Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$

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Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication

in any extension of F _2 and F _3

Stéphane Ballet, Julia Pieltant

To cite this version:

Stéphane Ballet, Julia Pieltant. Tower of algebraic function fields with maximal Hasse-Witt invariant

and tensor rank of multiplication in any extension of

_2 and

_3. Journal of Pure and Applied

Algebra, Elsevier, 2018, 222 (5), pp.1069-1086. �10.1016/j.jpaa.2017.06.007�. �hal-01063511v2�

Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in

any extension of F

and F

Stéphane Ballet

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France.

Case 907, 163 Avenue de Luminy, 13288 Marseille Cedex 9, France.

Julia Pieltant^1,∗

Inria – Saclay Île-de-France, LIX, École Polytechnique, 91128 Palaiseau Cedex, France.

Abstract

Up until now, it was recognized that a detailed study of the p-rank in towers of function fields is relevant for their applications in coding theory and cryptography.

In particular, it appears that having a largep-rank may be a barrier for a tower to lead to competitive bounds for the symmetric tensor rank of multiplication in every extension of the finite fieldFq, withqa power of p. In this paper, we show that there are two exceptional cases, namely the extensions ofF2andF3. In particular, using the definition field descent on the field with 2 or 3 elements of a Garcia- Stichtenoth tower of algebraic function fields which is asymptotically optimal in the sense of Drinfel’d-Vl˘adu¸t and has maximal Hasse-Witt invariant, we obtain a significant improvement of the uniform bounds for the symmetric tensor rank of multiplication in any extension ofF2andF3.

Keywords: Algebraic function field, tower of function fields, tensor rank, algorithm, finite field.

1. Introduction 1.1. General context

The determination problem of the tensor rank of multiplication in finite fields has been widely studied over the past 20 years, as shown by the growing number of

∗Corresponding author.

Email addresses:[email protected](Stéphane Ballet), [email protected](Julia Pieltant)

URL:http://iml.univ-mrs.fr/∼ballet(Stéphane Ballet), http://www.lix.polytechnique.fr/∼pieltant(Julia Pieltant)

1Present address:CNRS LTCI, Télécom ParisTech , 46 rue Barrault, 75634 Paris Cedex 13, France.

publications on this topic including amoung others[12],[18],[2],[11],[5],[16]. This problem is worthwhile both because of its theoretical interest and because it has several applications in the area of information theory such as cryptography and coding theory.

1.2. Tensor rank of multiplication

LetFqbe a finite field withqelements whereqis a prime power and letFqⁿbe a Fq-extension of degreen. The multiplication in theFq-vector spaceFqⁿis a bilinear map fromFqⁿ×Fqⁿ intoFqⁿ, thus it corresponds to a tensor t∈F^?_qⁿN

F^?_qⁿN Fqⁿ

whereF^?qⁿ denotes the dual ofFqⁿ overFq. Hence the product of two elements x andyofFqⁿis the convolution of this tensor withx⊗y∈Fqⁿ

NFqⁿ. If

t= Xλ

`=1

a_{`</sub>⊗b<sub>`}⊗c_`

wherea_{`</sub>∈F<sup>?</sup><sub>q</sub><sup>n</sup>,b<sub>`}∈F^?_qⁿ,c_`∈Fqⁿ, then

x·y=t(x⊗y) = Xλ

`=1

a_{`</sub>(x)b<sub>`}(y)c_{`</sub>. (1) Every expression (1) is called a bilinear multiplication algorithmU (resp. a symmetric bilinear multiplication algorithm ifa<sub>`}= b<sub>`</sub> for all`∈ {1, . . . ,λ}). The integer λ is called the tensor rank (resp. symmetric tensor rank) of the algo- rithmU, or the bilinear complexity (resp. symmetric bilinear complexity) of the algorithmU.

Let us set

µq(n):=min

U µ(U), respectively

µ^symq (n):=min

U^symµ(U^sym),

whereU is running over all bilinear multiplication algorithms (resp.U^symis run- nig over all symmetric such algorithms) inFqⁿoverFq.

It is interesting to study the minimal length of a symmetric multiplication algo- rithm since it turns out that it plays an important role in several other areas such as Riemann-Roch system of equations, arithmetic secret sharing, multiplication- friendly codes, etc, as mentioned in[9].

1.3. Basic notions related to function fields and notation

LetF/Fqbe an algebraic function field of one variable of genusg, with constant fieldFq. For any integerk≥1, we denote byP_k(F/Fq)the set of places of degree k, by B_k(F/Fq)the cardinality of this set and by P₍F/Fq) =∪kP_k(F/Fq) the set of all places inF/Fq. For any place P, we define F_P to be the residue class field of PandOP its valuation ring. Every elementt∈P such that P=tOP is called a

local parameter forP. The support ofD=P

Pa_PP is the set of the placesP such thata_P6=0.

