Yves Aubry

Marc Perret

MAXIMUM NUMBER OF RATIONAL POINTS ON HYPERSURFACES IN WEIGHTED PROJECTIVE SPACES OVER FINITE FIELDS

Keywords: 2010 Mathematics Subject Classification. Primary 14G05; Secondary 14G15. Key words and phrases. Rational points, finite fields, weighted projective spaces

An upper bound for the maximum number of rational points on an hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted projective spaces has been considered by Castryck, Ghorpade, Lachaud, O'Sullivan, Ram and the first author in 2017. A conjecture has been proposed there and proved in the particular case of the dimension 2. We prove here the conjecture in any dimension provided the second weight is also equal to one.

Introduction

Let F q be the finite field with q elements and P n (F q ) be the set of rational points over F q of the projective space of dimension n ≥ 1. Let us set p n := q n +• • •+q+1 for n ≥ 0 and p n := 0 for n < 0. We have clearly ♯P n (F q ) = p n .

Answering to a conjecture of Tsfasman made at the "Journées Arithmétiques de Luminy" in 1989, Serre proved in [START_REF] Serre | Journées Arithmétiques[END_REF] (and independently Sørensen proved later in [START_REF] Sørensen | Projective Reed-Muler codes[END_REF]) that if F is a nonzero homogeneous polynomial in F q [X 0 , . . . , X n ] of degree d ≥ 1, then the number of rational points over F q of the hypersurface V (F ) in P n defined by F satisfies the so-called Serre bound:

♯V (F )(F q ) ≤ dq n-1 + p n-2 .
Note that dq n-1 + p n-2 ≥ p n = ♯P n (F q ) if d ≥ q + 1 and thus the Serre bound is trivial in this case.

Moreover the Serre bound is reached: if d ≥ q + 1 then the polynomial X d-q-1 0 (X q 0 X 1 -X 0 X q 1 ) is homogeneous of degree d and the corresponding hypersurface has p m rational points, and if d ≤ q then the number of rational points on the hypersurface given by the polynomial

F = d i=1 (α i X 0 -β i X 1 ),
where (α 1 : β 1 ), . . . , (α d , β d ) are distincts elements of P 1 (F q ), attains the Serre bound. Note that Serre proved that the bound is reached for d ≤ q if and only if F is of the above form, that is V (F ) is the union of d hyperplanes containing a linear variety of codimension 2.

In 1997, Tsfasman and Boguslavsky in [START_REF] Boguslavsky | On the number of solutions of polynomial systems[END_REF] have considered the analogue question for a system of r polynomial equations. They propose a conjecture for the maximum number of points in P n (F q ) of the projective set given by the common zeros of r linearly independent homogeneous polynomials of degree d in F q [X 0 , . . . , X n ]. The Tsfasman-Boguslavsky conjecture for r = 1 is nothing else but the Serre bound. Boguslavsky succeded to prove in [START_REF] Boguslavsky | On the number of solutions of polynomial systems[END_REF] the case r = 2. In 2015, Datta and Ghorpade proved in [START_REF] Datta | On a conjecture of Tsfasman and an inequality of Serre for the number of points of hypersurfaces over finite fields[END_REF] that the Tsfasman-Boguslavsky conjecture is true if d = 2 and r ≤ n + 1 but is false in general if d = 2 and r ≥ m + 2. Moreover, in 2017 they proved in [START_REF] Datta | Number of solutions of systems of homogeneous polynomial equations over finite fields[END_REF] that the Tsfasman-Boguslavsky conjecture is true for any positive integer d, provided r ≤ n + 1. The case for r beyond n + 1 is specifically considered one year later by Beelen, Datta and Ghorpade in [START_REF] Beelen | Maximum number of common zeros of homogeneous polynomials over finite fields[END_REF] and they conjectured in 2022 in [START_REF] Beelen | A combinatorial approach to the number of solutions of homogeneous polynomial equations over finite fields[END_REF], and proved it in some cases, a general formula when d < q.

We are interested here in a generalization in another direction, namely the question of Tsfasman and Serre in the context of weighted projective spaces P(a 0 , . . . , a n ), i.e. the study, for any homogeneous polynomial F in F q [X 0 , . . . , X n ] of degree d (with respect to the weights a 0 , . . . , a n ), of the maximum number of rational points on the hypersurface V (F ) in P(a 0 , . . . , a n ). In [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF], the following quantity has been introduced:

e q (d; a 0 , a 1 , a 2 , . . . , a n ) := max F ♯V (F )(F q )
where the maximum ranges over the set of homogeneous polynomials F in F q [X 0 , . . . , X n ] of weighted degree d.

