetric bilinear form on NS(X × X) R , whose signature is given by the following Hodge Index Theorem (e.g. [START_REF] Hartshorne | Algebraic geometry[END_REF]).

Theorem 1 (Hodge index Theorem). The intersection pairing is non-degenerate on the space NS(X × X) R , and is definite negative on the orthogonal supplement of any ample divisor.

Let H = X × { * } and V = { * } × X be the horizontal and vertical classes on X × X. Their intersection products are given by H

• H = V • V = 0 and H • V = 1.
The restriction of the intersection product to Vect(H, V ) has thus signature (1, -1). Moreover Vect(H, V ) ∩ Vect(H, V ) ⊥ = {0}. By non-degeneracy, this gives rise to an orthogonal decomposition NS(X × X) R = Vect(H, V ) ⊕ Vect(H, V ) ⊥ . Let p be the orthogonal projection onto the non-trivial part Vect(H, V ) ⊥ , given by

p : NS(X × X) R -→ Vect(H, V ) ⊥ Γ -→ Γ -(Γ • V )H -(Γ • H)V. (1)
Since H + V is ample, the intersection pairing is definite negative on Vect(H, V ) ⊥ by Hodge index Theorem. Hence, Vect(H, V ) ⊥ can be turned into an euclidean space by defining a scalar product as the opposite of the intersection pairing.

Definition 2. Let E = Vect(H, V ) ⊥ . We define on E a scalar product, denoted by

•, • , as γ, γ = -γ • γ , ∀γ, γ ∈ E.
The associated norm on E is denoted by • .

In the sequel of this article, all the computations will take place in the euclidean space (E, •, • ). To begin with, let us compute this pairing between the iterates of the Frobenius morphism.

Lemma 3. Let F : X → X denotes the q-Frobenius morphism on X, and F k denotes the k-th iterate of F for any k ≥ 0, with the usual convention that F 0 = Id X . We denote by Γ k the Neron Severi class of the graph of F k . Then

p(Γ k ), p(Γ k ) = 2gq k ∀k ≥ 0 p(Γ k ), p(Γ k+i ) = q k (q i + 1) -X(F i q ) ∀k ≥ 0, ∀i ≥ 1. Proof -Since the morphism F k is a regular map of degree q k , one has Γ k • H = q k and Γ k • V = 1. Now, let k ∈ N and i ∈ N * . We consider the map π = F k × Id : X × X → X × X, which sends (P, Q) to (F k (P ), Q). We have π * (∆) = Γ k and π * Γ k+i = q k Γ i
, so that by projection formula for the proper morphism π:

Γ k • Γ k+i = π * (∆) • Γ k+i = ∆ • π * Γ k+i = q k ∆ • Γ i . If i ≥ 1, then ∆ • Γ i = X(F q i ). If i = 0, then Γ 0 = ∆ and ∆ • Γ 0 = ∆ 2 = -2g. The results follow since p(Γ k ) = Γ k -H -q k V by (1). Remark -Of course, dim E ≤ dim N S(X × X) R -2 ≤ 4g 2 .
But as the reader can check, we are working along this article only on the subspace F of E generated by the family p(Γ k ), k ≥ 0. By [START_REF] Zariski | Algebraic surfaces[END_REF] chapter VII, appendix of Mumford), any trivial linear combination of this family is equivalent to the triviality of the same linear combination of the family ϕ k , k ≥ 0, of the iteration of the Frobenius endomorphism ϕ on the Tate module V (Jac X) for any prime ∧ q = 1. We deduce that for a given curve X over Fq of genus g, we actually are working in an euclidean space FX of dimension equal to the degree of the minimal polynomial of ϕ on V (Jac X).

Absolute bounds

We observe, as a consequence of Lemma 3, that the non-negativity

0 ≤ Gram(p(Γ 0 ), p(Γ 1 )) = 2g q + 1 -X(F q ) q + 1 -X(F q ) 2gq = (2g √ q) 2 -(q + 1 -X(F q )) 2
is nothing else than Weil inequality. In this Section, we give other bounds using larger Gram determinants. For this purpose, we need preliminary notations, normalizations and results. Normalizations in Section 2.1 play two parts. They ease many formulae and calculations. They also make obvious that several features of the problem, such as the Gram matrix Gram(p(Γ i ); 0 ≤ i ≤ n), or the forthcoming n-th Weil domains W n , are essentially independent of q. The authors believe that these features deserve to be emphasized.

With regard to results, let n ∈ N * . To obtain the Weil bound of order n, let X be an algebraic curve of genus g defined over F q . We use both geometric and arithmetic facets of X as follows. We define from X in (4) a point P n (X) ∈ R n , whose abscissa x 1 given in (3) is essentially the opposite of X(F q ), hence have to be lower bounded. The algebraic-geometric facet of X implies, thanks to Hodge index Theorem, that some Gram determinant involving the coordinates of P n (X) is non-negative. This is traduced on the point P n (X) in Lemma 12 in that it do lies inside some convex n-th Weil domain W n , studied in Subsection 2.3. The arithmetic facet of X is used via Ihara constraints, that for any r ≥ 2 we have X(F q r ) ≥ X(F q ). We then define in Subsection 2.4 the convex n-th Ihara domain H g n in genus g. We state in Proposition 16 that for any curve X of genus g defined over F q , we have P n (X) ∈ W n ∩ H g n . We are thus reduced to minimize the convex function x 1 on the convex domain W n ∩ H g n . To this end, we establish in Subsection 2.4 the optimization criteria 17, decisive for the derivations of higher order Weil bounds.

This criteria is used to prove new bounds in Subsection 2.6. Finally, we obtain in Subsection 2.7 a refined form of Tsfasman upper bound for asymptoticaly exact towers.

The Gram matrix of the normalized classes of iterated Frobenius

In this Subsection, we choose to normalize the Neron-Severi classes of the iterated Frobenius morphisms as follows. For any n ≥ 1, and any curve X defined over F q of genus g = 0, we define a point P n (X) ∈ R n whose x i -coordinates is very closely related to X(F q i ).

Definition 4. Let X be an absolutely irreducible smooth projective curve defined over the finite field F q . For k ∈ N, we put

γ k = 1 2gq k p(Γ k ) ∈ E, (2)
so that by Lemma 3 we have γ k , γ k = 1. For i ∈ N * , we put

x i def. = γ k , γ k+i lemma 3 = (q i + 1) -X(F q i ) 2g √ q i . ( 3
)
For any n ≥ 1, we define the point From this Definition 4 and Lemma 3, the Gram matrix of the family γ 0 , . . . , γ n in E is

P n (X) = (x 1 , . . . , x n ) ∈ R n . (4
Gram(γ 0 , . . . , γ n ) =          1 x 1 • • • x n-1 x n x 1 . . . . . . x n-1 . . . . . . . . . . . . . . . x n-1 . . . . . . x 1 x n x n-1 • • • x 1 1         
(5)

Some identities involving Toeplitz matrices

The main result of this Subsection is Lemma 5, providing the existence of the factorization in Definition 6, in which the G - n factor plays a fundamental part in the following of the article.

A symmetric Toeplitz matrix is a symmetric matrix whose entries x i,j depend only on |i -j|, that is are constant along the diagonals parallel to the main diagonal [HJ90, §0.9.7], that is

T n+1 (x 0 , x 1 , . . . , x n ) =          x 0 x 1 • • • x n-1 x n x 1 . . . . . . x n-1 . . . . . . . . . . . . . . . x n-1 . . . . . . x 1 x n x n-1 • • • x 1 x 0          (6)
The symmetric Toeplitz matrix is said to be normalized if x 0 = 1. An Hankel matrix is a matrix whose entries x i,j depend only on i + j, that is are constant along the anti-diagonals parallel to the main anti-diagonal [HJ90, §0.9.8], that is

H n+1 (x 0 , x 1 , . . . , x 2n ) =          x 0 x 1 x 2 • • • x n x 1 . . . . . . x n+1 x 2 . . . . . . . . . . . . . . . . . . x 2n-1 x n x n+1 • • • x 2n-1 x 2n          (7)
Remark -It should be noticed that in this article, any matrix is indexed by its size, while this doesn't hold for determinants, for instance see formulae ( 9) and (10).

Lemma 5. For any n ≥ 1, we abbreviate T n (x 0 , . . . , x n-1 ) by T n . The determinant of this Toeplitz matrix factorizes as

Det (T 2n ) = Det (T n + H n (x 2n-1 , . . . , x 1 )) × Det (T n -H n (x 2n-1 , . . . , x 1 ))
and

Det (T 2n+1 ) = Det T n + H n (x 2n , . . . , x 2 ) t X n 2X n x 0 × Det (T n -H n (x 2n , . . . , x 2 )) ,
where X n = (x n , . . . , x 1 ).

Proof -Even size. The operations on the columns C j ← C j + C 2n-(j-1) for 1 ≤ j ≤ n, followed by the operations on the lines L n+i ← L n+i -L n-(i-1) for 1 ≤ i ≤ n lead to the determinant of a 2 × 2 blocks upper triangular matrix whose value is the expected product.

Odd size. The operations on the columns C j ← C j + C 2n+1-(j-1) for 1 ≤ j ≤ n, followed by the operations on the lines L n+1+i ← L n+1+i -L n+1-i for 1 ≤ i ≤ n lead to the result in the same manner. Definition 6. For n = 0, we put G 0 = G - 0 = G + 0 = 1, and for n ≥ 1, we put

G n (x 1 , . . . , x n ) def. = Det(T n+1 (1, x 1 , . . . , x n )) Lemma 5 = G - n (x 1 , . . . , x n ) × G + n (x 1 , . . . , x n ),
where the last factorization is the one proved in Lemma 5 in the same order.

Remark -The determinant Gn is the determinant of a matrix of size n + 1. Note that for n = 2m + 1 odd, both G - n and G + n are polynomials of degree m + 1 in x1, . . . , xn, while for n = 2m even, G - n has degree m + 1, while G + n have degree m. We will see later in the article that Weil bound of order n for X(Fq) depends heavily on the hypersurface

{G - n = 0}.
In order to raise any ambiguity and for later purpose, let us write down these determinants for n = 1, 2 and 3:

G 1 (x 1 ) = 1 x 1 x 1 1 = (1 + x 1 ) G - 1 (x 1 ) × (1 -x 1 ) G + 1 (x 1 ) (8)
G 2 (x 1 , x 2 ) = 1 x 1 x 2 x 1 1 x 1 x 2 x 1 1 = 1 + x 2 x 1 x 1 + x 1 1 G - 2 (x 1 ,x 2 ) × (1 -x 2 ) G + 2 (x 1 ,x 2 ) (9) G 3 (x 1 , x 2 , x 3 ) = 1 x 1 x 2 x 3 x 1 1 x 1 x 2 x 2 x 1 1 x 1 x 3 x 2 x 1 1 = 1 + x 3 x 1 + x 2 x 1 + x 2 1 + x 1 G - 3 (x 1 ,x 2 ,x 3 ) × 1 -x 3 x 1 -x 2 x 1 -x 2 1 -x 1 G + 3 (x 1 ,x 2 ,x 3 ) (10)
Lemma 7. For any n ≥ 1 and = ± or nothing, we abbreviate

G n (x 1 , . . . , x n ) by G n .
Then one has

2G n = G + n-1 × G - n+1 + G - n-1 × G + n+1 .
Proof -Let S - n+1 be the matrix obtained from T n by adding t (x n , . . . , x 1 ) to the first column, and S + n+1 be the matrix obtained from T n by removing t (x n , . . . , x 1 ) to the first column. Then by multilinearity, there exists a polynomial R n such that:

Det S - n+1 = G n + R n and Det S + n+1 = G n -R n .
On the other hand, using the same transformations as in the proof of Lemma 5, one get

Det(S - n+1 ) = G + n-1 G - n+1 and Det(S + n+1 ) = G - n-1 G + n+1
Adding Det(S - n+1 ) and Det(S + n+1 ) allows us to conclude.

The domain of positive definite normalized Toeplitz matrices

In this Subsection, we study for any n ≥ 1 some domain W n ⊂ R n , whose closure W n also plays a fundamental part in this article.

Definition 8. Let n ∈ N * . We denote by W n the set of (x 1 , . . . , x n ) ∈ R n , such that the symmetric normalized Toeplitz matrix T n+1 (1, x 1 , . . . , x n ) is positive definite.
The domain W n can be characterized in several useful ways.

Proposition 9. For n ≥ 1, the domain W n is a convex subset of ]-1, 1[ n , which can be written as follows.