For any divisorD ofF/Fq, the Riemann-RochFq-vector-space associated toD is the set

L(D) ={f ∈F/Fq| D+ (f)≥0} ∪ {0}.

The Riemann-Roch Theorem states that the dimension dimDof the vector space L(D)is related to the degree of the divisorDand to the genus ofF/Fqby:

dimD=degD −g+1+dim(κ− D), (2) whereκdenotes a canonical divisor ofF/Fq. In this relation, the complementary term i(D):=dim(κ− D)is called the index of speciality of D. Note that in any case, we have i(D)≥0. In particular, a divisorD is called a non-special divisor when the index of speciality i(D)is zero andDis called a special divisor if i(D)>0.

1.4. Known results 1.4.1. General results

The bilinear complexityµq(n)of the multiplication in then-degree extension of a finite fieldFq is known for certain values ofn. In particular, Winograd[20]and de Groote[13] have shown that this complexity is≥2n−1, with equality hold- ing if and only ifn≤¹₂q+1. Using the principle of the Chudnovsky-Chudnovsky algorithm[12]applied to elliptic curves, Shokrollahi has shown in[17]that the bi- linear complexity of multiplication is equal to 2nfor ¹

2q+1<n< ¹₂(q+1+ε(q)) whereεis the function defined by:

ε(q) =

greatest integer≤2pqprime toq, ifqis not a perfect square 2pq, ifqis a perfect square.

Eventually, the original algorithm of D.V. and G.V. Chudnovsky introduced in [12]leads to the following theorem by[3]:

Theorem 1. Let q be a prime power. The tensor rankµq(n)of multiplication in any finite extensionFqⁿ ofFqis linear with respect to the extension degree; more precisely, there exists a constant C_q such that for any n, it holds that:

µ^sym_q (n)≤C_qn.

Moreover, one can give explicit values forC_q; in particular forq=2 orq=3:

Proposition 2. The best known values for the constant C_q defined in the previous theorem are:

C_q=







19.6 if q=2 [11],[5] 27 if q=3 [3]

Remark. The estimateC₂=19.6 is obtained by combining the general uni- form boundµ^sym₂ (n)≤⁴⁷⁷₂₆n+⁴⁵₂ from[5]forngreater than 19, and the values of µ^sym₂ (n)given in[11, Table 1]forn≤18 (see Appendix).

Let us present the best finalized version of this algorithm in its symmetrical version, which is a generalization of the algorithm of Chudnovsky-Chudnovsky type introduced by Arnaud in[2]and developed later by Cenk and Özbudak in [11]. This generalization uses several coefficients in the local expansion at each place P_i instead of just the first one. Due to the way to obtain the local expansion of a product from the local expansion of each term, the bound for the symmetric bilinear complexity involves the complexity notionMb_q(u)introduced by Cenk and Özbudak in[11]and defined as follows:

Definition 3. We denote byMb_q(u)the minimum number of multiplications needed in Fqin order to obtain coefficients of the product of two arbitrary u-term polynomials modulo x^uinFq[x].

Note that in[16], Randriambololona gives an even more general version of the Chudnovsky-Chudnovsky algorithm, which encompass the case of non-necessarily symmetric algorithms. This generalization is not relevant here, since we focus on the symmetric tensor rank; thus we introduce the generalized symmetric algorithm Chudnovsky-Chudnovsky type described in[11].

Theorem 4. Let

• q be a prime power,

• F/Fqbe an algebraic function field,

• Q be a degree n place ofF/Fq,

• D be a divisor ofF/Fq,

• P={P₁, . . . ,P_N}be a set of N places of arbitrary degree,

• t₁, . . . ,t_N be local parameters for P₁, . . . ,P_Nrespectively,

• u₁, . . . ,u_N be positive integers.

We suppose that Q and all the places inPare not in the support ofDand that:

a) the map

Ev_Q:

L(D) → Fqⁿ'F_Q f 7−→ f(Q) is onto,

b) the map

Ev_P:

L(2D) −→ € Fq^degP1

Šu₁

׀ Fq^degP2

Šu₂

× · · · ׀ Fq^degPN

Šu_N

f 7−→ ϕ1(f),ϕ2(f), . . . ,ϕN(f)

is injective, where each applicationϕiis defined by

ϕi:

L(2D) −→ € Fq^degPi

Šu_i

f 7−→ €

f(P_i),f⁰(P_i), . . . ,f^(ui−1)(P_i)Š

with f = f(P_i) +f⁰(P_i)t_i+f⁰⁰(P_i)t²_i +. . .+f^(k)(P_i)t_i^k+. . ., the local expansion at P_i of f in L(2D), with respect to the local parameter t_i. Note that we set

f⁽⁰⁾:=f . Then

µ^symq (n)≤

N

X

i=1

µ^symq (degP_i)Mb_qdegPi(u_i).