It has been conjectured in 2017 in [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF] that:

Conjecture 1.1. If a 0 = 1 and lcm(a 1 , a 2 , . . . , a n )|d, and if we order the weights such that a 1 ≤ a 2 ≤ . . . ≤ a n then

e q (d; 1, a 1 , a 2 , . . . , a n ) = min{p n , d a 1 q n-1 + p n-2 }.
In the case of the projective line P(a 0 , a 1 ), it has been shown in [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF] that e q (d; a 0 , a 1 ) = min{p 1 , d/a} where a = lcm(a 0 , a 1 ), so the conjecture holds in this case. Moreover, the conjecture has been proved in [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF] for projective planes P(1, a 1 , a 2 ) with a 1 and a 2 coprime and a 1 < a 2 : if F ∈ F q [X 0 , X 1 , X 2 ] is a nonzero weighted homogenous polynomial of degree d ≤ a 1 (q + 1) which is a multiple of a 1 a 2 then ♯V (F )(F q ) ≤ d a 1 q + 1. The proof goes by imitating the proof of Serre by introducing a new notion of line, namely homogenized linear bivariate equations or the line at infinity.

Our purpose here is to prove Conjecture 1.1 in any dimension n provided a 1 = 1.

We recall in Section 2 the basic facts about weighted projective spaces and a lower bound for e q (d; a 0 , . . . , a n ). Then we study in Section 3 some morphisms between weighted projective spaces and we establish a relation between the numbers of zeros of a polynomial and its pullback. Section 4 is devoted to the proof of an upper bound for the number of rational points on an hypersurface in a weighted projective space. Finally we state and prove the main result in Section 5.

2.

A lower bound for the number of rational points 2.1. Weighted projective spaces. Let a 0 , . . . , a n be positive integers coprime with the characteristic of F q and S be the polynomial ring F q [X 0 , . . . , X n ] graded by deg(X i ) = a i . The weighted projective space P(a 0 , . . . , a n ) is defined by P(a 0 , . . . , a n ) = Proj S, and can be seen as the geometric quotient

A n+1
Fq \ {0}/G m,Fq of the punctured affine space A n+1

Fq \ {0} over F q under the action of the multiplicative group G m,Fq over F q given by λ.(x 0 , . . . , x n ) = (λ a 0 x 0 , . . . , λ an x n ).

If the a i 's are all equal to 1, then we recover the usual (or straight) projective space: P(1, . . . , 1) = P n .

The corresponding equivalent class is denoted by [x 0 : • • • :

x n ] without any reference to the corresponding weights a 0 , . . . , a n and is called a weighted projective point. We say that the point is

F q -rational if [x 0 : • • • : x n ] = [x q 0 : • • • : x q n ] .
It can be shown (see [START_REF] Perret | On the number of points of some varieties over finite fields[END_REF]) that every F q -rational point has at least one representative (in fact exactly q -1) in F n+1 q \ {(0, . . . , 0)}. In particular we have ♯P(a 0 , . . . , a n )(F q ) = p n .

For many more details about weighted projective spaces, one can consult the article of Beltrametti and Robbiano (see [START_REF] Beltrametti | Introduction to the theory of weighted projective spaces[END_REF]) for a theory over an algebraically closed field of characteristic 0, the article of Dolgachev (see [START_REF] Dolgachev | Weighted projective varieties, Group Actions and Vector Fields[END_REF]) for a theory over a field of characteristic prime to all the a i 's, and the Appendix of [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF] for a survey of the different points of view.

2.2.

A lower bound. Let F be a homogeneous polynomial in S of degree d, so that

F (λ a 0 X 0 , . . . , λ an X n ) = λ d F (X 0 , . . . , X n ) for all λ ∈ F * q
where F q is an algebraic closure of F q , and let V (F ) be the hypersurface defined by F in P(a 0 , . . . , a n ).

We define, as in the introduction, the quantity:

e q (d; a 0 , . . . , a n ) := max F ∈S d \{0} ♯V (F )(F q )
where S d stands for the space of weighted homogeneous polynomials in S of weighted degree d. Remark that the previous quantity is only defined for

d ∈ a 0 N + • • • + a n N.
Consider now the polynomial

F = d/ars i=1 (α i X ars/ar r -β i X ars/as s )
where r, s ∈ {0, . . . , n} are distincts indices, a rs = lcm(a r , a s ), d is a multiple of a rs satisfying d ≤ a rs (q + 1) and the (α i , β i )'s are distinct elements of P 1 (F q ). It has been proved in [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF] that ♯V (F )(F q ) = (d/a rs )q n-1 + p n-2 . So, if a := min{lcm(a r , a s ), 0 ≤ r < s ≤ n} and a | d, then it implies that e q (d; a 0 , . . . , a n ) ≥ min{p n , d a q n-1 + p n-2 }.