1. First, one has:

W n = {(x 1 , . . . , x n ) ∈ R n | G i (x 1 , . . . , x i ) > 0, ∀ 1 ≤ i ≤ n} .
2. Recursively, W 1 = ]-1, 1[ and for any n ≥ 2

W n = {(x 1 , . . . , x n-1 , x n ) ∈ W n-1 × R | G n (x 1 , . . . , x n ) > 0} = (x 1 , . . . , x n-1 , x n ) ∈ W n-1 × R | G - n (x 1 , . . . , x n ) > 0 and G + n (x 1 , . . . , x n ) > 0 .
3. W n is also the set of points between the graphs of two functions from W n-1 to R:

for ε = ±, there exists a polynomial G ε n ∈ Q[x 1 , . . . , x n-1 ], such that G n = -G n-2 x n + G n and W n = (x 1 , . . . , x n-1 , x n ) ∈ W n-1 × R | G - n (x 1 , . . . , x n-1 ) G - n-2 (x 1 , . . . , x n-2 ) < x n < G + n (x 1 , . . . , x n-1 ) G + n-2 (x 1 , . . . , x n-2 )
.

Proof -To prove the convexity, we remark that both sets of normalized symmetric Toeplitz matrices and of symmetric positive definite matrices are convex, so that W n is convex. Moreover,

W n ⊂ ]-1, 1[ n because if T n+1 (1, x 1 , . . . , x n
) is positive definite, then all its principal minors are positive. In particular, for any 1 ≤ i ≤ n, the 2 × 2 minors 1 x i x i 1 is positive. Item (1) follows from the well known fact that an n × n symmetric matrix is definite positive if and only if all its n leading principal minors (obtained by deleting the i last rows and columns for 0 ≤ i ≤ n -1) are positive [HJ90, Theorem 7.2.5]. Hence, the first characterization of item (2) follows from item 1 . To prove the second characterization of item (2), we use Lemma 7 stating that 2G

n-1 = G - n-2 G + n + G + n-2 G - n .
By induction, this allow us to prove that both factors are positive if, and only if, the product

G n (x 1 , . . . , x n ) = G - n (x 1 , . . . , x n )G + n (x 1 , . . . , x n )
is positive for any (x 1 , . . . , x n-1 ) ∈ W n-1 .

To prove item (3), for = ±, formulae defining the polynomials G n in Lemma 5 and Definition 6 imply, developing along their first column and taking advantage of the very particular forms (6) and (7), that both polynomials have degree 1 in x n , of the form

G n = -G n-2 x n + ε G n (x 1 , . . . , x n-1 ) for some G n ∈ Q[x 1 , . . . , x n-1 ]. Since by item 2 we have G ε n-2 > 0, the set W n is thus equal to the set of (x 1 , . . . , x n-1 , x n ) ∈ W n-1 × R, such that: G - n (x 1 , . . . , x n-1 ) G - n-2 (x 1 , . . . , x n-2 ) < x n < G + n (x 1 , . . . , x n-1 ) G + n-2 (x 1 , . . . , x n-2 )
, and the proof is complete.

The following Proposition is useful for later purpose in Section 2.6, where convexity plays a quite important part.

Proposition 10. The locus {(x 1 , . . . , x n ) ∈ W n-1 × R | G - n (x 1 , . . . , x n ) > 0} is convex, while the locus {(x 1 , . . . , x n ) ∈ W n-1 × R | G + n (x 1 , . . . , x n ) > 0} is concave. Proof -By item 3 of Proposition 9, W n is the set of (x 1 , . . . , x n-1 , x n ) ∈ W n-1 × R such that
x n is between two functions from W n-1 to R. By convexity of W n from Proposition 9, the lower function must be concave has a function on W n-1 while the upper one must be convex. The Proposition follows easily.

The closure of W n is given by the following Proposition.

Proposition 11. The closure W n of W n in R n corresponds to the set of (n+1)×(n+1) normalized symmetric Toeplitz matrices which are positive semi definite. Moreover, we have

W n = {(x 1 , . . . , x n ) ∈ R n | G n,I (x 1 , . . . , x i ) ≥ 0, ∀I ⊂ {0, 1, . . . , n}} ,
where, for I ⊂ {0, 1, . . . , n}, we denote by G n,I (x 1 , . . . , x n ) the principal minor of the normalized symmetric Toeplitz matrix T n+1 (1, x 1 , . . . , x n ) obtained by deleting the lines and columns whose indices are not in I.

Proof -This is a consequence of the fact that a matrix is positive semi definite if and only if all the principal minors (the ones obtained by deleting the same subset of lines and columns) are non-negative (i.e. ≥ 0) [HJ90, last exercise after Theorem 7.2.5].

The curves locus

In this Subsection, we introduce Ihara's constraints and the resulting n-Ihara domain H g n for genus g. We then gather the key results for the derivation of higher order Weil bounds. The first one is Proposition 16, the second one is criteria 17.

2.4.1

The n-the Weil domain W n Lemma 12. Let X be an absolutely irreducible smooth projective curve defined over F q . Then the point P n (X) = (x 1 , . . . , x n ), as defined in (4), belongs to W n .

Proof -By (3), we have x i = γ k , γ k+i where the γ k are defined in (2), so that it is easily seen that for any I ⊂ {0, 1, . . . , n}, we have

G n,I (x 1 , . . . , x n ) = Gram(γ i , i / ∈ I),
for G n,I s defined in Proposition 11. The non-negativity of G n,I follows as a Gram determinant in an euclidean space, hence the Lemma by Proposition 11.

Definition 13. W n is called the n-th Weil domain.

As a first illustration of the informations contained in these Weil domains, we prove the following simple result. From (9), the non-negativity of the Gram determinants G - 2 writes x 2 ≥ 2x 2 1 -1. Taking (3) into account, this gives immediately Proposition 14. For any curve X of genus g = 0, we have

X(F q 2 ) -(q 2 + 1) ≤ 2gq - 1 g X(F q ) -(q + 1) 2 .
This Proposition means that, for a given non rational curve X, any lower bound for the deviation of X(F q ) to q + 1 yields to a better upper bound than Weil's one for X(F q 2 ). In the same way, for any given order n, the non-negativity of the 3 × 3 determinant Gram(γ 0 , γ 1 , γ n ) gives a quite ugly upper bound of similar nature for X(F q n ) in terms of X(F q ) and X(F q n-1 ).

Remark -Note that this Proposition is a refinement of the following well known particular case: if X is either Weil maximal or minimal over Fq, then X(Fq) -(q + 1) = ±2g √ q, and this Proposition asserts that then X(F q 2 ) -(q 2 + 1) ≤ 2gq -4g 2 q g = -2gq, so that X is Weil minimal over F q 2 . Note also that curves such that this inequality is an equality are those curves such that the corresponding point P2(X) = (x1, x2) ∈ W2 lies on the bottom parabola x2 = 2x 2 1 -1 of the Ihara domain drawn in figure 1 below. The above particular case is that of curves corresponding to the corner points (-1, 1) and (1, 1).

Ihara constraints : the n-th Ihara domain H g n in genus g

If (x 1 , . . . , x n ) ∈ R n comes from a curve X over F q by formulae (3), then we have seen in Lemma 12 that (x 1 , . . . , x n ) lies on the closure W n of W n . But there are also other constraints, resulting from the arithmetical inequalities X(F q i ) ≥ X(F q ) for any i ≥ 1. These inequalities, by (3), write ∀i ≥ 2,

x i ≤ x 1 q i-1 2 + q i-1 -1 2gq i-2 2 . ( 11
) Let α = 1 √ q . ( 12
)
For g ≥ 0, and n ≥ 2, i ≥ 2, we define

h g i (x 1 , x i ) = x i -α i-1 x 1 - 1 2gα 1 α i-1 -α i-1 . ( 13
)
Then it is easily seen that ( 11) is equivalent to

h g i (x 1 , x i ) ≤ 0. ( 14
)
We now define Definition 15. Let n ≥ 1 and g ≥ 0. We define:

1. The n-th Ihara domain in genus g as

H g n = {(x 1 , . . . , x n ) ∈ R n | h g i (x 1 , x i ) ≤ 0, ∀ 2 ≤ i ≤ n} (15)
2. The n-th Ihara line in genus g as

L g n = {(x 1 , . . . , x n ) ∈ R n | h g i (x 1 , x i ) = 0, ∀ 2 ≤ i ≤ n} . (16)
Each L g n is a line which is identified with R using the first coordinate x 1 as a parameter. We denote by

P g n (x 1 ) = x 1 , αx 1 + q -1 2g , α 2 x 1 + q 2 -1 2g √ q , . . . , α n-1 x 1 + q n-1 -1 2gq n-2 2 (17)
the point of L g n with parameter x 1 . In the same way, for g = ∞, we define the n-th Ihara infinite line by

L ∞ n = (x 1 , . . . , x n ) ∈ R n | x i = α i-1 x 1 , 2 ≤ i ≤ n .
It follows from this the following Proposition.

Proposition 16. Let X be an absolutely irreducible smooth projective curve defined over the finite field F q and n ≥ 2. Then P n (X) = (x 1 , . . . , x n ) ∈ W n ∩ H g n , where the x i 's are defined from X by (3).

From this Proposition 16 and the remark below Definition 6, the strategy is the following.

Strategy. -For each n ≥ 1, we minimize the coordinate function x 1 (P ) for P lying inside the compact convex domain W n ∩ H g n . By Proposition 16, this leads to a lower bound for x 1 (P n (X)) for any curve X of genus g over F q , hence by (3) to an upper bound for the number X(F q ). It turns that for given q and genus g, this bound is better and better for larger and larger n, up to an optimal one. We are thus face to an optimization problem. Since the domain W n ∩ H g n on which we have to minimize the convex function x 1 is also convex by Proposition 9, one has the following necessary and sufficient characterization of the minimum from [HU96, Théorème 2.2] where in our case the active constraints are G - n = 0 and h g i = 0 for 2 ≤ i ≤ n, and where h g i are defined in (13). Let P 0 ∈ W n ∩ L g n . Suppose that:

• for any I ⊂ {1, . . . , n + 1} and ε = ± such that G ε n,I (P 0 ) = 0, there exists µ ε I ≥ 0,

• for any 2 ≤ i ≤ n, there exists µ i ≥ 0, such that

∇x 1 (P 0 ) - I⊂{1,...,n+1}
µ - I ∇G - n,I (P 0 ) -I⊂{1,...,n+1}

µ + I ∇G + n,I (P 0 ) + n i=2 µ i ∇h g i (P 0 ) = 0. (18) Then, x 1 (P 0 ) = min{x 1 | (x 1 , . . . , x n ) ∈ W n ∩ H g n }.
We deduce from that the following criteria.

Criteria 17 (for minimizing x 1 ).

If P 0 = (x 1 , . . . , x n ) ∈ W n ∩ L g n satisfies 1. G - n (P 0 ) = 0; 2. ∂ i G - n (P 0 ) ≥ 0 for all 2 ≤ i ≤ n; 3. n i=1 ∂ i G - n (P 0 )α i-1 > 0,
then P 0 minimizes the first coordinate x 1 on W n ∩ H g n .

Remark -Corollary 21 bellow states that the first requirement implies the third one.

Proof -Suppose that the assumptions of the criteria hold true. Then the system

           n i=1 ∂ i G - n (P 0 )α i-1 µ - ∅ = 1 µ 2 = ∂ 2 G - n (P 0 )µ - ∅ . . . µ n = ∂ n G - n (P 0 )µ - ∅
has a solution µ - ∅ > 0, µ 2 ≥ 0, . . . , µ n ≥ 0. As easily seen, this solution also satisfy

        1 0 . . . . . . 0         -µ - ∅         ∂ 1 G - n (P 0 ) ∂ 2 G - n (P 0 ) . . . . . . ∂ n G - n (P 0 )         + µ 2        -α 1 0 . . . 0        + • • • + µ n        -α n-1 0 . . . 0 1        =       0 . . . . . . 0       , hence (18)
holds with the choice µ I = 0 for any I = ∅ and µ + ∅ = 0, so that x 1

(P 0 ) = min{x 1 | (x 1 , . . . , x n ) ∈ W n ∩ H g n },
and the criteria is proved.

Weil bound and Ihara bound as bounds of order 1 and 2

We prove here that our viewpoint enables to deduce Weil bound and Ihara bound. We think that this Subsection is of interest since it shows on simple cases how this method works, especialy in the case of Ihara bound thanks to figure 1.