In particular, we will consider in this paper a specialization of this algorithm which is described in Section4and requires the additional hypothesis that there exists a non-special divisor of degreeg−1; this will motivate the study of ordinary towers.

1.4.2. Asymptotic bounds for the extensions ofF2andF3

The following asymptotic bound for the bilinear complexity was introduced in [18]:

M^sym_q :=lim sup

k→∞

µ^symq (k)

k .

Recently, with the help of the torsion-limit technique and Riemann-Roch sys- tems, Cascudo, Cramer and Xing improved in[9] the upper bounds for M^sym_q in the case whereqis small (q∈ {2, 3, 4, 5}). In particular, from optimal towers with asymptotically few 2-torsion points relatively to the genus they obtained:

M^sym₂ ≤7.23 and M^sym₃ ≤5.45.

1.5. Motivations – New results established in this paper

As mentioned in[9]by Cascudo, Cramer and Xing, and in [7]by Bassa and Beelen, it is widely accepted that towers of algebraic function fields having a large p-rank are less efficient for certain applications to information theory. Indeed, for example, in [7], it reads: “A detailed study of the p-rank in towers is rele- vant for their applications, see [8]. For example, although both of the towers introduced in [15] and [14] have the same limit and hence are equally influ- ential for applications in coding theory, a detailed study of their p-rank reveals that in fact the latter² is more appropriate for other kinds of applications, e.g.

secure multiparty computation and fast bilinear multiplication.” In particular, it appears that having a largep-rank may be a barrier for a tower to lead to com- petitive bounds for the symmetric tensor rank of multiplication in every extension

2which turns out not to be ordinary, according to[9]

of the finite fieldFq, with qa power of p. In this paper, we show that there are two exceptional cases, namely the extensions of F2 andF3. More precisely, we will show that an ordinary tower may lead to better uniform results for the ten- sor rank of multiplication in any extension ofF2andF3than a non-ordinary one because of the link between maximal p-rank and existence of a non-special divi- sor of degree g−1. Indeed, we know that the existence of a non-special divisor of degree g−1 in the function fieldF/Fq is of crucial importance in the perfor- mance of Chudnovsky-Chudnovsky type algorithms over Fq designed from F/Fq

[4], [16]. When the definition field Fq is such thatq≥4, then according to [4] there always exists a non-special divisor of degreeg−1. Nevertheless, the prob- lem persists in the case where the definition field isF2orF3. In[5], to avoid this obstacle, we substituted non-special divisors of degreeg−1 for zero-dimensional divisors whose degree is as close as possible to g−1, in the descent overF2 of the original Garcia-Stichtenoth tower presented in[14]and defined overF16; non- special divisors of degreeg−1 being the borderline case of zero-dimensional divi- sors. However, Bassa and Beelen established in[7]that the second optimal Garcia- Stichtenoth tower introduced in [15] and defined overFq² is ordinary. On the other hand, it was shown in[6] that there always exists a non-special divisor of degree g−1 in any ordinary function field. In this article, we combine these two results to a modified version of the optimal tower studied by Bassa and Beelen which improves the results obtained in[5]. Namely we define intermediate steps for the tower and descend the definition field fromF16 toF2, and fromF9toF3

respectively, which lead us to the two following bounds:

µ^sym₂ (n)≤15.23n+9

2 and µ^sym₃ (n)≤7.732n.

Note that the difficulty to obtain non-asymptotic estimations of the 2-torsion points in all steps of the tower used in[9]is an obstruction to obtain uniform bounds as we get in this paper.

2. Definitions and related properties of thep-rank

Definition 5. The p-rankγ(F), also called Hasse-Witt invariant, of a function fieldF with constant field Fp, the algebraic closure of the finite field Fp, is defined as the dimension overFp of the group of divisor classes of degree zero of order p.

If the function field is defined over a finite fieldFq, we define its p-rank as the p-rank of the function field FFq, obtained by extending the constant field to the algebraic closure ofFq.

It can be shown that :

Proposition 6. IfF/Fqbe a function field of genus g(F), then0≤γ(F)≤g(F). Definition 7. A function fieldF/Fqis called ordinary ifγ(F) =g(F).

A tower of function fieldsT = Fn/Fq

n∈Nis said ordinary if for any n≥0,Fnis such thatγ(Fn) =g(Fn), i.e. if any step of the tower is an ordinary function field.

Let us recall the following result from[6]:

Corollary 8. If F is an ordinary function field of genus g>0 defined over F2 or overF3, then there is always a degree g−1zero-dimensional divisor inF.