3. Some morphisms between weighted projective spaces 3.1. The morphisms π i . For i = 0, . . . , n, we consider the following morphims π i :

π i : P(a 0 , . . . , a i-1 , 1, a i+1 , . . . , a n ) -→ P(a 0 , . . . , a n ) [x 0 : • • • : x n ] -→ [x 0 : • • • : x a i i : • • • : x n ].
Our purpose in this Section is to study the behaviour of the rational points with respect to this morphisms. For this purpose, let us fix some generator δ of the multiplicative group F * q . For any given i ∈ {0, . . . , n}, set r i = (a i , q -1) the gcd of a i and q -1 and consider the map ϕ a i :

ϕ a i : F * q -→ F * q z -→ z a i .
Recall that the map ϕ a i is a group homomorphism with kernel Ker(ϕ a i ) =< δ q-1 r i >=: µ a i , the subgroup of F * q of a i -th roots of unity in F * q which has order r i , and with image Im(ϕ a i ) =< δ a i >=: ∆ a i , the subgroup of F * q of a i -th powers which has order q-1 r i . Let P be the whole set of rational points over F q of P(a 0 , . . . , a n ). We have a partition P = R i ∪ T i ∪ I i with respect to the i-th coordinate, where

R i := {[y 0 : • • • : y n ] ∈ P(a 0 , . . . , a n )(F q ) | y i = 0} ∪ {O i }, T i := {[y 0 : • • • : y n ] ∈ P(a 0 , . . . , a n )(F q ) | y i = 1} \ {O i }, I i := {[y 0 : • • • : y n ] ∈ P(a 0 , . . . , a n )(F q ) | y i ∈ F * q \ ∆ a i } and O i := [0 : • • • : 0 : 1 : 0 : • • • : 0]
is the point where the "1" appears only for index "i".

Let us scrutinize more narrowly the sets I i and T i . In order to do this, consider, for j ∈ {1, . . . , q -1}, the sets Z i (j) defined by

Z i (j) := {[y 0 : • • • : y n ] ∈ P(a 0 , . . . , a n )(F q ) | y i = δ j }.
Lemma 3.1. We have:

(i) Z i (j 1 ) = Z i (j 2 ) if j 1 ≡ j 2 (mod r i ). (ii) Z i (r i ) = T i . (iii) I i = ∅ if r i = 1 and
I i = Z i (1) ∪ . . . ∪ Z i (r i -1) otherwise.
Proof. We begin by proving that δ r i = λ a i for some λ ∈ F * q , which will be used in the proof of the three items. Indeed, there exist by Bézout Theorem some integers u, v such that r i = ua i +v(q-1), so that δ

r i = (δ u ) a i ×(δ q-1 ) v = λ a i for λ = δ u .
Suppose now that j 2 = j 1 + mr i for some integer m and consider some [y 0 :

• • • : y n ] ∈ Z i (j 2 ). By writing δ r i = λ a i , it is easily checked from δ j 2 = (δ r i ) m × δ j 1 = (λ m ) a i × δ j 1 that [y 0 : • • • : y i-1 : δ j 2 , y i+1 : • • • , y n ] = [(λ -m ) a 0 y 0 : • • • : (λ -m ) a i-1 y i-1 : δ j 1 : (λ -m ) a i+1 y i+1 : • • • : (λ -m ) an y n ]
which lies in Z i (j 1 ), so that Z i (j 2 ) ⊂ Z i (j 1 ). The reverse inclusion follows similarly.

The second item can be proved likewise by writing δ r i = λ a i , since then [y 0 :

• • • : y i-1 : δ r i : y i+1 : • • • : y n ] = [(λ -1 ) a 0 y 0 : • • • : (λ -1 ) a i-1 y i-1 : 1 : (λ -1 ) a i+1 y i+1 : • • • : (λ -1 ) an y n ].
Finally, the set I i contains of course the union Z i (1) ∪ . . . ∪ Z i (r i -1). Conversely, given some P = [y 0 : • • • : y i-1 : δ h : y i+1 : • • • : y n ] ∈ I i with 1 ≤ h ≤ q -1 not divisible by a i , then writing the Euclidean division of h by r i gives the existence of integers m and j such that h = r i m + j with 0 ≤ j ≤ r i -1. Thus, writing

δ h = (δ r i ) m × δ j = (λ m ) a i × δ j , we get [y 0 : • • • : y i-1 : δ h : y i+1 : • • • : y n ] = [(λ -m ) a 0 y 0 : • • • : (λ -m ) a i-1 y i-1 : δ j : (λ -m ) a i+1 y i+1 : • • • : (λ -m
) an y n ], so that P ∈ Z i (j) for this j ∈ {1, • • • , r i -1} which concludes the proof.

The following proposition describes the number of pre-images of points by the morphism π i according to the set of the partition that they belong to.

