Gu Éna Ël Renault
email: guenael.renault@calfor.lip6.fr

Computation of the Splitting Field of a Dihedral Polynomial *

Keywords: I.1 [Computing Methodologies]: Symbolic and algebraic manipulations Algorithm, Theory Galois theory, triangular set, splitting field, dihedral group

Let g be a univariate separable polynomial of degree n with coefficients in a computable field K and let (α1, . . . , αn) be an n-tuple of its roots in an algebraic closure K of K.

Obtaining an algebraic representation of the splitting field K(α1, . . . , αn) of g is a question of first importance in effective Galois theory. For instance, it allows to manipulate symbolically the roots of g. In this paper, we focus on the computation of the splitting field of g when its Galois group is a dihedral group. We provide an algorithm for this task which returns a triangular set encoding the relations ideal of g which has a degree 2n since the Galois group of g is dihedral. Our algorithm starts from a factorization of g in K[X]/ g and constructs the searched triangular set by performing n 2 computations of normal forms modulo an ideal of degree 2n.

INTRODUCTION

The computation of the splitting field of a polynomial plays an important role in Galois theory and more generally in algebra. It is the smallest field where all the roots of the polynomial lie. Computing a suitable representation of this field allows us to manipulate all the roots of the polynomial. Let g be a polynomial of degree n with coefficients in a computable field K and whose Galois group is dihedral, * which implies that it is irreducible and separable. We are interested here with the computation of the splitting field of g and the representation of the action of the Galois group over the roots of g. Our aim is to exploit the knowledge on the Galois group of g.

The splitting field of g can be represented as a simple extension of the base field but, since here we want to compute with all the roots of g it is better to considerate another representation. The natural representation for this task is the following quotient algebra K(α1, . . . , αn) K[x1, . . . , xn]/I where I is the kernel of the surjective morphism from K[x1, . . . , xn] to K(α1, . . . , αn) which maps xi to αi. The ideal I is called a relations ideal of g. Remark that I is zerodimensional and maximal. A Gröbner basis of I allows computations in this quotient algebra by means of linear algebra operations (see e.g. [START_REF] Cox | Ideals, varieties, and algorithms[END_REF][START_REF] Becker | Gröbner bases[END_REF]) and then to make symbolic operations with the roots of g.

When n = 5 and K = Q, Spearman and Williams give in [START_REF] Spearman | Dihedral quintic polynomials and a theorem of Galois[END_REF] a first solution to this problem: they provide closed formulae which express all the roots of g as rational functions of any two roots α1, α2. Thus, if we have a radical representation of α1 and α2, we have the ones for all the other roots. But, when n 6, the result of Spearman an Williams is not generalized. Moreover, the radical representation of the roots is not suitable for symbolic computation (see [START_REF] Lazard | Solving quintics by radicals[END_REF]Section 9]) when n 6. Thus, we focus on the computation of the representation of the splitting field of g with the above representation.

It is well known that the ideal I has a triangular reduced Gröbner basis for a lexicographical order (see e.g. [START_REF] Tchebotarev | Gründzüge des Galois'shen Theorie[END_REF][START_REF] Lazard | Solving zero-dimensional algebraic systems[END_REF][START_REF] Anai | Computation of the splitting fields and the Galois groups of polynomials[END_REF][START_REF] Valibouze | Étude des relations algébriques entre les racines d'un polynôme d'une variable[END_REF][START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF]). This Gröbner basis can be obtained from the polynomial g by computing successive factorizations in algebraic extensions of K (see e.g. [START_REF] Anai | Computation of the splitting fields and the Galois groups of polynomials[END_REF]). Another method for this task is based on computations and factorizations of resolvents (see e.g. [START_REF] Valibouze | Étude des relations algébriques entre les racines d'un polynôme d'une variable[END_REF][START_REF] Ducos | Construction de corps de décomposition grâce aux facteurs de résolvantes[END_REF]). When such a triangular basis is known, we can compute the symmetric representation of the action of the Galois of g over α1, . . . , αn (see [START_REF] Anai | Computation of the splitting fields and the Galois groups of polynomials[END_REF][START_REF] Abdeljaouad-Tej | Computation of the decomposition group of a triangular ideal[END_REF]). None of the aforementioned methods take advantage of the fact that the Galois group of the studied polynomial is dihedral.

In this paper, we focus on the computation of this triangular Gröbner basis. In our specific case, this triangular set {f1, f2, f3, . . . , fn} verifies a theorem of Galois which states that polynomials f3, . . . , fn are linear in their principal variable. Thus, as soon as we know an irreducible factor g2 of g over its stem field (an extension of K generated by one of its roots) we can take f1 = f and f2 = g2, then it rests to compute the polynomials f3, . . . , fn. If we know the action of the Galois group of g over approximations (complex or p-adic) of its roots we can compute these relations by interpolation (see [START_REF] Mckay | Finding relations among the roots of an irreducible polynomial[END_REF][START_REF] Yokoyama | A modular method for computing the Galois groups of polynomials[END_REF][START_REF] Renault | A modular algorithm for computing a relations ideal of a polynomial[END_REF]). Here, we do not assume the knowledge of the explicit action of the Galois over approximations (we only know the name of the group) so we cannot use these methods.

An other framework for the computation of such a basis is presented in [START_REF] Orange | Calcul efficace d'un corps de dcomposition[END_REF]: the main idea is to begin the process by a factorization of g over its stem field and to end the algorithm by computations using the algorithm GaloisIdeal (see [START_REF] Valibouze | Étude des relations algébriques entre les racines d'un polynôme d'une variable[END_REF]) with a tricky use of the galoisian informations obtained from the factorization in order to avoid some computations. This framework describes a method for the construction of a table-based algorithm for the computation of relations ideal of polynomial with a fixed degree.

The algorithm we provide here starts also from a factorization of g over its stem field but, it does not depend neither on the degree of g and nor on the field of its coefficients. We prove that the knowledge on the fact that the Galois group of g is dihedral allows to reduce the end of the process to computations of normal forms modulo an ideal of degree 2n obtained from the pre-process of factorization. We also prove that the number of normal forms computed by our algorithm is dominated by n 2 , and that they are computed modulo a zero-dimensional ideal of degree bounded by 2n.

In the particular case of degree n = 5, we prove that this basis can be given from the factorization of g over its stem field without any other computation of normal forms. This can be viewed as an improvement of the result of [START_REF] Spearman | Dihedral quintic polynomials and a theorem of Galois[END_REF].

The paper is organized as follows. In section 2, we present results about Galois ideals relating them to triangular sets. These results are used in Section 3 which is devoted to the proof of the principal results leading to the algorithm and its complexity study. Section 4 presents two examples of use of this algorithm.

Notations

In this paper, the following notations are used:

• K is a computable field and K is an algebraic closure of K.

• For i ∈ [[1, n]], the multivariate polynomials ring K[x1, . . . , xi] is denoted by K[X i ]. • For i ∈ [[2, n]], K[X i ] is equipped with the lexicograph- ical monomial order x1 < x2 < . . . < xi. For f ∈ K[X i ],
we denote by HM(f ) the greatest monomial of f .