Moreover, directly from Definition5, we can deduce the following lemma:

Lemma 9. Let r≥0 be an integer. If we set F/Fq^r:=H/Fq⊗Fq^r, then H/Fq is ordinary if and only ifF/Fq^ris ordinary.

PROOF. Note that the genus does not change under constant field extension or descent. It follows from Definition5that p-rank does not change under constant field extension or descent since thep-rank of a function fieldF/Fq defined over a finite fieldFqis equal to thep-rank ofFFq.

To conclude this section, we recall the following result which is proven in[7, Lemma 6, 2.]:

Lemma 10. IfH/Fis a finite extension of function fields with same constant fieldFq, then

g(H)−γ(H)≥g(F)−γ(F). In particular, ifHis ordinary then so isF.

3. Good ordinary sequences of function fields defined overF2orF3

In this section, we present sequences of algebraic function fields defined overF2

orF3, constructed from the well-known Garcia-Stichtenoth tower defined in[15], which will be used to obtain new bounds for the tensor rank of multiplication.

3.1. Definition of the Garcia-Stichtenoth’s tower

Let us consider a finite fieldFq² withq=p^r, for p a prime number and r an integer. We consider the Garcia-Stichtenoth’s elementary abelian towerT₀overFq²

constructed in[15]and defined by the sequence(F₀,F₁,F₂, . . .)where F₀:=Fq²(x₀)

is the rational function field overFq², and for anyi≥0,F_i+1:=F_i(x_i+1)withx_i+1 satisfying the following equation:

x^q_i+1+x_i+1= x_i^q x_i^q−1+1.

Let us denote byg_ithe genus ofF_iinT₀/Fq²and recall the following formulæ:

g_i=

¨ (qⁱ⁺¹² −1)² for oddi,

(qⁱ²−1)(qⁱ⁺²² −1) for eveni. (3)

Thus, according to these formulæ, it is straightforward that the genus of any step of the tower satisfies:

(q²ⁱ −1)(qⁱ⁺¹² −1)<g(F_i)<(qⁱ⁺²² −1)(qⁱ⁺¹² −1). (4) Moreover, a tighter upper bound will be useful and can be obtained by expanding expressions in (3):

g(Fi)≤qⁱ⁺¹−2qⁱ⁺¹² +1. (5) If the characteristicp=2 andr=2, i.e.q=4, then we can densify the Garcia- Stichtenoth’s tower with steps defined over the finite fieldFq² by considering the following completed tower:

T₁/F16 : F_0,0⊆F_0,1⊆F_0,2=F_1,0⊆F_1,1⊆F_1,2=F_2,0⊆ · · ·

such that F_i⊆F_i,s⊆F_i+1 for any integer s∈ {0, 1, 2}, with F_i,0:=F_i and F_i,2:=F_i+1. Indeed:

Proposition 11. There exists a tower T₁defined overF16whose recursive equation is defined overF2. More precisely, the tower T₁ is the densified Garcia-Stichtenoth’s tower overF16and is defined by T₁= F_i,s

i≥0s∈{0,1}

where for any i≥0:

F_i,0:=F_i and F_i,1:=F_i(t_i+1) with t_i+1satisfying the equation:

t²_i+1+t_i+1= x_i⁴

x³_i +1 for i=0, . . . ,n−1. (6) PROOF. Letx₀be a transcendental element overF2and let us set

F₀:=F16(x₀). We define recursively fori≥0

(i) x_i+1such thatx⁴_i+1+x_i+1= _x3^xi4

i+1 fori=0, . . . ,n−1, (ii) t_i+1such that t_i²₊₁+t_i+1= _x3^x4i

i+1 fori=0, . . . ,n−1, (or alternativelyt_i+1=x²_i+1+x_i+1).

Thus, we can define recursively the towerT₁by setting:

F_i,1=F_i,0(t_i+1) =Fi(t_i+1) and Fi+1,0=Fi+1=Fi(x_i+1).

Let us remark that it is possible to densify the general Garcia-Stichtenoth’s tower overFq² for any characteristic p and for any integer r since each exten- sionF_i+1/F_iis Galois of degreeq=p^rwith full constant fieldFq². However, in the general case the equation (6) for the intermediate steps is not defined overFp but overFq. For example, forp=3 andr=2, we obtain an equation which is defined overF9.

Notation. In the sequel, we will denote by B_k(F/K)the number of places of degreek of an algebraic function fieldF/K defined over a finite fieldK; we will also denote byg_i,sthe genus ofF_i,s/F16inT₁/F16.