Weil bound as a bound of order 1

With this viewpoint, the usual Weil bound comes from Cauchy-Schwartz inequality applied to γ 0 and γ 1 . Indeed, one have by ( 8)

Gram(γ 0 , γ 1 ) = 1 x 1 x 1 1 ≥ 0, that is |x 1 | ≤ 1.
By (3), we have just recovered the Weil bound

| X(F q ) -(q + 1)| ≤ 2g √ q (Weil "first order" bound)
Just for fun, we can also easily recover the fact that a curve which is maximal over F q must be minimal over the F q 2i and maximal of the F q 2i+1 . Indeed, being maximal over F q means by (3) that

x 1 = -1, therefore Gram(γ 0 , γ 1 , γ 2 ) = 1 -1 x 2 -1 1 -1 x 2 -1 1 = -(1 -x 2 ) 2 ≥ 0, so that x 2 = 1, that is by (3) that X is minimal over F q 2 . Then Gram(γ 0 , γ 2 , γ 3 ) = 1 1 x 3 1 1 -1 x 3 -1 1 = -(1 + x 3 ) 2 ≥ 0, so that x 3 = -1,
that is X is maximal over F q 3 , and so on. . . In the same way, if X is minimal over F q , that is if

x 1 = 1, then Gram(γ 0 , γ 1 , γ 2 ) = 1 1 x 2 1 1 1 x 2 1 1 = -(x 2 -1) 2 ≥ 0, that is x 2 = 1,
and X still minimal over F q 2 . And so on again...

Ihara bound as a bound of order 2

For n = 2, the domain W 2 recursively corresponds by item 2 of Proposition 9 and (9) to the set of (x 1 , x 2 ) ∈ R 2 satisfying:

     G 1 (x 1 ) = 1 -x 2 1 > 0 G + 2 (x 1 , x 2 ) = 1 -x 2 > 0 G - 2 (x 1 , x 2 ) = 1 + x 2 -2x 2 1 > 0 , that is -1 < x 1 < 1 2x 2 1 -1 < x 2 < 1 .
We represent the second Weil domain W 2 on figure 1.

L g 2 , g > L g 2 , g = √ q( √ q-1) 2 L g 2 , g < √ q( √ q-1) 2 µ 2 A g 2 x 1 x 2 -1 1 -1 1 Figure 1: The Weil domain W 2 . If X is a curve over F q of genus g ≥ √ q( √ q-1)

2

, then P 2 (X) = (x 1 , x 2 ) as defined by (3) lies on the grey domain W 2 ∩ H g 2 , so that x 1 is lower bounded by some constant µ 2 = min{x 1 (P ), P ∈ W 2 ∩H g 2 } > -1, improving Weil bound by Proposition 16.

The second Ihara line L g 2 with positive slope α = 1 √ q meets the domain W 2 if and only if the genus is greater than the genus g 2 for which L g 2 2 contains the point (-1, 1) (see figure 1), that is

1 = α × (-1) + 1 2g 2 α 1 α -α ⇐⇒ g 2 = √ q √ q -1 2 .
For g > g 2 , the constraint h g 2 (x 1 , x 2 ) ≤ 0 restricts the domain W 2 to the grey domain W 2 ∩ H g n , as can be seen on figure 1. The point A g 2 such that [A g 2 , B g 2 ] = W 2 ∩ L g 2 lies on the curve G - 2 = 0, and minimizes 3 the first coordinate x 1 on W 2 ∩ H g 2 . µ 2 is thus the solution in [-1, 0] of the quadratic equation G - 2 (x 1 , x 1 √ q + q-1 2g ) = 0. Solving it, we find

µ 2 = α 2 -α 2 + 8 q-1 2g -1 4
, so that using (3), we recover the well known Ihara bound X(F q ) -(q + 1) ≤ (8q + 1)g 2 + 4q(q -1)g -g 2 (Ihara "second order Weil" bound)

3 It can also be trivially proved using criteria 17 since ∇G - 2 = -4x 1 1 and x1(A g 2 ) < 0.

Weil bounds of higher finite orders

Following the same line, one can study Weil bounds of order n for n ≥ 3. We compute in Section 2.6.1 the exact formulae for the Weil bound of order 3 and for the genus bound g 3 from which this new bound is better than the Weil bound of order 2.

For n ≥ 4, computations to obtain explicit formula for the n-order Weil bound becomes intractable. Therefore we choose to develop an algorithm which, given a genus g and a size field q, computes the best upper order bound for the number of F q -rational points of a curve of genus g, together with the corresponding order n. We need to prove in Section 2.6.2 some preliminary results to justify this algorithm.

To illustrate the efficiency of the algorithm we display in Section 2.6.3 a table of numerical results for orders n ≥ 4 and few given q, g.

Weil bound of order 3

For n = 3, we have by (10) that

G + 3 (x 1 , x 2 , x 3 ) = 1 -x 3 x 1 -x 2 x 1 -x 2 1 -x 1 = -(1 -x 1 )x 3 + 1 -x 1 -(x 1 -x 2 ) 2 , G - 3 (x 1 , x 2 , x 3 ) = 1 + x 3 x 1 + x 2 x 1 + x 2 1 + x 1 = (1 + x 1 )x 3 + (1 + x 1 ) -(x 1 + x 2 ) 2 ,
where by ( 8) we have G - 1 = 1 + x 1 and G + 1 = 1 -x 1 . Hence, by item 3 of Proposition 9, we have

G - 3 = -1 + x 1 -(x 1 + x 2 ) 2 , G + 3 = 1 -x 1 -(x 1 -x 2 ) 2 ,
so that by item 3 of Proposition 9, we have

W 3 = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ W 2 and -1 + (x 1 + x 2 ) 2 1 + x 1 < x 3 < 1 - (x 1 -x 2 ) 2 1 -x 1 .
The boundary ∂W 3 = W 3 \ W 3 is by item 3 of Proposition 9 the part of the 2dimensional graphs G ± 3 = 0 above W 2 , and both graphs meet by Lemma 7 along a curve above the plane curve {(x 1 , x 2 ) ∈] -1, +1[ 2 ; G 2 (x 1 , x 2 ) = 0}, where G 2 is given by (9). More precisely, above the part G + 2 = 0, that is above the locus x 2 = 1, the two surfaces G - 3 = 0 and G + 3 = 0 meet along the segment (x 1 , 1, x 1 ) for -1 ≤ x 1 ≤ 1. Above the curve G - 2 = 0, i.e. above the locus x 2 = 2x 2 1 -1, one has x 2 + x 1 = (x 1 + 1)(2x 1 -1) and x 2 -x 1 = (x 1 -1)(2x 1 + 1). Therefore, {G + 3 = 0} and {G - 3 = 0} above x 2 = 2x 2 1 -1 are respectively

x 3 + 1 = (x 1 + x 2 ) 2 x 1 + 1 = (x 1 + 1)(2x 1 -1) 2 = 4x 3 1 -3x 1 + 1 and x 2 = 2x 2 1 -1 x 3 -1 = (x 1 -x 2 ) 2 x 1 -1 = (x 1 -1)(2x 1 + 1) 2 = 4x 3 1 -3x 1 -1 and x 2 = 2x 2 1 -1, so that their intersection above x 2 = 2x 2 1 -1 is the curve (x 1 , 2x 2 1 -1, 4x 3 1 -3x 1 ) for -1 ≤ x 1 ≤ 1.
• Looking at figure 2, for g small the line L g 3 does not meet the third Weil domain W 3 . • As g increase, it intersects the domain inside the G + 3 = 0 part, and increasing again inside the G - 3 = 0 part. This happens for g greater than the value g 3 such that the line L g 3 3 crosses the boundary ∂W 3 where the two graphs G ± 3 = 0 above W 2 meet. In other terms, the value g 3 is such that there exist a point

P g 3 (x 1 ) ∈ L g 3 3 ∩ W 3 ∩ {G - 3 = 0} ∩ {G + 3 = 0}. ( 19
)
We now compute this value g 3 , and the corresponding point 13) and ( 16), there exists some x 1 ∈ R, such that (x 1 , 1 g 3 ) is a solution of the two polynomials in (x 1 , 1 g )

P g 3 3 (x 1 ) ∈ L g 3 3 ∩ W 3 ∩ {G - 3 = 0} ∩ {G + 3 = 0}. Using (
G - 3 x 1 , x 1 √ q + q -1 2g , x 1 q + q 2 -1 2g √ q = 0 and G + 3 x 1 , x 1 √ q + q -1 2g , x 1 q + q 2 -1 2g √ q = 0.
(20) Eliminating x 1 in these equations, 1 g 3 is a solution of their resultant in x 1 . We have avoided calculations and factorization by hand: with the help of magma, we obtain that g 3 is a root of

(g -1)(g -q) g - √ q(q -1) √ 2 g + √ q(q -1) √ 2 .
Of course, g = -√ q(q-1) √ 2

< 0 cannot be a solution coming from a curve X. If g = 1 or q, substituting these values for g in (20), we find that x 1 = q+1 2 √ q is the only corresponding solution (for both values g = 1 and g = q), which is not contained in [-1, +1], so that these two values for the genus cannot come from a curve over a finite field by Weil bound (of order 1). Therefore

g 3 = √ q(q -1) √ 2 ,
and substituting again this value in (20), we find (using magma) that

x 1 = - √ 2 2
is the only corresponding solution, so that, as seen above, x 2 = 2x 2 1 -1 = 0 and

x 3 = 4x 3 1 -3x 1 = √ 2 2 . Finally, g = g 3 = √ q(q-1) √ 2
is the only solution, corresponding to the point P 3 (-

√ 2 2 ) = (- √ 2 2 , 0, √ 2
2 ), for which (19) holds. • For g > g 3 , the intersection of the line L g 3 with the domain

W 3 is a segment [A g 3 , B g 3 ],
where the point A g 3 lies on G - 3 = 0.

The point of the proof of the next Theorem is to check that, for g > g 3 , the point P 0 = A g 3 satisfies the requirements of criteria 17, showing that this point minimizes x 1 on the domain W 3 ∩H g 3 , which contains all points coming from curves by Proposition 16.

Figure 2: a complete view of the third Weil domain W 3 (upper left) and three partial views of W 3 ∩ L g 3 (for -1 ≤ x 1 ≤ 0): one for g < g 3 (upper right), one for g = g 3 (bottom left) and one for g > g 3 (bottom right). The red (resp. pink) surface is the locus {G - 3 = 0} (resp. {G + 3 = 0}) above W 2 . The dark green almost vertical plane is

x 2 = αx 1 + q-1
2g , the light green almost horizontal plane is

x 3 = α 2 x 1 + q 2 -1 2g √ q .
Their intersection is the black third Ihara line L g 3 in genus g, which cuts the red surface only for g ≥ g 3 .

Theorem 18. Let X be a smooth projective absolutely irreducible curve over F q of genus g such that g > g 3 = √ q(q-1) √ 2 . Then, we have

X(F q ) -q -1 ≤ a(q) + b(q) g + c(q) g 2 -1 + 1 q + d(q) g g √ q, where              a(q) = 5 + 8 √ q -1 q 2 b(q) = q-1 q √ q q 2 -4q √ q + 2q + 4 √ q -1 c(q) = q-1 4q q 3 -5q 2 -8q √ q -5q -8 √ q + 1 d(q) = 2 √ q(q-1) 2 q .
Proof -Let g > g 3 . Referring to the last item in the discussion above the statement of the Theorem, we begin to prove that the unique -1 < µ 3 < 0, such that

A g 3 = P 3 (µ 3 ) ∈ W 3 ∩ L g 3 ∩ {G - 3 = 0}
, satisfies criteria 17. Then, we compute the exact value of µ 3 , which is, by criteria 17 together with Proposition 16, a lower bound for x 1 's coming from curve over F q . We deduce the Theorem using (3).

Let us prove that P 0 = P 3 (µ 3 )) satisfies the requirements of criteria 17. The first requirement holds true: A g 3 ∈ {G - 3 = 0}. We prove later in corollary 21 that the third one also holds true for any n ≥ 1. we have only to check the second requirement, that ∂ i G - 3 (P g 3 (µ 3 )) ≥ 0 for i = 2, 3. Since by ( 10) we have

G - 3 (x 1 , x 2 , x 3 ) = 1 + x 3 x 1 + x 2 x 1 + x 2 1 + x 1 ,
we deduce that for any (x 1 , x 2 , x 3 ) ∈ R 3 ,

∂ 2 G - 3 (P 3 (x 1 )) = -2(x 1 + x 2 ) ∂ 3 G - 3 (P 3 (x 1 )) = 1 + x 3 . If x 1 = µ 3 , then P 0 = A g 3 = P 3 (µ 3 ) ∈ W 3 ⊂] -1, +1[ 3 , hence ∂ 3 G - 3 (P 3 (µ 3 )) = 1 + x 3 (P 0 ) > 0,
so that it remains to prove that x 1 (P 0 ) + x 2 (P 0 ) ≤ 0.