• Given an ideal I of K[X n ], the set of zeroes of I in Kn is denoted by Z(I). For i ∈ [[1, n -1]], we denote by Z(I) i the projection of Z(I) on the first i coordinates.

• Given V a finite subset of Kn , the unique radical ideal of K[X n ] vanishing on V is denoted by I(V ).

• The natural actions of the symmetric group Sn over elements of Kn and K[x1, . . . , xn] are defined by

Sn × Kn -→ Kn (σ, α) -→ σ.α = (α σ(1) , . . . , α σ(n) ) Sn × K[X n ] -→ K[X n ] (σ, f ) -→ σ.f = f (x σ(1) , . . . , x σ(n) )
• Dn denotes the symmetric representation of the dihedral subgroup of degree n of Sn. In the case where n is odd, it is generated by the product of transpositions

τ = (2, 3) . . . (n -1, n)
and the cycle σ = (1, 2, 4 . . . , 2k, . . . , n -1, n, . . . , 2k -1, . . . , 5, 3).

When n is even,

τ = (2, 3) . . . (n -2, n -1)
and σ = (1, 2, 4 . . . , 2k, . . . , n, n -1, . . . , 2k -1, . . . , 5, 3).

For example, when n = 5 (resp. n = 8) we have τ = (2, 3)(4, 5) (resp. τ = (2, 3)(4, 5)(6, 7)) and σ = (

GALOIS IDEALS

In this section, we recall the definition and give some results about Galois ideals (see [START_REF] Valibouze | Étude des relations algébriques entre les racines d'un polynôme d'une variable[END_REF][START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF]).

In the whole section, g is a separable polynomial of degree n with coefficients in K and α = {α1, . . . , αn} an n-tuple of its roots in K.

Definition 1. An ideal I of K[X n ] is a Galois α-ideal if there exists a subset L of the symmetric group Sn containing the identity such that:

I = I(L.α) .
More generally an ideal I of K[X n ] is said to be a Galois ideal if there exists a tuple β of roots of a separable polynomial of degree n such that I is a Galois β-ideal.

Example 2. The Galois ideal I({α}) is called a relations ideal and is denoted by I(α). Since K[X n ]/I(α) is isomorphic to the splitting field of g, I(α) is maximal [START_REF] Valibouze | Étude des relations algébriques entre les racines d'un polynôme d'une variable[END_REF][START_REF] Anai | Computation of the splitting fields and the Galois groups of polynomials[END_REF]. In fact, it is the unique maximal Galois α-ideal. More generally, all maximal ideals of K[X n ] which contain a Galois ideal are relations ideals (see [START_REF] Valibouze | Étude des relations algébriques entre les racines d'un polynôme d'une variable[END_REF]).

We have the following obvious characterization of a Galois α-ideal:

Lemma 2.1. [4] An ideal I of K[X n ] is a Galois α-ideal
if and only if I is radical and its associated algebraic variety Z(I) satisfies {α} ⊂ Z(I) ⊂ Sn.α .

The Galois group of g, for a fixed numbering of its roots, is now defined from a Galois ideal.

Proposition-Definition 3. [24, Définition 1.12] There exists a subgroup G of Sn such that the algebraic variety V = Z(I(α)) verifies V = G.α. A symmetric representation of the Galois group of g is such a maximal subgroup G (for the inclusion) of Sn. It is denoted by Gal K (α) in the sequel.

Remark 4. As soon as a Gröbner basis of I(α) and the generators of Gal K (α) are known, we can represent the action of the Galois group of g over a symbolic representation of its roots. In fact, the group Gal K (α) is the stabilizer of I(α) (see [START_REF] Abdeljaouad-Tej | Computation of the decomposition group of a triangular ideal[END_REF]), thus it represents the K-automorphisms of the algebra K[X n ]/I(α). Hence, the objects that we want to compute are exactly a Gröbner basis of I(α) and its stabiliser Gal K (α). Lemma 2.1 shows that any Galois α-ideal is included in I(α). More generally, we have the following result: Lemma 2.2. Let σ be a permutation of Gal Q (α). For any Galois α-ideal I, we have:

∀R ∈ I, I + σ.R ⊂ I(α) .
Proof. We have I ⊂ I(α), thus R ∈ I(α). Since σ.α ∈ Z(I(α)) (see Proposition-Definition 3), we have (σ.R)(α) = R(σ.α) = 0, which implies that σ.R ∈ I(α) since I(α) is radical.

When K is perfect, a Galois α-ideal I ⊂ K[X n ] for which there exists a subgroup G of Sn such that Z(I) = G.α is generated by a separable triangular set (see [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF]). Following the proof of [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF], we provide further a more general result. Definition 5. A subset T = {f1, . . . , fn} of K[X n ] is said to be triangular if there exist n positive integers k1, . . . , kn such that:

∀i ∈ [[1, n]], HM(fi) = xi k i . A triangular set T = {f1, . . . , fn} is said to be separable if for all i ∈ [[1, n]],
and for all β ∈ Z( T )i-1, fi(β, xi), seen as a univariate polynomial in xi, is separable. Remark 6. Note that in our definition, we only consider triangular sets whom initials are equal to 1, so that the ideal generated by the considered triangular set is the saturated ideal of the triangular set (see [START_REF] Ph | On the theories of triangular sets[END_REF]). In particular, from [START_REF] Ph | On the theories of triangular sets[END_REF], the saturated ideal of a separable triangular set is radical. In our case, this means that the ideals generated by the separable triangular sets we consider are radical.

We show now how to generalize the result of [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF] about triangular sets and Galois ideals. The first result we need is about equiprojectable varieties (see [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF][START_REF] Dahan | Lifting techniques for triangular decompositions[END_REF] for a definition).

Lemma 2.3. An algebraic variety V of ( Ksep ) n (where Ksep is the separable closure of K) is equiprojectable if and only if the ideal I(V ) can be represented by a separable triangular set.

Proof. A proof is given in [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF] for the case where K is perfect but this result does not depend on the perfectness of the base field as soon as we suppose the field K[V ] separable which is the case here (for example, see [START_REF] Dahan | Sharp estimates for triangular sets[END_REF] where this result is used).

From this lemma we obtain the following generalization of [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF].

Proposition 2.4. Let I ⊂ K[X n ] be a Galois ideal. If there exists a subgroup G of Sn and an element α of the algebraic variety V = Z(I) such that V = G.α then there exists a separable triangular set generating I.

Proof. Let G be a subgroup of Sn. An algebraic variety V ⊂ Kn such that V = G.α, where α is an n-tuple of roots of a separable degree n polynomial, is equiprojectable (see [START_REF] Ph | Using Galois ideals for computing relative resolvents[END_REF]). Moreover, since K(α) is separable so is K[V ] and the result follows from Lemma 2.3. Remark 7. A Galois ideal whose associated algebraic variety satisfies the conditions of Proposition 2.4 is said to be pure. There exist Galois ideals which are not pure and triangular thus Proposition 2.4 is not an equivalence. Moreover, there exist Galois ideals which are not triangular (see [START_REF] Renault | Calcul efficace de corps de décomposition[END_REF]).