3.2. Descent of the definition field of the Garcia-Stichtenoth’s tower on the fieldsF2

andF3

First we state that when q = 3, one can descend the definition field of the tower T₀/Fq² from Fq² toFq since the recursive equation defining the tower has coefficients lying inFq. Thus, we have the following result:

Proposition 12. If q=p=3, there exists a tower E/Fq defined overFq given by a sequence:

G₀⊆G₁⊆G₂⊆G₃⊆ · · ·

defined over the constant fieldFqand related to the tower T₀/Fq²by F_i=FqG_i for all i,

namelyF_i/Fq²is the constant field extension ofG_i/Fq.

Now, we are interested in the descent of the definition field of the towerT₁/Fq²

fromFq²toFpif it is possible. In fact, for the towerT₁/Fq², one can not establish a general result but one can prove that it is possible in the case where the character- istic is 2 andr=2, i.e. q=4. Note that in order to simplify the presentation, we are going to set the results by using the variablep.

Proposition 13. If p=2and q=p², the descent of the definition field of the tower T₁/Fq² fromFq² toFp is possible. More precisely, there exists a tower T₂/Fpgiven by a sequence:

H_0,0⊆H_0,1⊆H_0,2=H_1,0⊆H_1,1⊆H_1,2=H_2,0⊆ · · · defined over the constant fieldFpand related to the tower T₁/Fq²by

F_i,s=Fq²H_i,s for all i≥0and s∈ {0, 1, 2}, namelyF_i,s/Fq² is the constant field extension ofH_i,s/Fp. PROOF. It is a straightforward consequence of Proposition11.

In order to draw consequences for the previously descended towers, let us recall the known results concerning the number of places of degree one of the towerT₀/Fq², established in[15]and[1].

Proposition 14. If q=p^r≥2, then for any n>2:

B₁(F_n/Fq²) =

qⁿ(q²−q) +2q² if p=2, qⁿ(q²−q) +2q if p>2.

Now, we deduce some straightforward properties concerning the towersT₂/F2

andE/F3.

Proposition 15. Let q= p²=4. For any integers i≥0and s∈ {0, 1, 2}, the alge- braic function fieldH_i,s/Fp in the tower T₂/FphasB₁(H_i,s/Fp)places of degree one, B₂(H_i,s/Fp)places of degree two andB₄(H_i,s/Fp)places of degree four and satisfies:

(i) H_i/Fp⊆H_i,s/Fp⊆H_i+1/FpwithH_i,0=H_iandH_i,2=H_i+1, (ii) the genus g_i,sofH_i,s/Fp satisfies:

(ii.a) g_i,s≤ _p^gi+12−s (ii.b) g_i,s≤p^s−2(qⁱ⁺²−2q²ⁱ⁺¹) +p^s−2 (iii) B₁(H_i,s/Fp) +2B₂(H_i,s/Fp) +4B₄(H_i,s/Fp)≥qⁱ(q²−q)p^s.

Moreover,Fp is algebraically closed in each algebraic function fieldH_i,sof the tower T₂/Fp.

Remark. Bound (ii.a) is tighter than Bound (ii.b), but when we will need an estimate forg_i,swhich does not depend on the parity of the stepiof the tower, Bound (ii.b) will be useful.

PROOF. Property (i) follows directly from Proposition 13. Each extension Hi+1/H_i,sis a Galois extension of degree[Hi+1:H_i,s] =2^2−s. Moreover, according to[19, Prop. 3.7.8]the full constant field ofH_i,sisFp since at least one place of H₀is totally ramified inH_i,s. Indeed, the place at infinity ofF₀is totally ramified in the towerT₀/Fq². Hence, the same holds for the place at infinity ofH₀inT₂/Fp. Since the algebraic function fieldF_i,s is a constant field extension ofH_i,s, for any two integersi≥0 and s∈ {0, 1, 2}, F_i,s andH_i,s have the same genus, so by the Hurwitz Genus Formula[19], one has:

g_i,s≤ g_i+1

p^2−s (7)

with g(H_i+1/Fp) =g(F_i+1/Fq²) =g_i+1given by (3). Finally, applying Bound (5) ong_i+1, we get (ii.b). Moreover, forα∈Fq²\{ω∈Fq²|ω^q+ω=0}, letP_αdenote the place of degree one in the rational function fieldF₀which is the zero ofx₀−α, thenP_αsplits completely inF_i+1/F₀by[15, Lemma 3.9]. Let us setd:= [F_i+1:F₀] andd⁰ := [F_i+1 :F_i,s]. If ` denotes the number of places of F<sub>i,s</sub> lying over the placeP<sub>α</sub>ofF<sub>0</sub>, it is well known that`≤ _d^d0 with equality holding if and only ifP_α splits completely inFi+1/F₀; so`= _d^d0 = [F_i,s:F₀]which proves that the place P_α splits completely also inF_i,s/F₀. Thus, there are exactlyqⁱp^s places of degree one aboveP_αinF_i,s, so there are at leastqⁱ(q²−q)p^splaces of degree one inF_i,s, since

Fq²\{ω∈Fq²|ω^q+ω=0}

=q²−q.