Proposition 3.2. Let P be a rational point of P(a 0 , . . . , a n ).

(i) If P ∈ R i then P has exactly one pre-image rational over F q by π i . (ii) If P ∈ T i then P has exactly r i pre-images rational over F q by π i . (iii) If P ∈ I i then P has no pre-image rational over F q by π i .

y i-1 : δ (q-1)k r i : y i+1 : • • • : y n ] for k = 1, . . . , r i (the elements δ (q-1)k r i
are the a i -th roots of unity in F * q i.e. the elements of the group µ a i ).

(iii) The points [y 0 : • • • : y n ] with y i ∈ ∆ a i have no rational pre-image by π i since y i is not a a i -th power in F * q .

3.2. Number of zeros of the pullback. Let F be a homogeneous polynomial in F q [X 0 , . . . , X n ] of (a 0 , . . . , a n )-weighted degree d ≤ q + 1, i.e.

F (λ a 0 X 0 , . . . , λ an X n ) = λ d F (X 0 , . . . , X n ) for any λ ∈ F * q . Let π * i F (X 0 , . . . , X n ) := (F • π i )(X 0 , . . . , X n ) = F (X 0 , . . . , X a i i , .
. . , X n ) be the pullback of F , an homogeneous polynomial of (a 0 , . . . , a i-1 , 1, a i+1 , . . . , a n )weighted degree d. We consider the hypersurface V P(a 0 ,...,an) (F ) of zeros of F in P(a 0 , . . . , a n ) whose number of rational points over F q is denoted by N(F ). We also consider the hypersurface V P(a 0 ,...,a i-1 ,1,a i+1 ,...,an) (π * F ) of zeros of π * F in P(a 0 , . . . , a i-1 , 1, a i+1 , . . . , a n ) whose number of rational points over F q is denoted by N(π * i F ).

Let us set:

A(F ) := ♯(V P(a 0 ,...,an) (F ) ∩ A)

for A ∈ {R i , T i , I i , Z i (j)}. So, N(F ) denotes the number of rational points of V P(a 0 ,...,an) (F ) and R i (F ), T i (F ), I i (F ) and Z i (j)(F ) denote the number of those rational points lying on R i , T i , I i and Z i (j) respectively.

Proposition 3.3. We have :

(i) N(F ) = R i (F ) + T i (F ) + I i (F ). (ii) N(π * i F ) = r i T i (F ) + R i (F ). (iii) Consider the automorphism σ i : [y 0 : • • • : y n ] -→ [y 0 : • • • : y i-1 : δy i : y i+1 : • • • : y n ] of P(a 0 , . . . , a n ). If r i := (a i , q -1) = 1 then: (a) for j = 1, . . . , r i -1, we have T i (F • σ j i ) = Z i (j)(F ), (b) for j = r i -1, we have T i (F • σ j i ) = T i (F ) (c) and R i (F ) = R i (F • σ j i ) for 1 ≤ j ≤ r i -1.
Proof. The first equality comes from the partition P = R i ∪ T i ∪ I i .

The second one from Proposition 3.2 and the fact that if P is a rational point over F q of V P(a 0 ,...,a i-1 ,1,a i+1 ,...,an) (π * F ) then π i (P ) is a point of V P(a 0 ,...,an) (F ) which is rational over F q .

The third one follows from the fact that the automorphism σ i sends T i to Z i (1) and Z i (j) to Z i (j + 1) for 1 ≤ j ≤ r i -1, and by Lemma 3.1 sends Z i (r i -1) to T i , and leaves R i stable.

We are enable now to prove a relation on the numbers of points between two floors. Proposition 3.4. Let F be a homogeneous polynomial in F q [X 0 , . . . , X n ] with respect to the weights (a 0 , a 1 , . . . , a n ). For i ∈ {0, . . . , n}, let

π i : P(a 0 , . . . , a i-1 , 1, a i+1 , . . . , a n ) -→ P(a 0 , . . . , a n ) [x 0 : • • • : x n ] -→ [x 0 : • • • : x a i i : • • • : x n ] and π * i F (X 0 , . . . , X n ) := (F • π i )(X 0 , . . . , X n ) = F (X 0 , . . . , X a i i , . . . , , X n ) be the pullback of F .
Let also δ be a primitive element of F * q , and σ i : [y 0 : • • • : y n ] -→ [y 0 : • • • : y i-1 : δy i : y i+1 : • • • : y n ] inside P(a 0 , . . . , a n ). Denote by r i = (a i , q -1) the gcd of a i with q -1.