Separable triangular sets have many other properties (see [START_REF] Ph | On the theories of triangular sets[END_REF]), we use the following one in the sequel. Lemma 2.5. Let T = {f1, . . . , fn} be a separable triangular set of K[X n ] and R be a polynomial of K[X n ] such that HM(R) = xj with j 2. Suppose that the ideal

T , R = K[X n ]) and f1, . . . , fj-1 is a maximal ideal of K[X j-1 ].
Then, the ideal T , R is generated by the triangular set T = {f1, . . . , fj-1, R, fj+1, . . . , fn} which is separable.

Proof. Denote by Tj-1 the set {f1, . . . , fj-1}. Since Tj-1 is maximal, the quotient ring A = K[X j ]/ Tj-1 is a field. This implies that A[xj] is a principal ideal domain. For p ∈ K[X j ], we denote by p its image in A[xj]. Let F be a representative of the gcd F of fj and R in A [xj]. Since F can be rewritten as an algebraic combination of R and fj and f1, . . . , fj-1, one has f1, . . . , fj-1, F ⊂ f1, . . . , fj-1, fj, R . Consider now an element p of f1, . . . , fj-1, fj, R and let p its image in A[xj]. Thus, p is a multiple of F which implies that p can be written as an algebraic combination of f1, . . . , fj-1, F . Thus, f1, . . . , fj-1, F = f1, . . . , fj-1, fj, R . This implies that f1, . . . , fj-1, F, . . . , fn = f1, . . . , fj-1, fj, R, . . . , fn . By assumption HM(R) = xj which implies that either F is the unit in A, or R = F . Since T , R = K[X n ], one has F = R.

It remains to prove that T = f1, . . . , fj-1, R, . . . , fn is separable. Note that by assumption, Tj-1 is separable. Since R is linear in xj, R is separable in A [xj]. Consider now for j + 1 i n, the image of fi in K[X i-1 ]/ f1, . . . , fj-1, R, . . . , fi-1 . Remark that if it is not separable, then T can not be separable since Z(T ) ⊂ Z(T ).

MAIN RESULTS

In this section, we fix g ∈ K[x] a polynomial of degree n 5 with Dn as a symmetric representation of its Galois group (thus this polynomial is irreducible and separable). Such a polynomial is said to be a dihedral polynomial of degree n. We present an algorithm for the computation of a relations ideal of g from its factorization over its stem field. Here we only know the name of the Galois group of g. We show how to fix the representation of this group by numbering the factors of g over its stem field (which is equivalent to fix the order of roots orbits of g). Then, from this particular representation of the Galois group we deduce a process in order to construct a triangular set of a relations ideal of g by group action (same sorts of group actions are used in [START_REF] Renault | A modular algorithm for computing a relations ideal of a polynomial[END_REF][START_REF] Orange | Calcul efficace d'un corps de dcomposition[END_REF] in order to avoid computations). Proposition 3.1. Let α1 be a root of g. The factorization of g over its stem field K(α1) is given by: (x -α1)g2(α1, x) . . . g n+1 2 (α1, x) n odd (x -α1)g2(α1, x) . . . g n 2 (α1, x)(x -b n 2 +1 (α1)) n even where gi(t, x) = x 2 + bi(t)x + ai(t) and ai, bi are univariate polynomials of degree at most n -1.

Proof. A symmetric representation of the Galois group of g over the field K(α1) is

StabD n ({1}) = {s ∈ Dn | s(1) = 1}.
This group is explicitly given by:

StabD n ({1}) =  (2, 3) . . . (n -1, n) n odd (2, 3) . . . (n -2, n -1) n even
The orbits of the action of StabD n ({1}) over {1, . . . , n} are:

{1}, {2, 3}, . . . , {n -1, n} n odd {1}, {2, 3}, . . . , {n -2, n -1}, {n} n even.
There is a one-to-one correspondence between the orbits of the canonical action of the Galois group of a polynomial over its roots and the set of roots of its irreducible factors, so the result follows.

Let gi be the factors of g over its stem field with a fixed numbering as in Proposition 3.1. We consider the ideal of K[X n ] generated by the following separable triangular set T1:

n odd: n even:

8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : f 1 = g(x 1 ) f 2 = g 2 (x 1 , x 2 ) f 3 = x 3 + x 2 + b 2 . . . f 2i = g i+1 (x 1 , x 2i ) f 2i+1 = x 2i+1 + x 2i + b i+1 . . . f n-1 = g (n+1)/2 (x 1 , x n-1 ) fn = xn + x n-1 + b (n+1)/2 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : f 1 = g(x 1 ) f 2 = g 2 (x 1 , x 2 ) f 3 = x 3 + x 2 + b 2 . . . f 2i = g i+1 (x 1 , x 2i ) f 2i+1 = x 2i+1 + x 2i + b i+1 . . . f n-2 = g n/2 (x 1 , x n-2 ) f n-1 = x n-1 + x n-2 + b n/2 fn = xn + b n/2+1
where the polynomials bi are univariate in x1.

Remark that the set T1 depends on the numbering of the factors of g in its stem field. Thus, there exist several different sets constructed as T1. Actually, if Ω denotes the set of permutations of Stab Sn ({1}) defined by

n odd : 2 3 4 5 . . . n -1 n . . . 2k1 2k1 + 1 2k2 2k2 + 1 . . . 2k (n+1)/2 2k (n+1)/2 + 1 n even : 2 3 4 5 . . . n -2 n -1 . . . 2k1 2k1 + 1 2k2 2k2 + 1 . . . 2k n/2 2k n/2 + 1 where ki ∈ [[1, n+1 2
]], then the set S = {{ω.f : f ∈ T1} : ω ∈ Ω} represents all possible triangular sets with the same form as T1 and constructed with the factors gi (a permutation of Ω corresponds to a numbering of these factors). Proposition 3.2. There exists a triangular set T in S such that the ideal T is an α-Galois ideal where α is an n-tuple of the roots of g satisfying Gal K (α) = Dn.

Proof. Let α be an n-tuple of different roots of g verifying Gal K (α) = Dn. In the proof of Proposition 3.1, we have seen that for all integer i in [[1, n+1 2 ]], the set Ri = {α2i, α2i+1} corresponds to the roots of a quadratic factor of g in K(α1)[x]. If we number these factors so that

Roots(gi) = Ri for each i in [[1, n+1 2
]], then we can construct a triangular set T contained in S so that α is a zero of I = T and Z(I) ⊂ Sn.α. As T is clearly triangular and separable (since T1 is) , we have the result by Lemma 2.1 and Remark 6.

In particular, we have the following result: Corollary 3.3. With the same notations as in Proposition 3.2, if the degree n of g is equal to 5, then all the ideals generated by the triangular sets of S are α-Galois ideals with Gal K (α) = D5.