To conclude, we deduce from [15] (where the same formula with s=0 is proven forF_i,0) that the number of places of degree one ofF_i,s/Fq² is such that B₁(F_i,s/Fq²)≥(q²−q)qⁱp^s. Eventually, F_i,s being a degree four constant field ex- tension ofH_i,s, it is clear that for any two integersi≥0 ands∈ {0, 1, 2}, it holds that

B₁(H_i,s/Fp) +2B₂(H_i,s/Fp) +4B₄(H_i,s/Fp)≥(q²−q)qⁱp^s.

Similar results than those of Proposition15can be obtained for the towerE/F3, namely:

Proposition 16. Let q= p =3. For any integer i≥0, the algebraic function field G_i/Fq in the tower E/Fq has the same genus g_i than the corresponding step F_i/Fq²

of the tower T₀/Fq². Moreover, the number of places of degree one and two of each function fieldG_i/Fqis related to the number of rational places ofF_i/Fq²by:

B₁(F_i/Fq²) =B₁(G_i/Fq) +2B₂(G_i/Fq) thus, the following bound holds:

B₁(G_i/Fq) +2B₂(G_i/Fq)≥qⁱ(q²−q). (8) To conclude this section, let us recall that in[7], the authors established the ordinarity of the classical tower overFq²:

Theorem 17. For any prime power q, the tower T₀/Fq²is ordinary.

Thus, we can deduce that the ordinarity of T₀/Fq² provides the same property to the towersT₂/F2andE/F3:

Proposition 18. The towers T₂/F2and E/F3are ordinary.

PROOF. Since constant field descent preserves ordinarity from Lemma9, the tower E/F3is ordinary and so are the stepsF_i,0of the towerT₂/F2. Moreover Lemma10 implies that the intermediate stepsF_i,1are also ordinary since each one belongs to a finite extensionF_i+1,0/F_i,1with same constant field, whereF_i+1,0is ordinary.

A straightforward consequence of this last proposition and Corollary 8is the following result:

Corollary 19. For any function fieldFin the towers T₂/F2and E/F3, there exists a non-special divisor of degree g(F)−1.

4. New bounds for the tensor rank 4.1. Preliminary results

To obtain our new estimates forµ^sym2 (n)andµ^sym3 (n)from the towers described in the previous section, we will need some technical results which are proven below.

Theorem 20. Let n and d be two fixed integers. LetF/Fq be an algebraic function field of genus g with at least B_kplaces of degree k for any k|d. If the three following conditions are satisfied:

(a) B_n(F/Fq)>0,

(b) there exists a non-special divisor of degree g−1, (c) P

k|dk(B_k+b_k)≥2n+2g−1, where the integers b_k are chosen such that 0≤b_k≤B_k,

then

µ^symq (n)≤X

k|d

µ^symq (k)(B_k+b_k) +X

k|d

µ^symq (k)b_k, so

µ^symq (n)≤η



 X

k|d

k(B_k+b_k) +X

k|d

k b_k



 withη:=max

k|d

µ^symq (k)

k .

PROOF. The algorithm recalled in Theorem4is applied on a setP=∪k|dPkwith Pk a subset ofP_k(F/Fq)with cardinality B_k. Among each P∈Pk, b_k are used with multiplicityu=2; all such places form a subsetRofP. The othersB_k−b_k places ofPk are used with multiplicityu=1. From the existence of a non-special divisorG of degreeg−1 provided by Hypothesis (b) and the existence of a place Q of degree n, one constructs an effective divisorD such that degD=n+g−1 and dimD = n. Precisely, one can choose any divisor D which is equivalent to Q+G, but whose support is disjoint from the support ofQ+G. For such a divisor D, the mapEv_Q is bijective: indeed its kernelL(D −Q)is trivial sinceD −Qis equivalent toG which is non-special of degree g−1 and so is zero-dimensional;

thusEv_Qis injective and actually bijective by dimension reasons. Moreover, from Hypothesis (c) it holds thatL

2D − P

P∈PP+P

R∈RR

={0}since

deg 2D −(X

P∈P

P+X

R∈R

R)

!

=2 degD −X

k|d

k(B_k+b_k)<0

thusEv_Pis injective. From[11], it holds thatMb_q(2)≤3, so Theorem4then gives the following bound:

µ^sym_q (n)≤ X

P∈P

µ^sym_q (degP) +2X

R∈R

µ^sym_q (degR).