Then, the number N(F ) of rational points over F q of the hypersurface of the weighted projective space P(a 0 , a 1 , . . . , a n ) defined by F satisfies

N(F ) ≤ 1 r i r i -1 j=0 N(π * i (F • σ j i )).
Proof. If r i = 1, then the set I i is empty and by (i) and (ii) of Proposition 3.3, we have N(F ) = R i (F ) + T i (F ) = N(π * i F ) which gives the result. Suppose now that r i = 1. By (i) of Proposition 3.3, we have:

r i N(F ) = (r i T i (F ) + R i (F )) + (r i I i (F ) + (r i -1)R i (F )).
On one hand, we have by (ii) of Proposition 3.3 that r i T i (F ) + R i (F ) = N(π * i F ) and on the other hand, by Lemma 3.1, we can write

I i (F ) ≤ r i -1
j=1 Z i (j)(F ). Thus, we have:

r i I i (F )+(r i -1)R i (F ) ≤ r i r i -1 j=1 Z i (j)(F ) +(r i -1)R i (F ) = r i -1 j=1 (r i Z i (j)(F ) + R i (F )) .
Moreover, by Proposition 3.3 (iii), we have:

r i Z i (j)(F ) + R i (F ) = r i T i (F • σ j i ) + R i (F • σ j i )
and we obtain with Proposition 3.3 (ii):

r i Z i (j)(F ) + R i (F ) = N(π * i (F • σ j i )
). Thus we deduce that:

r i I i (F ) + (r i -1)R i (F ) = r i -1 j=1 N(π * i (F • σ j i ))
and we obtain the desired formula.

Remark 3.5. Note that under the additionnal assumption that (a i , a j ) = 1 for any 1 ≤ i = j ≤ n, we have equality in the above Proposition 3.4. This comes from the fact that, under this assumption, the sets Z i (j) for 1 ≤ j ≤ r i -1 form a partition of I i , hence both inequalities in the above proof are equalities. It remains to show that the sets Z i (j) for 1 ≤ j ≤ r i -1 are pairwise disjoint. Indeed, suppose that there is some common point with

F q -coordinates [y 0 : • • • : y i-1 : δ j 1 : y i+1 : • • • : y n ] = [y ′ 0 : • • • : y ′ i-1 : δ j 2 : y ′ i+1 : • • • : y ′ n ] ∈ Z i (j 1 )∩Z i (j 2 )
, with say 1 ≤ j 1 ≤ j 2 ≤ r i -1. Since this point does not lie in R i , there is at least one position k = i, such that y k = 0 = y ′ k . Since they are equal, there is some λ

∈ F * q such that (y ′ 0 , • • • , y ′ i-1 , δ j 2 , y ′ i+1 , • • • , y ′ n ) = (λ a 0 y 0 , • • • , λ a i-1 y i-1 , λ a i δ j 1 , λ a i+1 y i+1 , • • • , λ an y n ).
From the k-th and the i-th position, we get y

′ k = λ a k y k and δ j 2 = λ a i δ j 1 . It follows first that λ a k = y ′ k y k ∈ F * q , second that λ a i = δ j 2 -j 1 . But from a Bézout relation ua k + va i = 1, we deduce that λ = (λ a k ) u × (λ a i ) v = ( y ′ k y k ) u × (δ j 2 -j 1 ) v ∈ F * q .
Hence, we have λ = δ m for some m ∈ N, so that δ j 2 -j 1 = λ a i = δ ma i . It follows that j 2j 1 ≡ ma i (mod q -1). Since r i = (a i , q -1) divides both a i and q -1, it divides j 2j 1 ∈ {0, • • • , r i -1}, hence j 1 = j 2 and we are done.

An upper bound for the number of rational points

We prove in this section that an hypersurface in a weighted projective space cannot have more rational point than in a standard projective space. The proof is based on an unscrewing and uses Proposition 3.4. P(a 0 , a 1 , a 2 , . . . , a n ) P(1, a 1 , a 2 . . . , a n ) . . . Let F be a homogeneous polynomial in F q [X 0 , . . . , X n ] of (a 0 , a 1 , . . . , a n )-weighted degree d ≤ q + 1. Then the number N(F ) of rational points over F q of the hypersurface of the weighted projective space P(a 0 , a 1 , . . . , a n ) given by the set of zeros of F satisfies:

N(F ) ≤ dq n-1 + p n-2 .
Proof. Let F be a homogeneous polynomial in F q [X 0 , . . . , X n ] of (a 0 , a 1 , . . . , a n )weighted degree d. We consider the successive pullbacks π * 0 (F • σ j 0 0 ) with j 0 ∈ {0, . . . , r 0 -1}, and π * 1 (π * 0 (F • σ j 0 0 ) • σ j 1 1 ) with j 1 ∈ {0, . . . , r 1 -1}, and so on, of F . By Proposition 3.4, considering the morphism π 0 : P(1, a 1 , . . . , a n ) -→ P(a 0 , a 1 , . . . , a n ) [x 0 :

x 1 : • • • : x n ] -→ [x a 0 0 : x 1 : • • • : x n ]
following polynomial of degree d (the degree is d since we have supposed that the weights of X 0 and X 1 are equal to 1 in the graded ring F q [X 0 , . . . , X n ]):

F = d i=1 (α i X 0 -β i X 1 )
where (α 1 : β 1 ), . . . , (α d : β d ) are distinct elements of P 1 (F q ).