Proof. When n = 5, by Proposition 3.1 we have two non linear factors, so two possible numberings and thus two triangular sets in S. Let T1 and T2 be the two possible ideals corresponding to these two possible numbering. By Proposition 3.2, at least one of these ideals is an α-Galois ideal with Gal K (α) = D5, let T1 be this ideal. We have

T2 = {ω.f | f ∈ T1}
where ω = (5, 3)(2, 4), thus T2 is an (ω -1 .α)-Galois ideal with

Gal K (ω -1 .α) = ω -1 . Gal K (α).ω .
Since ω -1 Gal K (α)ω = ω -1 D5ω = D5 we obtain the result.

Remark 8. Corollary 3.3 can be seen as a symbolic reformulation of the result of Spearman and Williams [START_REF] Spearman | Dihedral quintic polynomials and a theorem of Galois[END_REF].

If the ideal T of Proposition 3.2 is known, the following proposition shows how to construct a Gröbner basis of the relations ideal I(α) with Gal K (α) = Dn by only applying the action of a permutation on the variables of the polynomials of T . Actually, a triangular set of S which generates an α-Galois ideal with Gal K (α) = Dn fixes an order on the roots of g, thus we can apply a particular group action on this set.

=  (1 2)(3 4) . . . (n -2 n -1) n odd (1 2)(3 4) . . . (n -1 n) n even
be a permutation of Sn. Then, the set T containing the 3 polynomials f1, f2, fn, all the fi with even integer 1 < i < n and µ.fi with odd integer 1 < i < n -1, is a Gröbner basis of the ideal I(α).

Proof. We can suppose w.l.o.g. that T = T1 (as we have seen above, it depends only on the numbering of the factors gi). Since for all odd integer k = 2i -

1 with i ∈ [[2, n-1 2 ]],
we have:

µ.f k = µ.(x k + x k-1 + bi+1(x1)) = x k+1 + x k-2 + bi+1(x2) ,
the set T is given by n odd : n even :

8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : f 1 f 2 f 3 x 4 + x 1 + b 2 (x 2 ) . . . x 2i + x 2i-3 + b i (x 2 ) f 2i+1 . . . x n-1 + x 1 + b (n+1)/2 (x 2 ) fn 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : f 1 f 2 f 3 x 4 + x 1 + b 2 (x 2 )
. . .

x 2i + x 2i-3 + b i (x 2 ) f 2i+1 . . . x n-2 + x 1 + b (n-2)/2 (x 2 ) f n-1 fn
By construction, the set T is triangular, so it is a Gröbner basis (see [START_REF] Cox | Ideals, varieties, and algorithms[END_REF]) and we just have to prove that this set generates the relations α-ideal. For this, we first prove that the ideal generated by T is maximal and then that it is contained in I(α).

The permutation µ is in Dn, more precisely we have the (right) product µ = στ . Since K[X 2 ]/ f1, f2 is isomorphic to a field (since f1 = g and f2 corresponds to an irreducible factor of g over its stem field), all ideals of the form f1, f2, x3 + h3(x1, x2), . . . , xi + hi(x1, . . . , xi-1) is a maximal ideal of K[X i ]. Thus we can recursively use the lemma 2.5 in order to construct a new ideal which is generated by T :

I = T1 + µ.f3 + µ.f5 + . . . + µ.fm ,
where m is the greatest odd integer less than n -1. Then I is generated by the set T and is maximal. By lemma 2.2, we have I ⊂ I(α), hence I is the relations α-ideal. Now, if we want to apply Proposition 3.4 for the computation of a relations ideal of g, we need to know an effective method to choose a triangular set in S which verifies the hypothesis of this proposition. Proposition 3.5. Let T be a triangular set of S and T be the triangular set obtained by action of µ on T (as in Proposition 3.4). If we have the following inclusion T ⊂ T , then T is an α-Galois ideal with α an n-tuple of roots of g and verifying Gal K (α) = Dn Proof. Assume that T ⊂ T . As we have seen in the proof of Proposition 3.4, the ideal T is maximal, thus, if β denotes one of its zeros, it is a relations β-ideal. Because of the form of T , one can see that Gal K (β), which is a conjugate of Dn, contains these two permutations: Now, we give the algorithm which computes a triangular basis of a relations ideal of g from its factorization over its stem field K[x1]/ g . We recall that any quadratic factor g(t, x) of g is of the form g(t, x) = x 2 + b(t).x + a(t).

µ =  (1 2)(3 4) . . . (n -2 n -1) n odd (1 2)(3 4) . . . (n -1 n) n even τ =  ( 2
We first give the scheme of the algorithm. By Corollary 3.3 we can split the process in two parts, the first one for degree 5 where no computations of normal forms is needed, the second for degree at least 6. In the second part, we successively number the factors of g in order to satisfy the condition of Proposition 3.5 and we apply, at the same time, the action of permutation µ to construct two linear relations. At the end of the process we obtain a triangular set of a relations ideal of g.

Algorithm: DihedralRelationsIdeal

Require: A dihedral polynomial g of degree n 5 and the set F of its irreducible quadratic factors over its stem field. Ensure: The set T = {f 1 , . . . , fn} is a triangular Gröbner basis of a relations ideal I(α) of g with Gal K (α) = Dn.

n := Degree(g); f 1 (x 1 ) := g(x 1 );

if n = 5 then Let f 2 (t, x) = x 2 +b(t).x+a(t) and f 3 (t, x) = x 2 +d(t).x+c(t) be the two elements of F ;

T := [f 1 (x 1 ), f 2 (x 1 , x 2 ), x 3 + x 2 + b(x 1 ), x 4 + x 1 + b(x 2 ), x 5 + x 4 + d(x 4 )]; return T ; end if Let f 2 (t, x) = x 2 + b(t).x + a(t) and f 3 (t, x) = x 2 + d(t).x + c(t) be two elements of F such that NormalForm(f 3 (x 1 , x 4 ), [f 1 (x 1 ), f 2 (x 1 , x 2 ), x 3 + x 2 + b(x 1 ), x 4 + x 1 + b(x 2 )]) = 0; F := F \ {f 2 , f 3 }; T := [f 1 (x 1 ), f 2 (x 1 , x 2 ), x 3 + x 2 + b(x 1 ), x 4 + x 1 + b(x 2 ), x 5 + x 4 + d(x 1 ), x 6 + x 3 + d(x 2 )]; if n = 6 then return T ; end if i := 3; while |F | > 1 do i := i + 1; Let f (t, x) := x 2 + b(t).x + a(t) be an element of F such that NormalForm(f (x 1 , x 2i ), T ) = 0; F := F \ {f }; T := Concat(T , [x 2i-1 +x 2i-2 +b(x 1 ), x 2i +x 2i-3 +b(x 2 )]); end while i := i + 1; f (t, x) := x 2 + b(t).x + a(t) be the last element of F ; T := Concat(T , [x 2i+1 + x 2i + b(x 1 )]); if n is even then c := the coefficient of x n-1 in g; T := Concat(T , [x 2i+2 + x 2i+1 + . . . + x 1 -c]); end if return T ;
Theorem 3.6. The algorithm DihedralRelationsIdeal terminates and computes a triangular basis of a relations ideal of g. Moreover, the number of normal forms performed during the computation is bounded by

Ψ(n) = 8 < : 0 n = 5 1 n = 6 1 2 (3m 2 -7m + 6) n 7
where n is the degree of the polynomial g and m := n-1 2 .