Since P 3 (µ 3 ) = A g 3 ∈ {G - 3 = 0} ∩ L g 3 , we have (x 1 + x 2 ) 2 = (1 + x 1 )(1 + x 3 ) x 3 = α 2 x 1 + q 2 -1 2g √ q .
We deduce that P 3 (µ 3 ) lies on on the plane parabola

x 1 + x 2 = ± (1 + x 1 ) 1 + α 2 x 1 + q 2 -1 2g √ q ,
and are reduced to check that P 3 (µ 3 ) lies on on the lower branch

x 1 + x 2 = -(1 + x 1 ) 1 + α 2 x 1 + q 2 -1 2g √ q .
The point is that (x 1 , x 2 ) = (-1, 1) lies on on both branches of this parabola, and that the tangent line to the parabola at this point is the vertical one. Hence, the upper branch defines a concave subset

(x 1 , x 2 ) | x 1 ≥ -1, x 2 ≥ 1 and x 1 + x 2 ≤ (1 + x 1 ) 1 + α 2 x 1 + q 2 -1 2g √ q of R 2 , containing (-1, 1) and (0, 1 + q 2 -1 2g √ q ) with 1 + q 2 -1 2g √ q > 1.
It follows that for any -1 < x 1 < 0, a point (x 1 , x 2 ) lying on the upper parabolic branch lies also on the locus x 2 > 1, which cannot hold for

P 0 = P 3 (µ 3 )) ∈ W 3 ⊂ [-1, -1] 3 .
It follows that all requirements of criteria 17 hold true for the point P 3 (µ 3 ), which lies on the branch

x 1 + x 2 = -(1 + x 1 )(1 + α 2 x 1 + q 2 -1 2g √ q ),
so that µ 3 is a lower bound on W 3 ∩ H g 3 for g > g 3 . Since P 3 (µ 3 ) lies on L g 3 , it lies in particular on x 2 = αx 1 + q-1 2g , so that µ 3 is a solution of the quadratic equation

(x 1 + αx 1 + q -1 2g ) 2 = (1 + x 1 )(1 + α 2 x 1 + q 2 -1 2g √ q ),
where we recall that α = 1 √ q . Solving this equation proves that the only solution in [-1, 0] is

µ 3 = q-1 q - 2 √ q(q-1) 2 gq -a(q) + b(q) g + c(q) g 2 2 ,
hence the Theorem using (3).

The formula becomes nicer if we let g going to infinity. We obtain the following third order asymptotic bound for Ihara's constant A(q) (see [START_REF] Ihara | Some remarks on the number of rational points of algebraic curves over finite fields[END_REF]).

Corollary 19. The Ihara constant A(q) is bounded above by:

A(q) ≤ 5 + 8 √ q - 1 q 2 -1 + 1 q √ q.
Remark -For q large the preceding upper bound is equivalent to (

√ 5 -1) √ q 1, 236 √ q.
This is better than the upper bound of A(q) following from Ihara bound

A(q) ≤ √ 8q + 1 -1 2 ,
which is equivalent to √ 2q 1, 414 √ q for q large. We prove later in Theorem 22 that we can also recover Dinfeld-Vlȃdut ¸and Tsfasman bounds with this viewpoint.

hence tangent to {G - n = 0} at P g n (µ n ) = A g n by Proposition 20. But {G - n ≥ 0} ∩ W n-1 is convex by Proposition 10, so that L g n would never meet the interior {G - n = 0}, a contradiction with (22).

It turns out that proving that this points is also always a minimum of the coordinate function x 1 seems to be difficult for N ≥ 4. But this is an easy task numerically using criteria 17: this will be done in our magma routine in Section 2.6.3.

Numerical results for n ≥ 4

We have implemented a routine in magma and leading to the following results.

In the following tabular we give, for genus g such that 1 ≤ g ≤ 52, and for q ∈ {2, 3}, the best generalized Weil bound for the number of points on a curve of genus g over F q . The number in brackets corresponds to the optimal order Weil bound. For instance, if this number is 2, this means that the bound is the well known Ihara bound and that bounds of other orders are not better. g q = 2 q = 3 g q = 2 q = 3 g q = 2 q = 3 g q = 2 q = 3 1 5(3) 7(2) 14 16(6) 26(4) 27 25(8) 42(5) 40 34(9) 57(6) 2 6(3) 9(2) 15 17(7) 28(4) 28 26(8) 43(5) 41 35(9) 58(6) 3 7(4) 10(3) 16 18(7) 29(5) 29 27(8) 44(5) 42 35(9) 59(6) 4 8(4) 12(3) 17 18(7) 30(5) 30 27(8) 46(5) 43 36(9) 60(6) 5 9(5) 14(3) 18 19(7) 31(5) 31 28(8) 47(5) 44 37(9) 61(6) 6 10(5) 15(3) 19 20(7) 32(5) 32 29(8) 48(6) 45 37(9) 62(6) 7 11(5) 17(4) 20 21(7) 34(5) 33 29(8) 49(6) 46 38(9) 63(6) 8 11(5) 18(4) 21 21(7) 35(5) 34 30(8) 50(6) 47 38(9) 65(6) 9 12(6) 19(4) 22 22(7) 36(5) 35 31(8) 51(6) 48 39(9) 66(6) 10 13(6) 21(4) 23 23(7) 37(5) 36 31(8) 52(6) 49 40(9) 67(6) 11 14(6) 22(4) 24 23(8) 38(5) 37 32(8) 54(6) 50 40(9) 68(6) 12 15(6) 24(4) 25 24(8) 40(5) 38 33(9) 55(6) 51 41(9) 69(6) 13 15(6) 25(4) 26 25(8) 41(5) 39 33(9) 56(6) 52 42(9) 70(6)

Comparing with the results available on the web site http://www.manypoints.org/, we unfortunately observe that we do not beat any record, and that we reach the records exactly in those cases were it is held by Osterlé bounds! . So we are pretty sure that there is a link between Osterle bound and the one in this Section, even if we do not know how to relate the two approaches.

Drinfeld-Vlȃdut ¸and Tsfasman bounds as a bound of infinite order

In this Section, we recover the Drinfeld-Vlȃdut ¸and Tsfasman bounds for the asymptotic of the number of points on curves over F q , giving a new meaning for the defect δ defined in [START_REF] Michael | Some remarks on the asymptotic number of points, Coding theory and algebraic geometry[END_REF]. With our point of view, these bounds can be considered as Weil bounds of infinite order.

We consider a sequence (X n ) n≥1 of absolutely irreducible smooth projective curves over F q . Let (g n ) n≥1 be the genus sequence and for r ≥ 1 and let B r (X n ) be the number of points of degree r on X n :

B r (X n ) = P ∈ X n (F q ) | deg(P ) = r .
Following Tsfasman, we assume this sequence to be asymptotically exact: this means that lim n→+∞ g n = +∞ and that for any r ≥ 1, the sequence Br(Xn) gn n≥1 admits a limit. Then, as usual, we put

∀r ≥ 1, β r def = lim n→+∞ B r (X n ) g n .
In the following Theorem, the notations and the normalizations are the same as in Definitions 2 and 4.

Theorem 22. Let (X n ) n≥1 be a sequence of absolutely irreducible smooth projective curves over F q . If this sequence is asymptotically exact, then its defect δ, defined as

δ = 1 -∞ r=1 rβr √ q r -1 , satisfy δ = lim m→+∞ lim n→+∞ γ 0 Xn + γ 1 Xn + • • • + γ m-1 Xn 2 Xn m .
Proof -Thanks to the matrix following Definition 4, we compute

1 m m-1 i=0 γ i Xn 2 Xn = 1 m 1 • • • 1 × Gram γ 0 Xn , . . . , γ m-1 Xn ×    1 . . . 1    = 1 m m + 2 m-1 i=1 (m -i)x i = 1 + 1 m m-1 i=1 (m -i) (q i + 1) -X n (F q i ) g n q i 2 = 1 + 1 g n m-1 i=1 (m -i) q i 2 + q -i 2 - m-1 i=1 1 - i m r|i r q i 2 × B r (X n ) g n .
Letting n tends to +∞ leads to

lim n→+∞ 1 m m-1 i=0 γ i Xn 2 Xn = 1- rs≤m-1 1 - rs m rβ r q rs 2 = 1- m-1 r=1   m-1 r s=1 1 - rs m 1 q rs 2   rβ r . (23)
To conclude, for any r, m ≥ 1,we remark that

1 q r 2 -1 - m-1 r s=1 1 - rs m 1 q rs 2 = +∞ s=1 1 q rs 2 - m-1 r s=1 1 - rs m 1 q rs 2 = +∞ s= m-1 r 1 q rs 2 + 1 m m-1 r s=1 rs q rs 2
.

Being the remainder of a converging serie, the first term of the last expression tends to zero when m → +∞; as for the second term it also tends to zero by Cesaro. Therefore

lim m→+∞ m-1 r s=1 1 - rs m 1 q rs 2 = 1 q r 2 -1
, and the Theorem follows letting m tends to +∞ in (23).

Just pointing out that a norm is non-negative, Theorem 22 leads to the well-known bound:

∞ r=1 rβ r √ q r -1 ≤ 1. (Tsfasman)
As is well known, Tsfasman bound is itself a refinement of

lim sup n→+∞ X n (F q ) g n ≤ √ q -1. (Drinfeld-Vlȃdut ¸)
3 Relative bounds

From now on, we concentrate on the relative situation. Our starting point is a finite morphism f : X → Y of degree d, where X and Y are absolutely irreducible smooth projective curves defined over F q , whose genus are denoted by g X and g Y .

The relative Neron-Severi subspace

The morphism f × f from X × X to Y × Y induces two morphisms

(f × f ) * : NS(X × X) R -→ NS(Y × Y ) R and (f × f ) * : NS(Y × Y ) R -→ NS(X × X) R satisfying (f × f ) * • (f × f ) * = d 2 Id Y ×Y . We put: ϕ = 1 d (f × f ) * and ψ = 1 d (f × f ) * . ( 24
)
Each of these morphisms can be restricted to the euclidean spaces (E X , •, • X ) and (E Y , •, • Y ) associated to the curves X and Y as in Section 1. Recall that both are the image of the orthogonal projections given by (1):

p X : NS(X × X) R -→ E X and p Y : NS(Y × Y ) R -→ E Y .
In fact, we still restrict a little bit more the morphisms ϕ and ψ. Let F X : X → X and F Y : Y → Y be the q-Frobenius on X and Y . As usual, for k ≥ 0, we denote by Γ k F X the class in NS(X × X) R of the k-th iterated of F X . We do the same for Γ k

F Y inside NS(Y × Y ) R .
Definition 23. For i ≥ 0, we put:

γ i X = 1 2q i p X Γ i F X ∈ E X , γ i Y = 1 2q i p Y Γ i F Y ∈ E Y ,
and we denote by F X and F Y the two subspaces of E X and E Y defined by

F X = Vect γ i X , i ≥ 0 ⊂ E X , F Y = Vect γ i Y , i ≥ 0 ⊂ E Y .
Remark -Note that the normalization in this Section differs from the one chosen in Definition 3 by a factor 1 √ g . The nice feature of this new normalization is that with this choice, the pull-back of divisor classes between some subspaces of the Neron Severi spaces is isometric, see the next Proposition 24.

For i ≥ 0 and j ≥ 1, one has by Lemma 3, with this new normalization for γ i ,

γ i X X = √ g X , γ i X , γ i+j X X = (q j + 1) -X(F q j ) 2 q j , ( 25
)
γ i Y Y = √ g Y , γ i Y , γ i+j Y Y = (q j + 1) -Y (F q j ) 2 q j . (26)
In other words, the vectors γ i X for i ≥ 0 lie on in the euclidean sphere of radius √ g X in the finite dimensional euclidean vector space E X , and the data of the scalar products is equivalent to the datas of the numbers of rational points of X on the finite extensions of F q .

Proposition 24. Restricted respectively to F X and F Y , the morphisms ϕ and ψ satisfy the followings.