(Aubry) Institut de Mathématiques de Marseille -I2M, Aix Marseille Univ, UMR 7373 CNRS, France

Email address: yves.aubry@univ-tln.fr (Perret) Institut de Mathématiques de Toulouse -UMR 5219, CNRS, UT2J, F-31058 Toulouse, France

Email address: perret@math.univ-toulouse.fr

  Proof. (i) The point O i := [0 : • • • : 0 : 1 : 0 : • • • : 0] ∈ P(a 0 , . . . , a n ) has only one pre-image by π i , namely the point [0 : • • • : 0 : 1 : 0 : • • • : 0] ∈ P(a 0 , . . . , a i-1 , 1, a i+1 , . . . , a n ). Moreover, the point [y 0 : • • • : y i-1 : 0 : y i+1 : • • • : y n ] has only one pre-image by π i , expressly the point [y 0 : • • • : y i-1 : 0 : y i+1 : • • • : y n ]. (ii) The point [y 0 : • • • : y i-1 : 1 : y i+1 : • • • : y n ] has r i pre-images by π i , which are precisely the points [y 0 : • • • :

P( 1 ,Figure 1 .

 11 Figure 1. Screwing of weighted projective spaces

Acknowledgments:

The authors are very grateful to Fabien Herbaut for fruitful discussions. They would also like to thank Jade Nardi, Sudhir Ghorpade and Mrinmoy Datta for some comments on this question.

 [START_REF] Aubry | Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory[END_REF]This work is partially supported by the French Agence Nationale de la Recherche through the BARRACUDA project under Contract ANR-21-CE39-0009.

1

we have:

N(F 0 (j 0 ))

where F 0 (j 0 ) = π * 0 (F • σ j 0 0 ). Then, considering the morphism π 1 :

x n ] we have for 0 ≤ j 0 ≤ r 0 -1:

where

). Thus:

Continuing this process, we obtain

The last polynomials are homogeneous polynomials of degree d in the standard n-dimensional projective space P n = P(1, . . . , 1). Then we apply the Serre bound

and we get the result.

The main result

We are now enable to state and prove Conjecture 1.1 provided a 1 = 1 (it was already assumed in the conjecture that a 0 = 1). Theorem 5.1. For any degree d and for any nonnegative integers a 2 , . . . , a n , we have:

In other words, Conjecture 1.1 is true for any (a 1 , a 2 , . . . , a n ) with a 1 = 1 and without any assumption on the degree d.

Proof. By Theorem 4.1, we have e q (d; 1, 1, a 2 , . . . , a n ) ≤ min{p n , dq n-1 + p n-2 }. Now if d ≤ q + 1, then the bound of Theorem 4.1 is met by the


Andy Amoordon

Virginie Deniau

Christophe Gransart

Anthony Fleury
email: anthony.fleury@imt-lille-douai.fr

Jonathan Villain

A Threshold-Based Detection Approach To Detect Fake Access

Keywords:

Introduction

Wireless networks are nowadays widely used as they allow mobility and the cheap and rapid expansion of computer networks. They are also essential to connect certain mobile equipment and small devices such as connected vehicles, and sensors. However, as compared to wired networks (which are often isolated in the ground, buildings...), wireless networks are more exposed and are therefore more vulnerable to certain types of attacks. IEEE 802.11 networks are, for instance, vulnerable to jamming and fake access points attacks. To protect these types of networks, we seek, in our research, to implement a Network Intrusion Detection System (NIDS) 1 to detect jamming and fake access point attacks on IEEE 802.11 networks in infrastructure mode.

A jamming attack is the intentional emission of signals in a frequency band in order to decrease the Signal to Interference ratio (SIR) and therefore degrade the reception of communications. There are different types of jamming attacks [START_REF] Grover | Jamming and anti-jamming techniques in wireless networks: a survey[END_REF]. In our work, we considered frequency sweeping jammers because, even though their use is prohibited, they are relatively easy to obtain. A fake access point attack consists in the identity usurpation of a licit access point in order to trick users into thinking that the fake access point is the licit one. When a fake access point attack is successful, all the communication of devices in the 802.11 network passes through the fake access point rather than the licit access point. The attacker who is in control of the fake access point has therefore control over the communication and can read, modify or generate frames2 [START_REF] Amoordon | Characterizing Wi-Fi Man-In-the-Middle Attacks[END_REF].