Proof. By Proposition 3.4 and Proposition 3.5, it is clear that this algorithm terminates and gives the good result. All the normal forms are performed when we have to find the good numbering of the fi, so degree 5 is not affected. Finding f2 and f3 requires at most m! (m-2)! = m 2 -m, so there is exactly one normal form to compute in the cases n = 6. All the other normal forms are performed during the while loop. Any such loop performs at most |F | -1 normal form. As |F | = m-2 before the while loop, the total number of normal forms computations is bounded by:

m-2 X k=2 k -1 = 1 2 (m -3)(m -2)
which gives the result.

Remark 10. Consider an irreducible separable polynomial g of degree n whose their irreducible factors over its stem field have the same degrees than the ones of a dihedral polynomial of the same degree. If the above algorithm, applied to the factors of g, terminates then, by Proposition 3.5, the Galois group of g is proved to be Dn and we compute at the same time a relations ideal of this polynomial. Using Theorem 3.6, one can stop the while loop if the number of computed normal forms is greater than the bound we provide.

EXAMPLES

In this section we give two examples of computations of relations ideals using the results of this paper.

Generic D5 relations ideal

This first example is devoted to the computation of a relations ideal of the D5 generic polynomial fD 5 of Brumer (see [START_REF] Jensen | Generic polynomials[END_REF]Theorem 2.3.5]). This polynomial has its coefficients in the function field Q(s, t). The polynomial fD 5 is given by:

x 5 + (t -3)x 4 + (s -t + 3)x 3 + (t 2 -t -2s -1)x 2 + sx + t . Using Trager's algorithm (see [START_REF] Trager | Algebraic factoring and rational functiun integration[END_REF]) we can compute the factorization of fD 5 over its stem field K[α1]. The two non linear factors computed with Magma ( [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF]) are given by: This ideal can be said generic because every irreducible polynomial f ∈ Q[x] with D5 as Galois group is Tchirnhaus equivalent to a specialization of fD 5 . So, the basis of a relations ideal I of f verifying D5.I = I, is Tchirnhaus equivalent to a specialisation of the basis of this generic ideal.

Proposition 3 . 4 .

 34 Consider a triangular set T = {f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn)} of S such that T is an α-Galois ideal with Gal K (α) = Dn and µ

Remark 9 .

 9 The last relation of the set T can be replaced by the classical one xn + xn-1 + . . . + x1 + c where c is the coefficient of x n-1 in g.

  3)(4 5) . . . (n -1 n) n odd (2 3)(4 5) . . . (n -2 n -1) n even Thus Gal K (β) contains µτ = σ and Gal K (β) = Dn. Hence T is a β-Galois ideal with Gal K (β) = Dn.

  Corollary 3.3 gives without any other computation the triangular basis of a generic D5 relations ideal:

	x 2 +	1 t	(-α	4 1 + (-t + 2)α	3 1 + (-s -1)α	2 1 + (s -t 2 + 2t)α 1 -t)x
	-α 1 + 1
	x 2 +	1 t	(α	4 1 + (t -2)α	3 1 + (s + 1)α	2 1 + (-s + t 2 -t)α 1 + t(t -2))x
	+	1 t	((t -1)α	4 1 + (t 2 -4t + 2)α	3 1 + (st -s -t 2 + 3t -1)α	2 1
	+(-2st + s + t 3 -2t 2 )α 1 ) + s -t + 1
	x 5 1 + (t -3)x 4 1 + (s -t + 3)x 3 1 + (-2s + t 2 -t -1)x 2 1 + sx 1 + t
	x 2 2 -	1 t	x 2 x 4 1 +	t + 2 t	x 2 x 3 1 +	-s -1 t	x 2 x 2 1 +	s -t 2 + 2t t	x 2 x 1
	-x 2 -x 1 + 1
	x 3 + x 2 -	1 t	x 4 1 +	-t + 2 t	x 3 1 +	-s -1 t	x 2 1 +	s -t 2 + 2t t	x 1 -1
	x 4 -	1 t	x 4 2 +	-t + 2 t	x 3 2 +	-s -1 t	x 2 2 +	s -t 2 + 2t t	x 2 + x 1 -1
	x 5 + x 4 +		1 t	x 4 1 +	t -2 t	x 3 1 +	s + 1 t	x 2 1 +	-s + t 2 -t t	x 1 + t -2

An example in degree 8

Let g = x 8 -3x 5 -x 4 + 3x 3 + 1 be a polynomial given by the Database for Number Fields of J. Klüners and G. Malle (see [START_REF] Klüners | A database for field extensions of the rationals[END_REF]) with rational coefficients and Galois group D8. The factorization of g over its stem field Q(α1) can be computed with Magma, GP/PARI (see [START_REF]version 2.2.5[END_REF]) or KANT/KASH (see [START_REF] Kant | [END_REF]). The three quadratic factors are:

Using Algorithm

where b is the coefficient of x1 in the polynomial f2(x1, x2).

If we choose the first numbering we obtain the following relations ideal: x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x 1 -3

CONCLUSION

In this paper, we proposed a method for the computation of the relations ideal of a dihedral polynomial which permits us to avoid factorizations. We also did the complexity analysis of the underlying algorithm. we hope that a better study of the set S could improve this algorithm, this will be investigated in a future work.


Ivan Giorgio
email: ivan.giorgio@uniroma1.it

Alessandro Della Corte

Dynamics of 1D nonlinear pantographic continua

Keywords: Discrete mechanical systems, nonlocal continua, nonlinear springs, traveling waves, quasi-soliton solutions

In this paper a mechanical system consisting of a chain of masses connected by nonlinear springs and a pantographic microstructure is studied. A homogenized form of the energy is justified through a standard passage from finite differences involving the characteristic length to partial derivatives. The corresponding continuous motion equation, which is a nonlinear fourth-order PDE, is investigated. Traveling wave solutions are imposed and quasi-soliton solutions are found and numerically compared with the motion of the system resulting from a generic perturbation.