1. One has ϕ • ψ = Id F Y , the identity map on F Y .

For all

i ≥ 0, ϕ(γ i X ) = γ i Y . 3. For all γ ∈ F X and all δ ∈ F Y , γ, ψ (δ) X = ϕ(γ), δ Y . 4. For all i, j ≥ 0, ψ(γ i Y ), ψ(γ j Y ) X = γ i Y , γ j Y Y . 5. The morphism ψ is an isometric embedding (F Y , •, • Y ) → (F X , •, • X ) and ψ • ϕ is the orthogonal projection of F X on its subspace ψ(F Y ). Proof -Item (1) follows from the identity (f × f ) * • (f × f ) * = d 2 Id Y ×Y . Item (2) follows from the identity (f × f ) * Γ F i X = dΓ F i Y
. Item (3) follows from the projection formula for the proper morphism f × f : X × X → Y × Y . Item (4) follows then from items (1) and (3). Finally, item (5) follow from the preceding ones together with the following Lemma.

Lemma 25. Let (E, ., . E ) and (F, ., . F ) be two finite dimensional euclidean vector spaces. Suppose that E is generated by a family {u i , i ≥ 0} and F is generated by a family {v j , j ≥ 0}. Let ϕ : E → F and ψ : F → E be linear maps satisfying ϕ • ψ = Id F and such that

ψ(v i ), ψ(v j ) E = v i , v j F , ∀i, j ≥ 0, ( 27
) v, ϕ(u) F = ψ(v), u E , ∀u ∈ E, ∀v ∈ F. (28)
Then, ψ is an isometric embedding from F to E and ψ • ϕ is the orthogonal projection from E to ψ(F ).

Proof -By (27), the morphism ψ must be an isometric embedding. Let us prove that ψ • ϕ is the orthogonal projection on ψ(F ). Since ϕ • ψ = Id F , we have (ψ

• ϕ) 2 = ψ •ϕ•ψ •ϕ = ψ •ϕ; therefore ψ •ϕ is a projector. For v ∈ F , one has ψ •ϕ(ψ(v)) = ψ(v), again by ϕ • ψ = Id F ; in other terms ψ(F ) is stabilized by ψ • ϕ. Last, for u ∈ E, one has u -ψ • ϕ(u) ∈ ψ(F ) ⊥ . Indeed, for v ∈ F , one has: u -ψ • ϕ(u), ψ(v) E = u, ψ(v) E -ψ • ϕ(u), ψ(v) E = ϕ(u), v F -ϕ(u), v F by (28) = 0.
This concludes the proof.

Definition 26. Let F X/Y be the subspace ψ(F Y ), called the relative subspace for the morphism X → Y .

Then we have just proved that

F X = F X/Y ⊕ F ⊥ X/Y ∈ ∈ ∈ u = ψ • ϕ(u) + (u -ψ • ϕ(u))
Remark -The absolute case of Section 2 enters in this relative case by choosing any nonconstant rational function X → P 1 ; in this case, the relative space F X/P 1 is the sub-space of EX generated by the projections of the classes of iterations of the Frobenius morphism.

We are now able to easily compute some norms and scalar products. Lemma 27. For any i ≥ 0, the following decomposition holds true:

γ i X = ψ(γ i Y ) ∈F X/Y + γ i X -ψ(γ i Y ) ∈F ⊥ X/Y
∈ F X and one has:

∀i, j ≥ 0, ψ(γ i Y ) ⊥ γ j X -ψ(γ j Y ) (29) ∀i ≥ 0, γ i X -ψ(γ i Y ) X = √ g X -g Y (30) ∀i ≥ 0, ∀j ≥ 1, γ i X -ψ(γ i Y ), γ i+j X -ψ(γ i+j Y ) X = Y (F q j ) -X(F q j ) 2 q j . ( 31
)
Proof -The first norm calculation is just a consequence of Pythagore Theorem:

γ i X 2 X = ψ(γ i Y ) 2 X + γ i X -ψ γ i Y 2 X by Pythagore = γ i Y 2 Y + γ i X -ψ γ i Y 2 X
since ψ isometric and thus by ( 25) and ( 26)

g X = g Y + γ i X -ψ γ i Y 2 X .
Taking intby o account orthogonality, we also easily compute the scalar product:

γ i X -ψ(γ i Y ), γ i+j X -ψ(γ i+j Y ) X = γ i X , γ i+j X X -ψ γ i Y , ψ γ i+j Y X by orthogonality = γ i X , γ i+j X X -γ i Y , γ i+j Y Y since ψ isometric = (q j + 1) -X(F q j ) 2 q j - (q j + 1) -Y (F q j )
2 q j by ( 25) and (26).

The result follows.

Number of points in coverings

As applications of this Proposition, we prove in the very same spirit than in Section 2 the following three results. The first one is well known, the others are new. It worth to notice that the first one can be proved using Tate modules of the jacobians of the involved curves (see e.g. [START_REF] Aubry | Coverings of singular curves over finite fields[END_REF]), whereas up to our knowledge, the others cannot.

Proposition 28. Suppose that there exists a finite morphism X → Y . Then we have

| X(F q ) -Y (F q )| ≤ 2(g X -g Y ) √ q.
Proof -We apply Cauchy-Schwartz inequality to the vectors γ 0 X -ψ(γ 0 Y ) and γ 1 Xψ(γ 1 Y ). Thanks to Lemma 27 specialized to j = 1 and i = 0, 1, we obtain

X(F q ) -Y (F q ) 2 √ q 2 = γ 0 X -ψ(γ 0 Y ), γ 1 X -ψ(γ 1 Y ) X 2 ≤ γ 0 X -ψ(γ 0 Y ) 2 X × γ 1 X -ψ(γ 1 Y ) 2 X = (g X -g Y ) 2 ,
hence the Proposition.

The following Proposition is the relative form of Proposition 14. Of course, although less nice, such upper bounds can be given for any n.

Proposition 29. For any finite morphism X → Y with g X = g Y , we have

X(F q 2 ) -Y (F q 2 ) ≤ 2(g X -g Y )q - X(F q ) -Y (F q ) 2 g X -g Y . Proof -After computating the 3 × 3 Gram determinant G 3 (X/Y ) = Gram(γ 0 X -ψ(γ 0 Y ), γ 1 X -ψ(γ 1 Y ), γ 2 X -ψ(γ 2 Y ))
using Lemma 27, it is readily seen, with the very same proof, that factorization in Lemma 5 holds also in this relative case, so that there exists two (explicitly given as determinants) factors G + n (X/Y ) and G - n (X/Y ) (n = 2 is sufficient here) as in Definition 6, such that G n (X/Y ) = G + n (X/Y ) × G - n (X/Y ) also holds. Moreover, Proposition 9 also continue to hold, hence for n = 2, the result follows from G - 2 (X/Y ) ≥ 0.

Number of points in a fiber product

Let Y 1 X Z Y 2 p 1 f 1 p 2 f 2 (32)
be a commutative diagram of finite covers of absolutely irreducible smooth projective curves defined over F q . Applying the results of the beginning of this Section to the four morphisms involved in this diagram leads to another diagram between Euclidean spaces

F Y 1 F Z F Y 2 F X ψ Y 1 / Z ψ Y 2 / Z ψ X / Y 1 ψ X / Y 2 ϕ Y 1 / Z ϕ Y 2 / Z ϕ X / Y 1 ϕ X / Y 2
Let us introduce the following hypothesis:

(H)

The fiber product Y 1 × Z Y 2 is abolutely irreducible and smooth.

Proposition 30. Suppose that (H) holds, and that

X = Y 1 × Z Y 2 in diagram (32). Then ϕ X/Y 2 • ψ X/Y 1 = ψ Y 2 /Z • ϕ Y 1 /Z on F Y 1 .
Proof -The proof of formula (30) is mainly set theoretic. We write:

(p 2 × p 2 ) * • (p 1 × p 1 ) * Γ F Y 1 = (p 2 × p 2 ) * {[(y 1 , y 2 ), (y 1 , y 2 )] ∈ (Y 1 × Z Y 2 ) 2 ; y 1 = F Y 1 (y 1 )} = {(y 2 , y 2 ) ∈ Y 2 × Y 2 ; ∃y 1 ∈ Y 1 ; f 1 (y 1 ) = f 2 (y 2 ); f 2 (y 2 ) = F Y 2 (f 1 (y 1 ))} = deg f 1 × {(y 2 , y 2 ) ∈ Y 2 × Y 2 ; f 2 (y 2 ) = F Y 2 (f 2 (y 2 ))} = deg f 1 × (f 2 × f 2 ) * (Γ F Z ). Now, since Y 1 × X Y 2 is
X = Y 1 × Z Y 2 in diagram (32). Then we have in F X ψ X/Y 1 (γ i Y 1 ), ψ X/Y 2 (γ i Y 2 ), ψ X/Z (γ i Z ) ∈ Vect ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) , ψ X/Y 2 γ Y i 2 -ψ Y 2 /Z (γ i Z ) , γ i X -ψ X/Z (γ i Z ) ⊥ and ∀i, j ∈ N, ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) ⊥ ψ X/Y 2 γ j Y 2 -ψ Y 2 /Z (γ j Z ) .
Proof -First, we compute scalar products with ψ X/Y 1 (γ i Y 1 ). We have

ψ X/Y 1 (γ i Y 1 ), ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) X = γ i Y 1 , γ i Y 1 -ψ Y 1 /Z (γ i Z ) Y 1 since ψ X/Y 1 is isometric = 0, by Lemma 27 ψ X/Y 1 (γ i Y 1 ), ψ X/Y 2 γ Y i 2 -ψ Y 2 /Z (γ i Z ) X = ϕ X/Y 2 • ψ X/Y 1 (γ i Y 1 ), γ Y i 2 -ψ Y 2 /Z (γ i Z ) Y 2 by prop. 24 = ψ Y 2 /Z • ϕ Y 1 /Z (γ i Y 1 ), γ Y i 2 -ψ Y 2 /Z (γ i Z ) Y 2 by prop. 30 = ψ Y 2 /Z (γ i Z ), γ Y i 2 -ψ Y 2 /Z (γ i Z )
Y 2 by prop. 24 = 0, by Lemma 27 and

ψ X/Y 1 (γ i Y 1 ), γ i X -ψ X/Z (γ i Z ) X = ψ X/Y 1 (γ i Y 1 ), γ i X -ψ X/Y 1 (γ i Y 1 ) + ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) X = 0.
The same holds for ψ X/Y 2 (γ i Y 2 ).

Secondly, we compute scalar products with ψ X/Z (γ i Z ) = ψ X/Y • ψ Y /Z (γ i Z ), for = 1, 2. We have:

ψ X/Z (γ i Z ), ψ X/Y γ i Y -ψ Y /Z (γ i Z ) X = ψ X/Y • ψ Y /Z (γ i Z ), ψ X/Y γ i Y -ψ Y /Z (γ i Z ) X = ψ Y /Z (γ i Z ), γ i Y -ψ Y /Z (γ i Z ) Y ψ X/Y isometric = 0. by Lemma 27
Last, ψ X/Z (γ i Z ), γ i X -ψ X/Z (γ i Z ) X = 0 by Lemma 27. In the same way than the use of the orthogonal projections associated to F X/Y ⊂ F X in the preceding Section, we do the same here with F X/Y 1 + F X/Y 2 ⊂ F X .

Lemma 32. Suppose that (H) holds, and that X = Y 1 × Z Y 2 in diagram (32). For i ≥ 0, let γ i 12 be the orthogonal projection of γ i X onto F X/Y 1 + F X/Y 2 ⊥ inside F X . Then

γ i 12 = γ i X -ψ X/Y 1 (γ i Y 1 ) -ψ X/Y 2 (γ i Y 2 ) + ψ X/Z (γ i Z )
, and one has:

γ i 12 X = g X -g Y 1 -g Y 2 + g Z γ i 12 , γ i+j 12 X
= Y 1 (F q j ) + Y 2 (F q j ) -X(F q j ) -Z(F q j ) 2 q j .

Proof -Clearly γ i X -γ i 12 ∈ F X/Y 1 + F X/Y 2 . Thanks to orthogonality relations of Lemma 31 and rewriting γ i 12 a little bit like

γ i 12 = γ i X -ψ X/Z (γ i Z ) -ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) -ψ X/Y 2 γ i Y 2 ) -ψ Y 2 /Z (γ i Z ) ,
we prove that γ i 12 ∈ F X/Y 1 + F X/Y 2 ⊥ .

To compute the norm let us note that in the decomposition

γ i X -ψ X/Z (γ i Z ) = ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) + ψ X/Y 2 γ i Y 2 ) -ψ Y 2 /Z (γ i Z ) + γ i 12 .
The first bracket lies on in F X/Y 1 +F X/Y 2 , while the second one lies on in F X/Y 1 + F X/Y 2 ⊥ .