To detect the two attacks, we have adopted an anomalybased approach. This approach consists in the comparison of two situations (a normal situation and an attack situation) in order to identify differences and determine proper indicators to detect the attacks. Indicators can be determined manually using a threshold-based approach or automatically using supervised learning algorithms. In this paper, we present a threshold-based approach, its limitations and we explain why it is more interesting to adopt a supervised learning model approach.

The rest of the paper is organised as follows. In section 2, we detail our experimental setup for the normal and attack situations. In section 3, we detail the type of frames we have analyzed and the attributes and indicators we have selected. In section 4, we explain how we implemented the threshold-based approach on python and comment detection results for the fake access point attack. In section 5, we discussed the limitations of the threshold-based approach and give perspective for a new line of research based on a supervised learning model approach. Finally, in section 6, we conclude the paper.

Experimental Setup

To obtain data, we conducted laboratory experiments. We have configured three situations: a normal situation, a situation with a fake access point attack and a situation with a jamming attack.

Figure 1 describes the normal situation. In this situation, there is an observer, a client, an IEEE 802.11 access point, and a server. The client is connected to the access point via a Wi-Fi link on channel 13 (2.472 GHz). We have ensured that no other surrounding devices in the laboratory were communicating on channel 13. The server is connected to the access point via an ethernet cable (wired connection).

Data is transmitted at a rate of 100 Mbps, using iPerf3 3 , by the server to the client via the access point. The client is at a distance of 120 cm from the access point and at a distance of 100 cm from the observer. The observer is a computer with a Wi-FI card in monitor mode. Monitor mode allows a device to capture all frames sent within a frequency channel. The observer is capturing all frames transmitted by the client and the access point during the experiment. The experiment lasts two minutes.

Figure 2. Setup with jamming attack

In the jamming attack situation, as shown in Figure 2, we have added a jamming signal generated by a signal generator and emitted using a directional antenna at a distance of 100 cm from the client. The rest of the configuration remains unchanged.

In the fake access point attack situation, as shown in Figure 3, we have added a fake access point, at a distance of 120 cm between the client and the server. The fake access point emits the same identification information (beacon frames) at the same interval and on the same frequency channel (channel 13) as the licit access point. 3 iPerf3 is a tool for active measurements of the maximum achievable bandwidth on IP networks, https://iperf.fr/

Analysis and indicators selection

After the experiments, we have a two-minute capture for each situation. The observer has captured all frames transmitted on channel 13. According to the IEEE 802.11 specification, there are three types of frames: management frames, data frames, and control frames. Data frames are frames that contain user data while management frames are frames that are sent to handle discovery, connection, and disconnection with the access point. Control frames are frames sent to coordinate the emission of data frames. In our analysis, we have analysed only management and data frames.

Beacon frames are a subtype of management frames that are sent periodically by an access point to inform, devices in the channel, of its presence. According to the IEEE 802.11 specifications, beacon frames should be sent at an interval of 102.4 ms. A fake access point tries to send the same beacon frames at the same interval as the licit access point. Therefore when there is a fake access point attack operating on the same channel as the licit access point, devices in the channel should receive twice as many beacons. For this reason, we chose to study the distribution of attribute "beacon interval " during the two-minute experiment in the normal situation and the fake access point situation.

During the comparison, we have indeed noticed, as shown in Figure 4, that the mean beacon interval is around 51.2 ms in the fake access point situation and around 102.4 ms in the normal situation. Moreover, in beacon frames, there is a counter called the sequence number and this counter is incremented each a management frame is generated. When an attacker creates a fake access point, even though he can easily copy off some important "static" information of the licit's access point beacon frames, he cannot easily copy the sequence number. During the comparison, by further analyzing the attribute sequence number gap, we also noticed, as shown in Figure 5, that the gap range is higher when there is a fake access point attack.

We have concluded that based on our experiments, the following indicators can be used to detect a fake access point attack on IEEE 802.11 networks: Beacon Interval < 51.2 ms and Sequence Number gap > 8. Concerning the jamming attack, we were expecting to receive little to no frames. However, as per our observations, we, actually, continue to receive management frames in the jamming situation. We have tried to modify the orientation of the jamming to make it face the observer, for instance, but the latter still receives the management frames. After further analysis, we found that management frames in the 2.4 GHz band are sent at a lower bit rate than data and control frames and are sent as Direct-Sequence spread spectrum Modulated Signals. These signals are resilient to frequency sweeping jamming signals. For this reason, we could not observe a significant change in the beacon interval between the normal and the jamming attack situation.