Introduction

The present paper investigates the dynamics of a 1D continuum which is intended as the homogenized limit of a discrete mechanical system consisting of masses connected by nonlinear springs and a pantographic microstructure. The study of different kinds of microstructures, which are today much easier to obtain thanks to the possibility of driving the manufacturing process by means of computers [START_REF] Dell'isola | Synthesis of Fibrous Complex Structures: Designing Microstructure to Deliver Targeted Macroscale Response[END_REF], has led to the development of new models in continuum mechanics (apart from classical works such as [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF][START_REF] Toupin | Theories of elasticity with couple-stress[END_REF], the reader can find interesting results in [START_REF] Pietraszkiewicz | On natural strain measures of the non-linear micropolar continuum[END_REF][START_REF] Altenbach | On the linear theory of micropolar plates[END_REF][START_REF] Altenbach | Analysis of the viscoelastic behavior of plates made of functionally graded materials[END_REF][START_REF] Placidi | Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity[END_REF][START_REF] Madeo | Band gaps in the relaxed linear micromorphic continuum[END_REF][START_REF] Misra | Granular micromechanics based micromorphic model predicts frequency band gaps[END_REF][START_REF] Aminpour | A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis[END_REF][START_REF] Aminpour | On the Modelling of Carbon Nano Tubes as Generalized Continua[END_REF]). The pantographic microstructure (for a detailed description see [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF]) is particularly interesting because it is at the same time very simple in principle but still leads to homogenized continua that are beyond the scope of classical continuum models (see [14,15] for convergence theorems involving structures of this type). Moreover, it displays behaviors that are very promising from a purely mechanical point of view, among which a high toughness, an advantageous strength to weight ratio and a very good predictability of the fracture zones (see [16][17][18][19][20]).

The material nonlinearity of the springs obviously entails that the motion of the system is described by nonlinear equations. The universe on nonlinear PDEs escapes from almost any possible generalization. Very seldom one can provide closed-form solutions, and in general even qualitative analysis of the behavior of the solutions and of the structure of their set is far from trivial. An exception, however, has to be made for those PDEs having soliton solutions: this is indeed one of the cases in which a very regular (and thus very peculiar) behavior can be shown for PDEs otherwise showing a much more complex time evolution. Recent literature also investigates, in suitable cases, solutions representing (in various senses) an approximation of a genuine soliton (see for instance [21]). This is indeed the approach we chose for the present work. Of note, soliton solutions have a significant application potential in mechanics (for a general perspective including connections with mechanical and biological phenomena, see [22]), in particular in damage detecting techniques [23].

The paper is organized as follows: in Sec. 2 a one-dimensional continuum is conjectured as the homogenized limit starting from a nonlinear discrete microstructure. In Sec. 3 a solution in form of a traveling wave is obtained; manipulating this solution we explore the possibility to have a quasi-soliton solution after having shown that exact soliton solutions are not possible. Finally, in Sec. 4 we numerically compare the quasi-soliton solution with a generic perturbation and observe markedly lesser dispersion effects in the former case.

Non local 1D discrete systems

Let us consider a system consisting of an infinite number of material points P i having equal mass m, arranged in a straight chain and connected by springs. The kinematic descriptor of each P i is the displacement u i , which is assumed to be directed along the reference direction of the chain. We add to this system a set of flexible and inextensible beams connected by ideal hinges to form a pantographic micro-structure [START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model[END_REF][START_REF] Dell'isola | Plane bias extension test for a continuum with two inextensible families of fibers: A variational treatment with Lagrange multipliers and a perturbation solution[END_REF]. We thus consider two elastic interaction potentials between the masses:

1. an interaction due to the pantographic micro-structure (see [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF]) 1 ;

Φ pan,i = 1 2 K [u i-1 (t) -2u i (t) + u i+1 (t)] 2 (1)
2. a nonlinear interaction between each pair of adjacent masses due to the introduced nonlinear springs

Φ N L,i = 1 2 κ 1 [u i+1 (t) -u i (t)] 2 + 1 4 κ 3 [u i+1 (t) -u i (t)] 4 (2)
This kind of interaction potentials (together with a standard form for the kinetic energy) leads to an infinite set of equations by means of a standard application of the Hamilton principle:

mü i (t) + κ 1 [u i (t) -u i-1 (t)] -κ 1 [u i+1 (t) -u i (t)] + κ 3 [u i (t) -u i-1 (t)] 3 -κ 3 [u i+1 (t) -u i (t)] 3 - 2K[u i-1 (t) -2u i (t) + u i+1 (t)] + K[u i-2 (t) -2u i-1 (t) + u i (t)]+ K[u i (t) -2u i+1 (t) + u i+2 (t)] = 0 (3)
In this paper we only consider the longitudinal motion of the system. For what concerns the motion in the orthogonal direction, we just want to add a few very simple considerations. Within linear

i i+1 i-1 i+2 i-2
Fig. 1 Scheme of the discrete microstructure, made by a chain of masses connected by nonlinear springs and a set of beams constrained by ideal hinges. The hinges at the nodes (i, i + 1, etc.) do not interrupt the continuity of the beams, but just prescribe equal displacements for two material points belonging to them.

regime, the axial elongation of the homogenized system due to transverse displacement can of course be neglected; therefore, the material nonlinearity due to the springs plays no role. As for the pantographic microstructure, in the linear regime the energies associated to transverse and longitudinal displacements are uncoupled and have the same form (see [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF]). Of course, if one wants to extend the study to finite deformations, the equation for transverse and longitudinal displacement would become coupled and the whole problem harder to address.

If the distance between two adjacent masses tends to zero, it is possible to identify the differences involved in the above potentials as finite difference approximations of suitable derivatives. Therefore, we can conjecture that the homogenized version of the considered discrete system is the one given by the continuous potentials specified below:

1. By considering the energies of pantographic substructures we obtain:

1 2 K 4 u i-1 (t) -2u i (t) + u i+1 (t) 2 2 ---→ →0 1 2 k 2 ∂ 2 u(X, t) ∂X 2 2 dX ( 4
)
where X is the referential abscissa, k 2 ≈ K 3 ; therefore the homogenized continuous elastic potential in the range of small deformation can be defined as

φ pan = 1 2 k 2 ∂ 2 u(X, t) ∂X 2 2 (5)
2. by considering the energies of the nonlinear springs we obtain:

1 2 κ 1 [u i+1 (t) -u i (t)] 2 + 1 4 κ 3 [u i+1 (t) -u i (t)] 4 ---→ →0 1 2 k 1 ∂u(X, t) ∂X 2 + 1 4 k 3 ∂u(X, t) ∂X 4 dX (6)
in which k 1 ≈ κ 1 and k 3 ≈ κ 3 3 ; hence the homogenized continuous elastic nonlinear potential can be assumed as:

φ N L = 1 2 k 1 ∂u(X, t) ∂X 2 + 1 4 k 3 ∂u(X, t) ∂X 4 (7)
By using these elastic potentials and the kinetic energy:

E k = 1 2 ∂u(X, t) ∂t 2 (8)
again employing the Hamilton principle, the homogenized version of Eq. ( 3) becomes 2 :

u tt -k 1 u XX 1 + 3β(u X ) 2 + k 2 u XXXX = 0 (9)
with β := k 3 /k 1 . As far as the authors know the nonlinear fourth-order PDE Eq. ( 9) is not found in the literature. For the sake of generality, we will also consider the same equation with the second term of the nonlinear potential [START_REF] Placidi | Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity[END_REF] written in the abstract form Φ(u X ):

u tt -u XX k 1 + ∂ 2 Φ ∂(u X ) 2 + k 2 u XXXX = 0 (10)
The system here studied can be considered a generalized model for the beam in which non locality and material nonlinearities have been taken into account. This line of investigation can be framed in the rich literature existing on generalized beam theories [START_REF] Gabriele | A 1D nonlinear TWB model accounting for in plane cross-section deformation[END_REF][START_REF] Piccardo | A direct approach for the evaluation of the conventional modes within the GBT formulation[END_REF][START_REF] Piccardo | A complete dynamic approach to the Generalized Beam Theory cross-section analysis including extension and shear modes[END_REF][START_REF] Taig | An unconstrained dynamic approach for the Generalised Beam Theory[END_REF][START_REF] Ruta | A beam model for the flexural-torsional buckling of thin-walled members with some applications[END_REF], which includes also numerical tools that can be useful for the investigation of our system [START_REF] Greco | Consistent tangent operator for an exact Kirchhoff rod model[END_REF]. Isogeometric analysis is particularly suitable for the numerical study of this kind of systems and for its higher dimension generalizations, for its capability to comfortably include non local effects ( [START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF][START_REF] Cazzani | Constitutive models for strongly curved beams in the frame of isogeometric analysis[END_REF][START_REF] Cazzani | An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams[END_REF][START_REF] Cazzani | Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches[END_REF][START_REF] Cuomo | Isogeometric analysis of space rods: Considerations on stress locking[END_REF][START_REF] Greco | Multi-patch isogeometric analysis of space rods[END_REF][START_REF] Greco | An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love space rod[END_REF][START_REF] Greco | An isogeometric implicit G1 mixed finite element for Kirchhoff space rods[END_REF][START_REF] Greco | B-Spline interpolation of Kirchhoff-Love space rods[END_REF]). We finally remark that only small deformations of the pantographic microstructure have been considered to get Eq. ( 9) and [START_REF] Aminpour | A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis[END_REF]. The investigation of large deformations of the same microstructure would be of course of interest for real-world applications, but it is clearly harder due to the possible onset of instabilities of different kinds. In order to pursue this research direction, the tools developed in [START_REF] Pignataro | Effects of warping constraints and lateral restraint on the buckling of thin-walled frames[END_REF][START_REF] Gabriele | On the imperfection sensitivity of thin-walled frames[END_REF][START_REF] Aminpour | A one-dimensional beam model for single-wall carbon nano tube column buckling[END_REF][START_REF] Rizzi | On the postbuckling analysis of thin-walled frames[END_REF][START_REF] Rizzi | Initial postbuckling behavior of thin-walled frames under mode interaction[END_REF][START_REF] Rizzi | The effects of warping on the postbuckling behaviour of thin-walled structures[END_REF][START_REF] Pignataro | The effects of warping constraints on the buckling of thin-walled structures[END_REF] for studying static instabilities and in [START_REF] Luongo | Linear stability analysis of multiparameter dynamical systems via a numericalperturbation approach[END_REF][START_REF] Luongo | Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces[END_REF] for dynamic ones can be usefully employed.

The linear case

In Eq. ( 9) if β = 0, we have a linear medium whose governing equation is:

u tt -k 1 u XX + k 2 u XXXX = 0 (11)
which corresponds to the discrete set of equations (3) with κ 3 = 0. In the discrete model we have two sources of dispersive effects, i.e. the term due to the non-local potential (1) and the discretization itself. Since the Eq. ( 11) is linear we can seek solution in the form u(X, t) = Ae -j(kX-ωt) , being j the imaginary unit, k the wave number and ω the angular frequency; thus, the dispersion relation is:

k 4 k 2 + k 2 k 1 -ω 2 = 0 ( 12
)
2 It is easy to provide an (informal) justification for the passage from the discrete to the continuous case. Indeed, setting

∆ i [u] = u i+1 (t) -u i (t), one has: -κ 3 (∆ i [u] 3 -∆ i-1 [u] 3 ) = -κ 3 (∆ i [u] -∆ i-1 [u])(∆ i [u] 2 + ∆ i-1 [u] 2 + ∆ i [u]∆ i-1 [u])
and then rearranging the terms, in the limit for going to zero one gets

-κ 3 3 ∆ i [u] 2 - ∆ i-1 [u] 2 ∆ i [u] 2 2 + ∆ i-1 [u] 2 2 + ∆ i [u]∆ i-1 [u] 2 ---→ →0 that explicitly becomes: ω = k √ k 2 k 2 + k 1 √ ρ (13)
and therefore the velocity of propagation as function of ω can be expressed in the following way:

c = ω k = √ 2k 2 ω k 2 1 + 4k 2 ρω 2 -k 1 (14)
Similarly, we can proceed for the discrete system, imposing to the displacements the following time dependence:

u i+N = Ae -j(k N -ωt) , N ∈ Z,
where N is the oriented distance between the N -th and the i-th masses. This leads to the dispersion relation:

2(κ 1 + K) -mω 2 -2κ 1 cos(k ) -2K cos(2k ) = 0 ( 15
)
which gives

k =     arctan     -2κ 2 1 + 2κ 1 (κ 1 + 4K) 2 -4Kmω 2 -4K + 4Kmω 2 -κ 2 1 + 4Kmω 2 + κ 1 + 4K     + 2πc 1     (16)
and the wave speed:

c = ω k = ω     arctan     -2κ 2 1 + 2κ 1 (κ 1 + 4K) 2 -4Kmω 2 -4K + 4Kmω 2 -κ 2 1 + 4Kmω 2 + κ 1 + 4K         (17)
With the micro-macro identification k 2 ≈ K 3 , k 1 ≈ κ 1 and m ≈ , we can compare the two systems as shown in fig. 2. We note that as decreases, the interval of ω in which the continuous system well approximates the discrete one becomes larger. By Eq. ( 17), the continuous system is a good approximation of the discrete one as long as the wave length is larger than 4 . Finally, we remark that the linear case is significant not only when the stiffness k 3 vanishes, but also when |u X | is small.

Wave propagation

We search for solutions of Eq. ( 9) in the form of traveling waves: u(X, t) = f (X -c t) = f (η), with propagation speed c. By substituting in Eq. ( 10) we obtain the fourth order ODE:

f IV (η) + f (η) A -B ∂ 2 Φ ∂(f ) 2 = 0 ( 18
)
where A = c 2 -k1 k2 and B = 1 k2 . Setting f (η) = g(η) and integrating we get

g (η) + Ag(η) -B ∂Φ ∂g = C 1 (19)
whence multiplying by g (η) and integrating again discrete system, l=1.0e-03 discrete system, l=0.75e-03 discrete system, l=0.5e-03 discrete system, l=0.1e-03 continuum Fig. 2 Frequency-velocity plot for the linear version of the discrete system with different values of the scale length (dashed lines) and for the homogenized linear continuuum (continuous line).

g (η) = ± 2BΦ[g(η)] -A[g(η)] 2 + 2C 1 g(η) + 2C 2 (
which in case Φ is the second term of the nonlinear potential [START_REF] Placidi | Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity[END_REF] becomes

g (η) = ± B 2 [k 3 g(η)] 4 -A[g(η)] 2 + 2C 1 g(η) + 2C 2 (21)
Equation ( 18) is therefore solved (up to a constant) by the primitive of an elliptic function:

f (X) = ζ -1 (g(ξ))dξ with ζ(g) = ± dg B 2 k 3 g 4 -Ag 2 + 2C 1 g + 2C 2 (22)
We remark that the signs here are not intended as mutually exclusive over the whole domain, because a solution can of course assume different signs on different intervals provided its global regularity. We will use a solution of this form in the following section. When dealing with nonlinear PDEs such as [START_REF] Misra | Granular micromechanics based micromorphic model predicts frequency band gaps[END_REF] an interesting (and by now standard) question is whether it can admit soliton solutions [START_REF] Calogero | Spectral transform and solitons[END_REF]. In the literature the definition itself of soliton is not completely consistent among different fields of investigation. In the present work, we choose to search for solutions with the following (minimal) properties: i) they preserve their shape; ii) they vanish together with all their derivatives as η goes to infinity; iii) they emerge unchanged after an interaction between themselves.