Applying Pythagore again, we deduce that

γ i X -ψ X/Z (γ i Z ) 2 X = ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) + ψ X/Y 2 γ i Y 2 -ψ Y 2 /Z (γ i Z ) 2 X + γ i 12 2 X = ψ X/Y 1 γ i Y 1 -ψ Y 1 /Z (γ i Z ) 2 X + ψ X/Y 2 γ i Y 2 ) -ψ Y 2 /Z (γ i Z ) 2 X + γ i 12 2 X = γ i Y 1 -ψ Y 1 /Z (γ i Z ) 2 Y 1 + γ i Y 2 -ψ Y 2 /Z (γ i Z ) 2 Y 2 + γ i 12 2
X . Lemma 27 allows us to conclude since

γ i 12 2 X = γ i X -ψ X/Z (γ i Z ) 2 X -γ i Y 1 -ψ Y 1 /Z (γ i Z ) 2 Y 1 -γ i Y 2 -ψ Y 2 /Z (γ i Z ) 2 Y 2 = (g X -g Z ) -(g Y 1 -g Z ) -(g Y 2 -g Z ) .
In the same way, the calculation of γ i 12 , γ i+j 12 is a consequence of the computation of the scalar product γ i X -ψ X/Z (γ i Z ), γ i+j X -ψ X/Z (γ i+j Z ) .

Last we can prove the following results.

Theorem 33. Let X, Y 1 , Y 2 and Z be absolutely irreducible smooth projective curves in a cartesian diagram (32) of finite morphisms. Suppose that the fiber product Y 1 × Z Y 2 is also absolutely irreducible and smooth. Then

| X(F q ) -Y 1 (F q ) -Y 2 (F q ) + Z(F q )| ≤ 2(g X -g Y 1 -g Y 2 + g Z ) √ q.
Proof -First, for X = Y 1 × Z Y 2 the result is a direct consequence of Cauchy-Schwartz inequality applied to γ 0 12 and γ 1 12 thanks to Lemma 31 since (H) holds by assumption. Now, for general X satisfying the assumptions of the Theorem, we have by the universal property of the fibered product a finite morphism X → Y 1 × Z Y 2 . By triangular inequality, and using Proposition 28 together with the result for X, we have

| X(F q ) -Y 1 (F q ) -Y 2 (F q ) + Z(F q )| ≤ | X(F q ) -Y 1 × Z Y 2 (F q )| + | Y 1 × Z Y 2 (F q ) -Y 1 (F q ) -Y 2 (F q ) + Z(F q )| ≤ 2(g X -g Y 1 × Z Y 2 ) √ q + 2(g Y 1 × Z Y 2 -g Y 1 -g Y 2 + g Z ) √ q = 2(g X -g Y 1 -g Y 2 + g Z ) √ q
and the Theorem is proved.

It worth to notice that Theorem 33 cannot holds without any hypothesis. For instance, if X = Y 1 = Y 2 and the morphisms Y i → Z are equal, then the right hand side equals 2(g X -2g X + g Z ) √ q = -2(g X -g Z ) √ q, a negative number! In this case, the Theorem doesn't apply since Y 1 × Z Y 2 is not irreducible.

Remark -For Y1 ×Z Y2 to be absolutely irreducible, it suffices that the tensor product of function fields Fq(Y1) ⊗ Fq (Z) Fq(Y2) to be an integral domain. For (Q1, Q2) ∈ Y1 ×Z Y2 to be smooth it is necessary and sufficient that at least one of the morphism Yi → Z is unramified at Qi.

Questions

In this Section, we suggest a few questions raised by the viewpoint of this article.

1. Numerical calculations using our algorithm make it possible that the genus sequence (g n , n ∈ N) whose existence is asserted in Proposition 20 has, at least asymptotically, a quite nice behaviour. It seems that for fixed n ≥ 1, there do exists a constant 0 < c n < 1, such that for q large, g n ∼ c n √ q n , with moreover lim n→+∞ c n = 1. Of course, we have proved that c 2 = 1 2 and c 3 = 1 √ 2 . It numerically also seems that for any n ≥ 1 and any q, g n < √ q n+1 √ q -1 .

)

  Remark -Note first that X is Weil-maximal if and only if x1 = -1, and is Weil minimal if and only if x1 = 1. Second, that to give upper bounds for X(Fq) amount to give lower bounds for x1.

  assumed to be absolutely irreducible, we have deg f 1 = deg p 2 and deg f 2 = deg p 1 in a fiber product setting. Taking into account the normalization (24), and projecting onto E Y 2 as in Section 4, allow us to conclude.This allow us us to compute some other norms and scalar products.

	Lemma 31. Suppose that (H) holds, and that

Other known results can be proved from this viewpoint, for instance the rationality of the Zeta function, using the vanishing of Hankel determinants. In order to save place, we have chosen to skip this.

See the introduction of Section 2

If O is a convex open subspace of R n , for any x ∈ O and any y ∈ O, the segment ]x, y[ is included in O.

General features

The idea of the algorithm producing numerical values for higher order Weil bounds for given g, q is the following. It computes numerically in a first step an approximation of the unique -1 ≤ x 1 < 0 for which P n (x 1 ) lies on the intersection of L g n with G - n = 0, and it checks numerically in a second step that this point satisfies criteria 17. This algorithm is valid thanks to Proposition 20 that such an x 1 as in the first step do exists, at least for g large enough.

Proposition 20. There exists a sequence (g n ) n≥2 such that g > g n if, and only if, the intersection of the n-th Ihara line L g n in genus g with the interior of the n-the Weil domain W n is a non-empty x 1 -segment ]A g n , B g n [, with A g n lies on on the hypersurface of W n-1 × R having equation G - n = 0, and with -1 ≤ x 1 < 0.

Proof -We keep notations of Section 2.4 and we begin by studying the infinite line L ∞ n = {(x 1 , αx 1 , . . . , α n-1 x 1 ), x 1 ∈ R}, whose points are the P ∞ n (x 1 ) for x 1 ∈ R. This line L ∞ n and the domain W n have a non empty intersection since the origin of

To relieve the notations, for = ± or nothing, we put L n (x 1 ) = G n (x 1 , αx 1 , . . . , α n-1 x 1 ). Let us prove by induction that the abscissa µ ∞ n satisfy

For n = 1 this is true from (8): µ 1 = -1 is the only root of L 1 (x 1 ) = 1 + x 1 and satisfies L 0 (-1) > 0 since L 0 = 1. Suppose that assertion (21) holds true for some n ≥ 1. By Lemma 7, one has

where the underbrace (in)equalities come from the induction hypotheses. We first establish that

To this end, let m be the quotient of n + 2 by 2. By Lemma 5, one has

, where T m is a Toeplitz given by (6), while H m is Hankel given by (7). Specializing to the point (µ ∞ n , αµ ∞ n , . . . , α n µ ∞ n ), the Hankel part becomes a rank one matrix:

This matrix is clearly symmetric positive semi definite. But, for general A and B positive semi-definite symmetric matrices, one has det(A + B) ≥ det(A) (easy consequence of [HJ90, Corollary 4.3.3]). We deduce by induction hypothesis that, for some z ∈ R 2n ,

n belong to W n , this is the case for the point P ∞ n (µ). Hence one must have L m (µ) > 0 for all m ≤ n and L n+1 (µ) = 0. This proves that point P ∞ n+1 (µ) belongs to W n+1 and that µ = µ n+1 . This concludes the induction, hence in particular

= 0 as asserted, with moreover -1 ≤ µ ∞ n < 0. Now for finite genus, formula (13) shows that the union

g ∈ [0, +∞[ . Hence the intersection P n ∩ W n is a 2-dimensional convex subset in the closed half plane P n . Moreover, we have just seen above that the cut out with the line having parameter 1 g = 0 meet it on an non-empty segment with an end

convex, this segment is non empty and meet G - n = 0 for at least one point with x 1 < 0 for a connected subset of the parameter 0 ≤ 1 g < +∞ containing 0, hence have the form 0 ≤ 1 g < 1 gn for some 0 < g n < +∞. for such a g, the cut out with

It turns out that under the assumption that a point on L g n lies on the locus W n ∩ {G - n = 0} always satisfy the last requirement of criteria 17:

Proof -Denote by ϕ(x 1 ) = G - n (P g n (x 1 )) for x 1 ∈ R. This is a C ∞ function of the real variable x 1 .

By assumption, P g n (µ n ) ∈ {G - n = 0} ∩ L g n and g > g n , hence by Proposition 9 and Proposition 20 we have

, and there exists some ε > 0, such that

Hence by item 2 of Proposition 9, we have

where u = (1, α, . . . , α n-1 ) is a director vector of L g n . Now, suppose by contradiction that dϕ dt (µ n ) = 0. Then by (2.6.2) the line L g n would be perpendicular to

2. Is it true that the first requirement of criteria 17 implies the second one, in the same way that it do implies the third one as proved in corollary 21 ? It turns out that numerically, our algorithm shows that this holds true for all pairs (g, q) we have tested. We gave up this question when we became aware that our bounds seems to be equivalent to Oesterlé's.

3. Why do the upper bounds obtained in this article always coincide numerically with Osterlé's one ?

4. In view of the hope to write down explicitly Weil bounds of order n ≥ 4, it is necessary to solve a one variable polynomial equation on x 1 of degree n+1 2 , which is greater than 5 for n ≥ 8. This polynomial equation is, eliminating 1 g , the resultant

Do this polynomial have solvable Galois group over the rational function field

5. Is there a nice interpretation in this framework for the number of rational points of the Jacobian Jac(X) of a curve X ? In the affirmative, same question for the Prym variety associated to an unramified double cover X → Y ?

6. Do the viewpoint of this article can be extended to higher dimensional varieties ?

7. As explained in the article, the geometry and the arithmetic of X are traduced via Hodge Index Theorem to an euclidean property throught Gram determinants by P n (X) ∈ W n , and its arithmetic by P n (X) ∈ H g n through Ihara's constraints, so that upper bounds for X(F q ) are derived from the lower bound for x 1 on W n ∩H g n . Any other constraint coming either from the geometry or the arithmetic of X is likely to reduce the domain W n ∩ H g n , making possible that the x 1 function on the new domain as greater lower bound. For instance, do the records obtained in the table http://www.manypoints.org/, better than Osterlé bound, can be understood in this way ?


Laurent André
email: l.andre@brgm.fr

C Christov

Arnault Lassin

Mohamed Azaroual

Pitzer ion-interaction parameters for Al(III) in the {H + Na + K + Ca + Mg + Cl + H 2 O} system up to salts solubility at 298.15 K

come

Pitzer ion-interaction parameters for Al(III) in the H + Na + K + Ca + Mg + Cl + H2O system up to salts solubility at 298.15 K

INTRODUCTION

The fundamental research and economic interest of industrials for natural brines is continuously increasing all over the World. More and more substances are extracted from these brines, needing the development of specific exploitation processes. Geochemical modeling can help these industrial technologies providing robust geochemical codes and relevant databases for describing and predicting the critical chemical reactions/steps inside these specific and complex media (dissolution/ precipitation of salts, corrosion/scaling problems…).

The Pitzer approach is usually quoted to describe such complex systems but its related databases need to be continuously improved to integrate new chemical species and interaction parameters of complex aqueous systems. This study focuses on very acidic systems and on the efforts made to define the interactions of Al(III) species inside the H-Na-K-Mg-Ca-Cl-H 2 O system at 25°C. This study is the continuation of a work focused on Fe(III) species in brines [START_REF] Andre | Thermodynamic behavior of FeCl 3 -H 2 O and HCl-FeCl 3 -H 2 O systems -A Pitzer Model at 25[END_REF].

The Pitzer interaction parameters determined in this study allow the description of the thermodynamic behavior of ternary systems up to salt solubility. These parameters are implemented in a database adapted to the geochemical code PHREEQC [START_REF] Parkhurst | User's guide to PHREEQC (version 2) -A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF].

DESCRIPTION OF THE WORK

The building of the database able to describe the ion-ion interactions in saline solutions needs to define specific binary and ternary interaction parameters. The developed methodology is based on a progressive approach, beginning with the determination of the ion interaction coefficients in binary systems before working on ternary systems. The equilibrium constant of salts are also integrated in this approach in order to obtain a fully internally consistent database. Some of the interaction coefficients can be issued from literature, the others being estimated or re-estimated in order to match experimentally measured data, namely: water activity, osmotic coefficients and salt solubility [START_REF] Andre | A methodology to estimate Pitzer's interaction parameters[END_REF].

RESULTS

Binary systems

Osmotic coefficients of AlCl 3 solutions are plotted versus the salt molality on Fig. 1. It shows that the parameterizations for the AlCl 3 -H 2 O system issued from [START_REF] Christov | Thermodynamic study of the K-Mg-Al-Cl-SO 4 -H 2 O system at the temperature 298.15 K[END_REF] match the measured values of osmotic coefficient.