We, therefore, analysed only data frames and when studying the mean data frame interval, we noticed that, as shown in Figure 6, it drastically increases from 0.2 ms to 10000 ms. We concluded that based on our experiment and the data transmission rate (100 Mbps), the following indicator could reasonably be used to detect the presence of frequency sweeping jamming signals: data frame interval > 100 ms. Based on these indicators, we have implemented a first version of the NIDS using Python4 . The NIDS can detect the attack on a live stream or a capture file. In both cases, the NIDS analyses the live stream or capture file using a 10second read buffer. It then decides based on the indicators whether there is the presence of a fake access point or frequency sweeping jamming signals in the 10-second buffer. The procedure is repeated continuously for the live stream and until the end of the file for the capture file. The result is displayed using a graphical interface as shown in Figure 7. We observe, as shown in Figure 7, that in the normal situation, the IDS Status is set to NoAttack. The NIDS supports its decision by displaying the graphs of beacon interval and sequence number gap against time. When we analyze the two graphs, we observe some fluctuations in both cases but values do not cross our predefined thresholds for attacks.

In the presence of a fake access point, as shown in Figure 8 the values cross our predefined thresholds, we observe that for instance, the beacon interval has been divided by 2 and

Limitations and Future Work

The detection fake access points and jamming attacks is not new in scientific research. Several authors have presented different ways to detect these attacks including the threshold-based approach based on beacon interval, sequence number gap and data frame interval [START_REF] Guo | Sequence number-based MAC address spoof detection[END_REF][START_REF] Han | A Scheme of Detection and Prevention Rogue AP using Comparison Security Condition of AP[END_REF][START_REF] Alotaibi | Rogue access point detection: Taxonomy, challenges, and future directions[END_REF][START_REF] Pirayesh | Jamming attacks and anti-jamming strategies in wireless networks: A comprehensive survey[END_REF]. We want to differentiate our work by adopting a holistic approach to detect the attacks. Jamming attacks can be combined with a fake access point attack to increase the latter's effectiveness [START_REF] Amoordon | Characterizing Wi-Fi Man-In-the-Middle Attacks[END_REF]. Deauthentication attacks [START_REF] Amoordon | Characterizing Wi-Fi Man-In-the-Middle Attacks[END_REF] have a similar effect on fake access point attacks. We would like to have a single Network Intrusion Detection System that can detect the three attacks when they are perpetrated independently and simultaneously. Moreover, when detection is based on only one indicator, attackers can more easily adopt counter measurements. For this reason, we would also like to adopt a detection approach that analyses several indicators simultaneously. Finally, concerning the jamming attack, the effect of a jamming attack varies according to the transmission power. In this paper, we have studied the effect of jamming signals with strong transmission power. We would like to be able to study and detect the effect of various jamming power. However, it is tedious to implement a threshold detection approach that can detect the three attacks independently and simultaneously by analysing several indicators simultaneously. For this reason, we have planned to adopt a machine learning approach [START_REF] Shalev-Shwartz | Understanding Machine Learning: From Theory to Algorithms[END_REF] that will also allow us to consider multiple indicators, multiple attacks, multiple jamming effects, and combined attacks. We have also planned to integrate Software Defined Radio equipment (SDR) as an input source to our NIDS. Software Defined Radio equipment as opposed to Wi-Fi card in monitor mode will allow us to analyse signal characteristics that can be combined with frame header information to increase detection efficiency.

Conclusion

We have shown that it is possible to detect fake access points and frequency sweeping jamming attack situations on IEEE 802.11 networks using a threshold-based approach with indicators such as beacon interval, data frame interval, and sequence number gap. We have also highlighted that this detection approach has limitations and that it is difficult to implement when detecting combined attacks using several indicators. For these reasons, we have planned to adopt a machine learning approach in future work. We have also planned to integrate software radio equipment as an input source to our NIDS to increase detection efficiency.

Figure 1 .

 1 Figure 1. Setup in normal situation

Figure 3 .

 3 Figure 3. Setup with fake access point

Figure 4 .Figure 5 .

 45 Figure 4. Data analysis-Beacon Interval

Figure 6 .

 6 Figure 6. Data analysis-Data frame interval

Figure 7 .

 7 Figure 7. Monitoring the normal situation

Figure 8 .

 8 Figure 8. A Fake access point detection

A Network Intrusion Detection System is a system that analyzes incoming network traffic to detect anomalies, https://en.wikipedia.org/wiki/Intrusion d etection s ystem

Frame, https://en.wikipedia.org/wiki/Frame ( networking)

Python is a programming language that lets you work quickly and integrate systems more effectively, https://www.python.org/

Acknowledgements

This work is part of a thesis co-financed by the Region Hauts-de-France, University Gustave Eiffel and the GLOCAT project sponsored by the Region Hauts-de-France (Stimule project). The authors would like to thank the financiers for their support.