Point i) is automatically satisfied by a solution verifying Eq. ( 18); as for point ii), we start remarking that a very natural assumption on Φ is that it goes to zero when so does u X . We have therefore that g → 0 implies g → 0 if and only if C 2 = 0, which is thus a necessary condition for the existence of soliton solutions. From Eq. ( 19) we obtain that g → 0 if

∂Φ ∂g (g(η)) ----→ η→∞ - C 1 B (23)
Clearly, the case Φ = 1 4 k 3 ∂u(X,t) ∂X 4 considered above excludes (23) if g (and therefore u X ) goes to zero. This means that in our case there cannot be the typical compensation between dispersive and nonlinear effects producing soliton solutions. This result is consistent with the numerical simulations shown in Fig. 3 Numerical solutions of the ODE (18) showing unbounded growth of f . In the zoom the rapidly oscillating behavior of the derivatives is visible.

Fig. 3, where some numerical solutions of the ODE (18) (with different values of initial conditions) are plotted and a zoom is shown on bottom left. The value of the function f appears to be unbounded for η → ∞, while the derivatives are oscillating. One can notice that, for some initial data, the growth is very slow. This, combined with the fact that the derivatives are not unbounded (even if they do not converge asymptotically to zero) makes ineteresting the search for solutions that almost preserve shape, i.e. quasi-solitons (see the following section).

Soliton solutions are found in nonlinear PDEs formally quite similar to Eq. ( 9) as for instance the well known Boussinesq equation

u tt = u XX (1 + βu) + u 2 X -u XXXX ( 24
)
describing surface water waves in the hypotheses of weak nonlinearity and weak dispersion. The Boussinesq equation, however, displays the compensation above mentioned, which is formally described by the fact that there exist two differential operators L and M depending on a function u(X, t) such that L = [M, L] (where [M, L] is the commutator: M • L -L • M ) holds if and only if u is a solution of the Eq. ( 24)3 [START_REF] Zakharov | On stochastization of one-dimensional chains of nonlinear oscillators[END_REF]. It is interesting to observe that Eq. ( 10) cannot be reduced to the Boussinesq equation for any behavior of nonlinear spring described by the potential Φ. Indeed, to get Eq. ( 24) from Eq. ( 10) it

should be ∂ 2 φ ∂(u X ) 2 = u 2 X u XX
+ βu, which is not possible in the hypothesis that Φ is only depending on u X .

4 Quasi-soliton solutions While we have already seen that exact soliton solutions of Eq. ( 9) are not possible, we want to numerically investigate whether the equation, similarly to what has been observed in other cases can have quasi-soliton solutions (see [21]), i.e. localized traveling wave solutions which approximately preserve their shape and emerge unchanged after interaction among themselves. We constructed a candidate quasi-soliton solution σ(η) by gluing smoothly the function f (η) given in formula (22) over an interval I := (η 0 , η 1 ), at the extrema of which it vanishes with its first derivative, and the null function f 0 ≡ 0 over R \ I.

Imposing u(0, t) = σ(η) as a boundary condition for Eq. ( 9), we obtain the result shown in Figure 4. To allow a direct comparison, we also imposed as a boundary condition a generic function of the form sech(αt) with α = 6; the result is shown in Figure 5. It is directly observable that the traveling solution originated by σ preserves much better its shape than the generic solution originated by the hyperbolic secant. To quantitatively assess this, we plotted the time history of the peak amplitude in the two cases in Fig. 4. In is clear that the traveling solution originated by σ preserves almost perfectly its amplitude, while the generic solution with similar initial amplitude and speed does not. This justifies calling the former a quasi-soliton solution [21].

Conclusions

In the present paper the dynamics of a 1D system with a pantographic microstructure and a nonlinear set of springs has been studied. A novel nonlinear PDE describing the motion of the system in small deformation regime has been investigated. Traveling wave solutions has been searched and a general form for the nonlinear potential of the spring has been considered to show that the equation cannot reduce to a Boussinesq-type, a well-known equation having soliton solutions. Gluing smoothly a suitable restriction of the traveling wave solution with a null function, quasi-soliton solutions have been numerically identified and compared with generic solutions. Further development of the mathematical study of Eq. ( 9) will be interesting and, as for most of nonlinear problems, far from trivial. From a mechanical point of view, future studies of this type of systems should try to weaken the simplifying assumption here introduced, for instance considering, as already mentioned, large deformations of the microstructure, and also studying its possible damage and fracture concerning both the nodes and the beams themselves. In this connection the tools developed in [START_REF] Placidi | A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model[END_REF][START_REF] Placidi | A variational approach for a nonlinear 1-dimensional second gradient continuum damage model[END_REF][START_REF] Yang | Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film[END_REF][START_REF] Cuomo | A poroplastic model for hygro-chemo-mechanical damage of concrete[END_REF][START_REF] Contrafatto | An enriched finite element for crack opening and rebar slip in reinforced concrete members[END_REF][START_REF] Cuomo | A variational model based on isogeometric interpolation for the analysis of cracked bodies[END_REF][START_REF] Contrafatto | A concrete homogenisation technique at meso-scale level accounting for damaging behaviour of cement paste and aggregates[END_REF] can be useful.

Fig. 4

 4 Fig.4Evolution of a perturbation σ(η) (applied in X = 0) obtained gluing smoothly the function f (η) given in formula (22) on an interval I and the null function outside I. The shape is almost preserved.

Fig. 5 Fig. 6

 56 Fig.5Evolution of a perturbation α sech(6t) applied in X = 0, with α = 20 selected to have the same amplitude of the previous case. It is clearly visible that the shape is not preserved.

Fig. 7

 7 Fig.7Time-space 3D plotting of the collision between two identical initial perturbations σ traveling in opposite direction. The corresponding traveling wave solutions emerge practically unchanged after the interaction, almost maintaining shape and velocity.

Fig. 8

 8 Fig.8The same simulation shown in the previous figure but with a generic initial perturbation (sech(αt) with α = 6). The corresponding solutions lose amplitude and show dispersion.

It is straightforward to see that the quantity (u i+1 -u i ) -(u i -u i-1 ) = u i+1 -2u i + u i-1 is always zero if the beams undergo no deformation.

A pair L,M with such a property is called the Lax pair for a given PDE.