Rearranging summation to group places with the same degree, we get the result.

Here we state two special cases of Theorem20which are adapted to the study of the tensor rank onF2andF3respectively. The first one is adapted to the case where places of degree one, two and four are taking into account:

Proposition 21. Let p=2. IfF/F2is an algebraic function field of genus g with at least B_k places of degree k for k=1, 2 and 4, such that the three following conditions are satisfied:

(a) B_n(F/F2)>0,

(b) there exists a non-special divisor of degree g−1, (c) P

k|4k(B_k+b_k)≥2n+2g−1, where the integers b_k are chosen such that 0≤b_k≤B_k,

then

µ^sym₂ (n)≤9

2(n+g+1) +9 4

X

k|4

k b_k.

PROOF. It is a straightforward consequence of Theorem 20 with q=p=2 and d=4. We recall that µ^sym2 (2) = 3 and µ^sym2 (4) ≤ 9; so we obtain η=max_k|4^µ

sym 2 (k)

k ≤maxn

1;³

2;⁹

4

o=⁹₄ and the result follows from a choice of the B_k’s and the b_k’s such that P

k|4k(B_k+b_k) =2n+2g−1+ε, with ε∈ {0, 1, 2, 3}: we must consider the less favorable case where there only exists places of degree four and so we have to chooseε=3.

This second specialization corresponds to the case where only places of degree one and two are considered:

Proposition 22. Let q =3. IfF/F3 is an algebraic function field of genus g with at least B_k places of degree k for k=1, 2 such that the three following conditions are satisfied:

(a) Bn(F/F3)>0,

(b) there exists a non-special divisor of degree g−1, (c) P

k|2k(B_k+b_k)≥2n+2g−1, where the integers b_k are chosen such that 0≤b_k≤B_k,

then

µ^sym₃ (n)≤3(n+g) +3 2

X

k|2

k b_k.

PROOF. The same proof than the previous one withq=3 andd=2, and soη= ³₂ in Theorem20gives the result.

Lemma 23. Let q=4=p²and n≥19. There exists a stepH_i,s/F2of the tower T₂/F2

such that the three conditions of Proposition 21 are satisfied with b₁=b₂=b₄=0. Moreover, if Condition (c) is satisfied then the two others also are.

PROOF. Thanks to Corollary19, Condition (b) is satisfied for any step of the tower.

For i≤ ⁿ⁻¹³₄ , it holds that p²ⁱ⁺⁶≤pⁿ⁻¹² . Then we get that p²ⁱ⁺⁶

1−_p¹i+2+_p2i+3¹

≤pⁿ⁻¹² p²(pp−1), since 1− _pi+2¹ + _p2i+3¹ ≤1≤p²(pp−1). It follows that p²ⁱ⁺⁴−pⁱ⁺²+p≤pⁿ⁻¹² (pp−1), which leads to 2g_i,s+1≤pⁿ⁻¹² (pp−1)according to Proposition15(ii.b) withs∈ {0, 1}(one can always assume thats6=2 sinceH_i,2=H_i+1,0). Hence, from[19, Corollary 5.2.10] Condition (a) is satisfied for any stepH_i,ssuch thati≤ ⁿ⁻¹³₄ .

On the other hand, for i such that i>log_q(n)−¹₂, one has qⁱ⁺¹⁻¹² ≥n+1, so qⁱ⁺¹p^s−1(q−3)≥n+1 since p^s−1≥q⁻¹² =p⁻¹, which gives that qⁱ⁺¹p^s(q−3)≥2n+2 and so qⁱ⁺¹p^s(q−1−2) +q^{i−12 ps}≥2n+2. Thus it holds that: qⁱ⁺¹p^s(q−1)≥2n+2+2qⁱ⁺¹p^s−qⁱ⁻¹² p^s=2n+2p^s−2(qⁱ⁺²−q²ⁱ⁺¹) +2.

Eventually, one gets thatqⁱ⁺¹p^s(q−1)≥2n+2p^s−2(qⁱ⁺²−q²ⁱ⁺¹) +2p^s−2−1 since 2p^s−2−1≤2 fors∈ {0, 1}, and Condition (c) is satisfied according to the inequal- ities (ii.b) and (iii) established in Proposition15.

Thus, for n≥21 one can find at least one integer i in the interval i

log_q(n)−¹₂;ⁿ⁻¹³

4

i

, and so a corresponding step of the towerH_i,0for which Propo- sition21holds. Note that in any case, Condition (a) is satisfied for lower steps than Condition (c), so it may happened that the first suitable step that satisfy both con- ditions is notH_i,0itself but one of the previous step.