Ternary systems

For the K-Al-Cl system, the solubility measurements of sylvite in AlCl 3 solutions are selected from different authors (Fig. 2). The ternary parameters,  and  proposed by [START_REF] Christov | Thermodynamic study of the K-Mg-Al-Cl-SO 4 -H 2 O system at the temperature 298.15 K[END_REF] do not allow reproducing the experimental data. A new parameterization has been done, leading to a good fit of the data and a correct description of the invariant point between sylvite and AlCl 3 .6H 2 O (s) (Fig. 2) For describing the Na-Al-Cl system, Christov et al. [START_REF] Christov | Thermodynamic modeling of aqueous aluminum chemistry and solid-liquid equilibria to high solution concentration and temperature. I. The acidic H-Al-Na-K-Cl-H 2 O system from 0 to 100[END_REF] used ternary parameters from Palmer and Wesolowski [START_REF] Palmer | Aluminum speciation and equilibria in aqueous solution: II. The solubility of gibbsite in acidic sodium chloride solutions from 30 to 70°C[END_REF]. With these parameters, the match between experimental and calculated data is good. For the Mg-Al-Cl system, the parameterization given by [START_REF] Christov | Thermodynamic study of the K-Mg-Al-Cl-SO 4 -H 2 O system at the temperature 298.15 K[END_REF] does not allow fitting experimental data. With a revised  value, the description of experimental can be achieved (Fig. 3). For the Ca-Al-Cl system, solubility data of antarcticite and AlCl 3 .6H 2 O(s) are from Sarkarov and Mironova [START_REF] Sarkarov | Solubility in the aluminium chloride-calcium chloride-lithium chloride system[END_REF] and Gao et al [START_REF] Gao | A Practical Approach to Produce Mg-Al Spinel Based on the Modeling of Phase Equilibria for NH 4 Cl-MgCl 2 -AlCl 3 -H 2 O System[END_REF]. The  value needed to be adjusted to match the experimental solubility data allowing then to reproducing correctly the invariant point between the two salts.

For the H-Al-Cl system,  and  parameters established by [START_REF] Christov | Thermodynamic modeling of aqueous aluminum chemistry and solid-liquid equilibria to high solution concentration and temperature. I. The acidic H-Al-Na-K-Cl-H 2 O system from 0 to 100[END_REF] give an excellent fit between numerical modelling results and the experimental solubility data of AlCl 3 :6H 2 O (s) in HCl solutions.

CONCLUSIONS

The set of parameters established in this work is suitable to reproduce water characteristics of aluminum chloride solutions both at low and high ionic strengths. This new set of parameters can represent in particular the composition and the excess properties of aluminum in the H-Na-K-Ca-Mg-Cl-H 2 O system. All binary and mixed solution interaction parameters and equilibrium constants of Al(III) minerals are coherent, and therefore can be used to determine the activities of dissolved species up to the salt solubility concentrations.

Fig. 1 :

 1 Fig. 1: Calculated (line) and experimental (symbols) water activity for the AlCl 3(aq) -H 2 O binary system as a function of total molality of AlCl 3 at 25 °C. The maximum experimental ionic strength is about 20 eq/kg.

Fig. 2 :

 2 Fig. 2: Salt solubility diagrams in (KCl+AlCl 3 ) solutions at 25°C

Fig. 3 :

 3 Fig. 3: Salt solubility diagrams in (MgCl 2 +AlCl 3 ) solutions at 25°C


Qiaochu Li
email: qiaochu.li@utc.fr

Carine Jauberthie

Lilianne Denis-Vidal
email: lilianne.denis-vidal@math.univ-lille1.fr

Zohra Cherfi
email: zohra.cherfi@utc.fr

Nonlinear Dynamical Aerospace Models

Keywords: Parameter estimation, State estimation, Continuous-time systems, Nonlinear systems, Bounded noise, Interval analysis, Aerospace models

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INTRODUCTION

Complex systems are often subjected to uncertainties that make the modeling task awkward. These uncertainties can be unstructured when the equations of the system are not entirely known or structured when the equations are known but not the values of their parameters. In both cases, it is particularly difficult to get an accurate model of the perturbations and noises acting on the system. This may turn the usual stochastic framework inappropriate. Thus, we prefer to deal with set-membership framework in which perturbations and noises are assumed to be bounded but otherwise unknown. In this framework, we obtain "guaranteed solutions". This last expression means that for all conditions belonging to a bounded set, the obtained set contains all the solutions.

Guaranteed state and parameter estimation methods are an interesting alternative to stochastic model based estimation when perturbations and noises are assumed to be bounded but otherwise unknown. These methods have received a lot of attention in the last years and the literature on this topic shows interesting progress, for example [START_REF] Kieffer | Guaranteed recursive nonlinear state bounding using interval analysis[END_REF], [START_REF] Deville | Consistency techniques in ordinary differential equations[END_REF], [START_REF] Jaulin | A nonlinear set membership approach for the localization and map building of underwater robots[END_REF], [START_REF] Rauh | Modeling, design and simulation of systems with uncertaintites[END_REF], [START_REF] Pasca | Formally Verified Conditions for Regularity of Interval Matrices[END_REF] or for example (Jauberthie et al., 2013).

Moreover, experimental design is important for identifying mathematical models of modern aircraft dynamics from flight test data, for example. In the case of aerospace domain, the flight test input has a major impact on the quality of the data for modeling purposes. Good experimental design must account for practical constraints during the test. The overall goal is to design an experiment that produces data from which model parameters can be estimated accurately. Most importantly, in an estimation framework, the experimental conditions about noise and disturbances are usually properly modeled through appropriate assumptions about probability distributions ( [START_REF] Mehra | Optimal input signals for parameter estimation in dynamic systems -survey and new results[END_REF], [START_REF] Walter | Identification de modèles paramétriques à partir de données expérimentales[END_REF], [START_REF] Kiefer | General equivalence theory for optimum designs (approximate theory)[END_REF]). The conventional approach for the experimental design is based on stochastic models for uncertain parameters and measurement errors (see for example [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF]). However, other sources of uncertainty are not well-suited to the stochastic approach and are better modeled as bounded uncertainty. This is the case of parameter uncertainties that generally arise from design tolerances and from aging. In such cases, combining stochastic and bounded uncertainties may be an appropriate solution. Some works consider that the parameters belong to some prior domain, on which no probability function has to be defined (for example [START_REF] Pronzato | Robust experiment design via maximin optimization[END_REF], [START_REF] Belforte | Optimal worst case estimation for lpv-fir models with bounded errors[END_REF]). The first aim at optimizing is the worst possible performance of the experi-ment over the prior domain for the parameters. This maximin approach to synthesis the optimal input is described and the specific criterion are developped.

In a recent paper [START_REF] Jauberthie | Optimal input design for a nonlinear dynamical uncertain aerospace system[END_REF], it is supposed that the uncertainty on parameters can be modelled by bounded intervals and the concepts of interval analysis are used for the optimal input synthesis. In this paper the original approach of optimal input design for uncertain bounded parameter estimation is an extension of the works of E.A. [START_REF] Morelli | Flight test of optimal inputs and comparison with conventional inputs[END_REF]) using the dynamic programming. This approach combines the concepts of dynamical programming with the maximin approach and with the tools of interval analysis. In the presented work, we propose to apply an optimal input obtained in [START_REF] Jauberthie | Optimal input design for a nonlinear dynamical uncertain aerospace system[END_REF] for the same case study taken from aerospace domain. By using this optimal input, we obtain an original algorithm to achieve a guaranteed state and parameter estimation based on interval analysis.

This paper is organized as follows. In Section 2, the problem statement and the case study are presented. The case study is taken from aerospace domain and describes the longitudinal motion of a glider. Section 3 presents some basic tools of interval analysis. The notions of interval, box, interval matrix and inclusion function are given. Section 4 presents the fundamental algorithm to implement state and parameter estimation. In Section 5, the estimation results obtained on the case study are presented and discussed. Two cases of inputs are tested and the performance of the optimal input is highlighted. Finally, some conclusions are outlined in section 6.

PROBLEM FORMULATION AND CASE STUDY

This paper deals with estimating the unknown state and parameters for a nonlinear dynamic system of the following form:

ẋ(t, p) = f (x(t, p), p) + u(t)g(x(t, p), p)), y(t, p) = h(x(t, p), p), x(0) ∈ [X 0 ], p ∈ [P 0 ],
(1) where x(t, p) ∈ R n and y(t, p) ∈ R n y denote respectively the state variables and the measured outputs. The initial conditions x(0) are supposed to belong to an initial bounded box [X 0 ]. u(t) represents the input.

The vector p is the vector of parameters to be estimated and p ∈ R n p which is supposed to belong to an a priori box [P 0 ].

The time t is assumed to belong to [0,t max ]. The functions f , g and h are nonlinear functions. f and g are supposed analytic on M for every p ∈ [p 0 ], where M is an open set of R n such that x(t, p) ∈ M for every p ∈ [p 0 ] and t ∈ [0,t max ]).

The output error is assumed to be given by:

v(t i ) = y m (t i ) -y(t i , p), i = 1, ..., N.
(2) We assume that v(t i ) and v(t i ) are known as lower and upper bounds for the acceptable output errors. Such bounds may, for instance, correspond to a bounded measurement noise. The integer N is the total number of sample times.

Interval arithmetic is used to compute guaranteed bounds for the considered problem at the sampling times {t 1 ,t 2 , ...,t N }.

The case study that we consider in this work is given by an aerospace model which describes the longitudinal motion of a glider. The projection of the general equations of motion onto the aerodynamic reference frame of the aircraft and the linearization of aerodynamic coefficients give the following system:

                                                                 V = -g sin(θ -α) - 1 2m ρSV 2 (C 0 x +C xα (α -α 0 ) +C xδ m (δ m -δ m 0 )), α = 2 2mV + ρSlVC z α mV q + mg cos(θ -α) V - 1 2 ρSV 2 (C 0 z +C zα (α -α 0 ) +C zq ql V +C zδ m (δ m -δ m 0 )) , q = 1 2B ρSlV 2 C 0 m +C mα (α -α 0 ) +C mq ql V +C m α 2l 2mV 2 + ρSlV 2 C z α mV q +mg cos(θ -α) V - 1 2 ρSV 2 (C 0 z +C zα (α -α 0 ) +C zq ql V +C zδ m (δ m -δ m 0 )) +C mδ m (δ m -δ m 0 ) , θ = q.
(3) In these equations, the state vector x is given by (V, α, q, θ) , the observation y is full (i.e., y = x), the input u is δ m (δ m 0 represents the initial condition). The variable V denotes the speed of the aircraft, α the angle of attack, α 0 the trim value of α, θ the pitch angle, q the pitch rate, δ m the elevator deflection angle, ρ the air density, g the acceleration due to gravity, l a reference length and S the area of a reference surface. B represents a moment of inertia. The parameters to be estimated are C z α, C zq , C m α, C mq , which are assumed to be uncertain. The other coefficients correspond to the dynamic stability derivatives are supposed to be known.

Interval analysis provides tools for computing with sets which are described using outerapproximations formed by union of non-overlapping boxes. The following results are mainly taken from [START_REF] Jaulin | Applied interval analysis with examples in parameter and state estimation[END_REF]. The set of all real intervals of R is denoted IR.

Basic tools

A real interval [u] = [u, u]
Two intervals [u] and [v] are equal if and only if u = v and u = v. Real arithmetic operations are extended to intervals [START_REF] Moore | Interval analysis[END_REF].

Arithmetic operations on two intervals [u] and [v] can be defined by:

• ∈ {+, -, * , /}, [u] • [v] = {x • y | x ∈ [u], y ∈ [v]}.
An interval vector (or box) [X] is a vector with interval components and may equivalently be seen as a cartesian product of scalar intervals:

[X] = [x 1 ] × [x 2 ]... × [x n ].
The set of n-dimensional real interval vectors is denoted by IR n .

An interval matrix is a matrix with interval components. The set of n × m real interval matrices is denoted by IR n×m . The width w(.) of an interval vector (or of an interval matrix) is the maximum of the widths of its interval components. The midpoint m(.) of an interval vector (resp. an interval matrix) is a vector (resp. a matrix) composed of the midpoint of its interval components.

Classical operations for interval vectors (resp. interval matrices) are direct extensions of the same operations for punctual vectors (resp. punctual matrices) [START_REF] Moore | Interval analysis[END_REF].