Moreover one can check that forn=19,H₁is the first suitable step of the tower to apply Proposition21with b₁=b₂=b₄=0. Indeed, it holds that g(H₁/F2) =9 so Condition (a) is satisfied and since B₁(H₁/F2) = 4, B₂(H₁/F2) = 2 and B₄(H₁/F2) =12, Condition (c) is also satisfied forH₁but it is not the case forH_0,1. Similarly forn=20,H₁does not satisfy Condition (c), butH_1,1does satisfy both Conditions (a) and (c) since g(H_1,1/F2) =21, B₁(H_1,1/F2) =4, B₂(H_1,1/F2) =2 and B₄(H_1,1/F2) =25.

Lemma 24. Let q=3and n≥13. There exists a stepGi/F3of the tower E/F3such that the three conditions of Proposition22are satisfied with b₁=b₂=0. Moreover, if Condition (c) is satisfied then the two others also are.

PROOF. Thanks to Corollary19, Condition (b) is satisfied for any step of the tower.

Fori≤ⁿ⁻⁵₂ , Condition (a) is satisfied since one has: qⁱ≤^q

n−42

pq ≤ ^q

n−42

pq+1=qⁿ⁻⁴²

pq−1 2

and so (qⁱ⁺²² −1)(qⁱ⁺¹² −1)≤qⁿ⁻¹²

pq−1

2 which gives that the inequality 2g_i+1≤qⁿ⁻¹² (pq−1)of[19, Corollary 5.2.10]holds according to (4).

On the other hand, wheni≥2 log_q€_n

2−1Š

, Condition (c) is satisfied. Indeed, for suchione has: q²ⁱ ≥ ⁿ₂−1, so 4q²ⁱ ≥2n−5, which gives that 4qⁱ²+2q≥2n+1.

Adding 2qⁱ⁺¹, which equals (q²−q)qⁱ, to both sides it follows that:

(q²−q)qⁱ+2q≥2n+2qⁱ⁺¹−4q²ⁱ +1=2n+2(qⁱ⁺¹−2q²ⁱ +1)−1. Thus from (8) and (5) we get that inequality of Condition (c) holds withb₁=b₂=0.

To conclude, one can see that forn≥13, the interval”

2 log_q€_n

2−1Š

;ⁿ⁻⁵

2

—con- tains at least an integeri and soG_i/F3is a suitable step of the tower; moreover

the smallest such integer is the smallesti≥2 log_q€_n

2−1Š

, i.e. the smallest one for which Condition (c) is satisfied.

Till the end of this section, we will deal with the following notations:

n_2,i,sdef := maxn

m

2m+2g(H_i,s)−1≤X

k|4

kB_k(H_i,s/F2)o

and

n_3,idef := maxn

m

2m+2g(G_i)−1≤X

k|2

kB_k(G_i/F3)o .

Let us explain the relevance of these definitions, focusing on the case of the role ofn_3,i in the towerE/F3 (the same holds for the towerT₂/F2 when one replaces n_3,ibyn_2,i,s). The integern_3,iis the biggest one for which it holds that:

X

k|2

kB_k(G_i/F3)≥2n_3,i+2g_i−1

i.e. Fq^n3,i is the biggest extension ofF3for whichG_i/F3could be a suitable step of the tower to apply Proposition22withb₁=b₂=0. Ifn>n_3,i, then

X

k|2

kBk(Gi/F3)<2n+2gi−1 but one has

X

k|2

kBk(Gi/F3) +2(n−n_3,i)≥2n+2gi−1

which means thatG_iis still a suitable step of tower to apply Theorem22if we can choose theb_k’s such thatP

k|2k b_k≥2(n−n_3,i).

Thus, we are interested in the determination of a lower bound forn_3,iandn_2,i,s. It is the purpose of the two following lemmas:

Lemma 25. If p=2and q=p²=4, then n_2,i,s≥qⁱ⁺¹p^s+q²ⁱ⁺¹p^s−1.

PROOF. According to Proposition15(iii) and (ii.a), and Formula (5), we get:

X

k|4

kBk(H_i,s/F2)−2g(H_i,s) +1 ≥ (q²−q)qⁱp^s−2p^s−2(qⁱ⁺²−2qⁱ²⁺¹+1) +1

= qⁱ⁺²p^s−qⁱ⁺¹p^s−p^s−1(qⁱ⁺²−2q²ⁱ⁺¹+1) +1

= qⁱ⁺²p^s−1(p−1)−qⁱ⁺¹p^s+q²ⁱ⁺¹p^s−p^s−1+1

≥ qⁱ⁺¹p^s−1(q−p) +q²ⁱ⁺¹p^s−1

sinces∈ {0, 1, 2}andp−1=1

= qⁱ⁺¹p^s+q²ⁱ⁺¹p^s−1.