Let f : R n → R m , the range of the function f over an interval vector [u] is given by:

f ([u]) = { f (x) | x ∈ [u]}. The interval function denoted [ f ] is a function from IR n to IR m . It is an inclusion function for f if: ∀[u] ∈ IR n , f ([u]) ⊆ [ f ]([u]).
An inclusion function of f can be obtained by replacing each occurrence of a real variable by its corresponding interval and by replacing each standard function by its interval evaluation. Such a function is called the natural inclusion function. In practice the inclusion function is not unique, it depends on the syntax of f .

Set Inversion

Consider the problem of determining a solution set for the unknown quantities u defined by:

S = {u ∈ U | Φ(u) ∈ [y]} = Φ -1 ([y]) ∩ U, (4)
where [y] is known a priori, U is an a priori search set for u and Φ a nonlinear function not necessarily invertible in the classical sense. (4) involves computing the reciprocal image of Φ and is known as a set inversion problem which can be solved using the algorithm Set Inverter Via Interval Analysis (denoted SIVIA). The algorithm SIVIA proposed in [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF] is a recursive algorithm which explores all the search space without losing any solution. This algorithm makes it possible to derive a guaranteed enclosure of the solution set S as follows:

S ⊆ S ⊆ S.
The inner enclosure S is composed of the boxes that have been proved feasible. To prove that a box

[u] is feasible it is sufficient to prove that Φ([u]) ⊆ [y].
Reversely, if it can be proved that Φ([u]) ∩ [y] = / 0, then the box [u] is unfeasible. Otherwise, no conclusion can be reached and the box [u] is said undetermined. The latter is then bisected and tested again until its size reaches a user-specified precision threshold ε > 0. Such a termination criterion ensures that SIVIA terminates after a finite number of iterations.

GUARANTEED STATE AND PARAMETER ESTIMATION

This section concerns the integration of (1) and set inversion computation. Thus, the objective of this section is fist to obtain the state vector x at the sampling times {t 1 ,t 2 , ...,t N } corresponding to the measurement times of the outputs. Second follow the SIVIA procedure to get the validated sets of feasible parameters.

We note [x j ] the box [x(t j )] where t j represents the sampling time, j = 1, ..., N and x j represents the solution of (1) at t j .

Validated Integration for Nonlinear Systems

Rigorous solution for dynamical nonlinear systems can be solved efficiently by considering methods based on Taylor expansions [START_REF] Moore | Interval analysis[END_REF], [START_REF] Rihm | Interval methods for initial value problems in odes[END_REF], [START_REF] Berz | Verified integration of odes and flows using differential algebraic methods on high-order taylor models[END_REF] or (Nedialkov and Jackson, 2001). These methods consist in two parts: the first one verifies the existence and uniqueness of the solution by using the fixed point theorem and the Picard-Lindelöf operator. At a time t j+1 , an a priori box [ x j ] containing all solutions corresponding to all possible trajectories between t j and t j+1 is computed.

In the second part, the solution at t j+1 is computed by using a Taylor expansion, where the remainder term is [ x j ].

To obtain the set [ x j ], a classical technique consists in inflating this set until it verifies the following inclusion [START_REF] Lohner | Enclosing the solutions of ordinary initial and boundary value problems[END_REF], (Nedialkov and Jackson, 2001):

[x j ] + h j f ([ x j ]) ⊆ [ x j ], (5)
where h j denotes the integration step and [x j ] the first solution. In the proposed work, to state estimate, we use the package VNODE in which the previous validated integration method is implemented. The package VNODE, developed by N.S Nedialkov (Nedialkov et al., 2001), is a C ++ package for computing bounds of solutions in Initial Value Problem for ordinary differential equation. In the latest version, named VNODE-LP, algorithms corresponding to high order enclosure and Hermite-Obreschkoff method [START_REF] Nedialkov | Vnode-lp a validated solver for initial value problems in ordinary differential equations[END_REF] have been implemented. Thus VNODE-LP gives a way to obtain tighter enclosure.

Parameter Estimation

Parameters estimation from experimental measures are usually obtained within a stochastic framework in which known distribution laws are associated to interferences and measurement noise. Oppositely, in a bounded error context, measures and modeling errors are supposed to be unknown but to stay within known and acceptable bounds.

Errors between measured and predicted outputs may rely on many factors, among them: limited sensors accuracy, interferences, noise, structured uncertainties, etc. Some are quantifiable, some are not. We consider here the quantifiable error ν, which is added to the model output y. The experimental outputs y m are given by (Equation ( 2)):

y m (t i ) = y(t i , p) + ν(t i ), 1 ≤ i ≤ N. ( 6
)
In the presented work, the error ν is supposed to be within an interval whose lower bound is ν and upper bound is ν. An allowable error set E may be defined as a set of constraints:

E = {ν(t i ) | ν(t i ) ≤ ν(t i ) ≤ ν(t i )}. ( 7
)
These bounds may be considered constant over time as well as variable. They may be established from data given by constructors for electronic parts for example.

A parameter vector p is acceptable if and only if the error between y m and the model output y is bounded in a known way. To estimate system parameters, we have to get the set P of all parameters p enclosed in the a priori search set [P 0 ] such that error between real data and model outputs belongs to E:

P ={p ∈ [P 0 ] | y m (t i ) -y(t i , p) ∈ [v i , v i ], ∀i = 1, • • • , N}, = {p ∈ [P 0 ] | [ν(t i )] ∈ E, ∀i = 1, • • • , N} . (8
) The characterization of the set P may be defined as a set inversion problem (4). By simplicity of notation, we note this set:

P = [ν -1 ](E) ∩ [P 0 ]. (9)
A guaranteed enclosure of P may be computed by using the SIVIA algorithm presented in Section 3.2.

Parameter and state estimation

To perform the state and parameter estimation, we propose the following algorithm. This algorithm has been implemented in C++. It combines the strategy of bisections used in SIVIA and the validated integration used by VNODE. A threshold ε is considered for the bisections in SIVIA. The choice of this threshold depends on the a priori initial box of parameters to be estimated. In this algorithm, the function bisect-Box divides a box into two sub-boxes and the function VNODELP is the call to the software VNODE-LP. while i <= N do 6:

x e (i) := V NODELP(x e (i -1)); 7:

j := i; 8: i := i + 1; 9:
end while P uncertain := P uncertain ∪ [p];

10: if [h]([x e (1 : j)]) ⊆ [y(1 : j)]
16:

else 17: bisectBox([p]) → {[p] 1 , [p] 2 | [p] 1 ∪ [p] 2 = [p]};
18:

P list := P list ∪ [p] 1 , P list := P list ∪ [p] 2 ; 19:
end if 20: end while

APPLICATION

In this section, the state and parameter estimation of the aerospace system is performed by using the proposed algorithm. The initial conditions are supposed to belong to: (10)

[X 0 ] =   
The parameters are supposed to be included in:

[P 0 ] =   
1.71 1.89 4.75 5.25 -5.25 -4.75 -23.1 -20.9

   . ( 11
)
The output error ( 2) is supposed to be bounded by:

[ν] =   
-0.0447 0.0447 -0.0044 0.0044 -0.0044 0.0044 -0.0044 0.0044

   . ( 12
)
The measurements have been simulated by using the parameters equal to (1.8, 5, -5, -22) and initial states [X 0 ]. The test duration is fixed at one second. The stop criterion for SIVIA is ε = [0.01, 0.05, 0.05, 0.1] that means that the stop threshold for the first parameter is 0.01, the second and third are 0.05 and the last one is 0.1. Two cases of inputs are considered for the tests: the first one concerns a constant input and the second one is an optimal input proposed in [START_REF] Jauberthie | Optimal input design for a nonlinear dynamical uncertain aerospace system[END_REF] with six stages. The optimal input is the following:

u(t) = δ m0 + a 6 0 H(t -t 0 6 ) -2a 6 1 H(t -t 1 6 )
+2a 6 2 H(tt 2 6 ) -2a 6 3 H(tt 3 6 )+ 2a 6 4 H(tt 4 6 ) -2a 6 5 H(tt 5 6 ) (13) with a 6 i = 1.6 degrees with i = 0, • • • , 5 and , t 0 6 = 0 s, a 1 6 = 0.1667 s, a 2 6 = 0.3334 s, a 3 6 = 0.5001 s, a 4 6 = 0.6668 s a 5 6 = 0.8335 s. The function H is the Heaviside function.

The optimized input is given in the Figure 1: As seen in these figures, the first three parameters have not been well estimated. The diameter of each interval remained almost as proposed. The parameter C mq has been well obtained compared with other parameters.

By using the optimal input, we obtain Figures 8, 9, 10, 11, 12 and 13. We compare the results from the two inputs. The term %p indicates the percentage of unacceptable and uncertain interval sets we eliminated. Results have been done with constant input and optimal input. Clearly, the optimal input improves significantly the estimated parameters' domain. The last 3 rows of Table 1 show an improvement in estimation results. Through Table 2, we show the clear improvement of the acceptable domain for the parameters by using an optimal input. The first one (for parameters C z α, C zq , C m α) and the second one (for parameters C zq , C m α, C mq ) are divided by 100.

CONCLUSION

In this contribution, a procedure for parameter and state estimation in a bounded-error context has been pointed out. Two different inputs have been implemented and the estimation results have been compared. We can see that the coefficient C z α is difficult to be correctly estimated. The efficiency of the proposed algorithm combined with an optimized input has been pointed out. The presented method has potential for being used for active diagnosis problems in continuous-time systems or hybrid systems.

Our future works concern an improvement in the estimation parameter problem for these models and the potential application of this method to the active diagnosis. In fact, this last objective will be to use these tools to achieve an active diagnostic methodology that is to find a sequence of actions to refine the diagnosis.

As seen in the results for parameter estimation, the obtained results are clearly closed to the choice of input, thus another direction of our future work concerns the development of a methodology of optimal input design in a bounded error context for parameter estimation which is a new perspective.

  is a closed and connected subset of R where u represents the lower bound of [u] and u represents the upper bound. The width of an interval [u] is defined by w([u]) = uu, and its midpoint by m([u]) = (u + u)/2.

Algorithm 1

 1 Parameter estimation ([h], [y], P admis , ε) Require: [x](0), [p](0); Ensure: P admis , P uncertain , P re jected ; 1: initialization: P list := [p](0), x e (0) := ([x](0), [p](0)); 2: while P list :

P

  admis := P admis ∪ [p]; 12: else if [h]([x e (1 : j)]) ∩ [y(1 : j)] := / 0 then 13: P re jected := P re jected ∪ [p]; 14: else if w([p]) < ε then 15:

Figure 1 :

 1 Figure 1: Optimized input for six stages. The order of the Taylor expansion is chosen automatically by the VNODE-LP. The parameter estimation results, for a constant input, are given in Figures 2, 3, 4, 5, 6 and 7. In these figures, the red boxes represent the acceptable sets for parameters, the blue boxes represent the rejected boxes and the yellow boxes represent the undetermined boxes. The black border cube represent the box [P 0 ].

Figure 2 :

 2 Figure 2: Acceptable sets C z α, C zq , C m α with constant input.

Figure 3 :

 3 Figure 3: Acceptable sets C zq , C m α, C mq with constant input.

Figure 4 :

 4 Figure 4: Rejected sets C z α, C zq , C m α with constant input.

Figure 5 :

 5 Figure 5: Rejected sets C zq , C m α, C mq , with constant input.

Figure 6 :

 6 Figure 6: Undetermined sets C z α, C zq and C m α with constant input.

Figure 7 :

 7 Figure 7: Undetermined sets C zq , C m α, C mq , with constant input.

Figure 8 :

 8 Figure 8: Acceptable sets C z α, C zq , C m α with optimal input.

Figure 9 :

 9 Figure 9: Acceptable sets C zq , C m α, C mq with optimal input.

Figure 10 :

 10 Figure 10: Rejected sets C z α, C zq , C m α with optimal input.

Figure 11 :

 11 Figure 11: Rejected sets C z α, C zq , C m α with optimal input.

Figure 12 :

 12 Figure 12: Undetermined sets C z α, C zq , C m α with optimal input.

Figure 13 :

 13 Figure 13: Undetermined sets C z α, C zq , C m α with optimal input.

Table 1 :

 1 Eliminated percentage of initial box.

	Parameter %p constant %p optimal
	C z α	0.00	0.00
	C zq	0.00	75.00
	C m α	25.00	87.50
	C mq	65.62	93.75
	The volume of obtained acceptable boxes are pre-
	sented in the following table:

Table 2 :

 2 Volume of obtained acceptable boxes.

	Parameters	Constant input Optimal input
	C z α, C zq and C m α	0.1215	9.4482e-04
	C zq , C m α and C mq	1.7325	0.0116.