I Janovitz-Freireich

A Szántó

B Mourrain
email: mourrain@sophia.inria.fr.

L Rónyai

Moment Matrices, Trace Matrices and the Radical of Ideals

Keywords: Moment Matrices, Matrices of Traces, Radical Ideal, Solving polynomial systems

Let f1, . . . , fs ∈ K[x1, . . . , xm] be a system of polynomials generating a zero-dimensional ideal I, where K is an arbitrary algebraically closed field. Assume that the factor algebra A = K[x1, . . . , xm]/I is Gorenstein and that we have a bound δ > 0 such that a basis for A can be computed from multiples of f1, . . . , fs of degrees at most δ. We propose a method using Sylvester or Macaulay type resultant matrices of f1, . . . , fs and J, where J is a polynomial of degree δ generalizing the Jacobian, to compute moment matrices, and in particular matrices of traces for A. These matrices of traces in turn allow us to compute a system of multiplication matrices {Mx i |i = 1, . . . , m} of the radical √ I, following the approach in the previous work by Janovitz-Freireich, Rónyai and Szántó. Additionally, we give bounds for δ for the case when I has finitely many projective roots in P m K .

Introduction

This paper is a continuation of our previous investigation in [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] to compute the approximate radical of a zero dimensional ideal which has zero clusters. The computation of the radical of a zero dimensional ideal is a very important problem in computer algebra since a lot of the algorithms for solving polynomial systems with finitely many solutions need to start with a radical ideal. This is also the case in many numerical approaches, where Newton-like methods are used. From a symbolicnumeric perspective, when we are dealing with approximate polynomials, the zero-clusters create great numerical instability, which can be eliminated by computing the approximate radical.

The theoretical basis of the symbolic-numeric algorithm presented in [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] was Dickson's lemma [START_REF] Dickson | Algebras and Their Arithmetics[END_REF], which, in the exact case, reduces the problem of computing the radical of a zero dimensional ideal to the computation of the nullspace of the so called matrices of traces (see Definition 14): in [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] we studied numerical properties of the matrix of traces when the roots are not multiple roots, 1 but form small clusters. Among other things we showed that the direct computation of the matrix of traces (without the computation of the multiplication matrices) is preferable since the matrix of traces is continuous with respect to root perturbations around multiplicities while multiplication matrices are generally not.

It turns out that the computationally most expensive part of the method in [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] is the computation of the matrix of traces. We address this problem in the present paper, and give a simple algorithm using only Sylvester or Macaulay type resultant matrices and elementary linear algebra to compute matrices of traces of zero dimensional ideals satisfying certain conditions.

More precisely, we need the following assumptions: let f = [f 1 , . . . , f s ] be a system of polynomials of degrees

d 1 ≥ • • • ≥ d s in K[x]
, with x = [x 1 , . . . , x m ], generating an ideal I in K[x], where K is an arbitrary algebraically closed field. We assume that the algebra A := K[x]/I is finite dimensional over K and that we have a bound δ > 0 such that a basis S = [b 1 , . . . , b N ] of A can be obtained by taking a linear basis of the space of polynomials of degree at most δ factored by the subspace generated by the multiples of f 1 , . . . , f s of degrees at most δ. By slight abuse of notation we denote the elements of the basis S which are in A and some fixed preimages of them in K[x] both by b 1 , . . . , b N . Thus we can assume that the basis S consists of monomials of degrees at most δ. Note that we can prove bounds

δ = m+1 i=1 d i -m (or δ = m i=1 d i -m if s = m) if I
has only finitely many projective common roots in P m K and have no common roots at infinity, using a result of Lazard [START_REF] Lazard | Résolution des systèmes d'équations algébriques[END_REF] (see Theorem 3). Furthermore, we also assume that A is Gorenstein over K (see Definition 1). Note that in practice we can easily detect if A is not Gorenstein (see Remark 10). Also, a random change of projective variables can eliminate roots at infinity with high probability when they are in finite number, but we will address the necessity of this assumption in an upcoming paper.

The main ingredient of our method is a Macaulay type resultant matrix Mac ∆ (f ), which is defined to be a maximal row-independent submatrix of the transpose matrix of the degree ∆ Sylvester map (g 1 , . . . , g s ) → Definition 5). Using our assumptions on A, we can compute a basis S of A using Mac ∆ (f ), and we also prove that a random element y of the nullspace of Mac ∆ (f ) provides a non-singular N × N moment matrix M S (y) with high probability (similarly as in [START_REF] Lasserre | A unified approach to computing real and complex zeros of zero-dimensional ideals[END_REF]). This moment matrix allows us to compute the other main ingredient of our algorithm, a polynomial J of degree at most δ, such that J is the generalization of the Jacobian of f 1 , . . . f s in the case when s = m. The main result of the paper now can be formulated as follows:

s i=1 f i g i ∈ K[x] ∆ for ∆ ≤ 2δ + 1 (see
Theorem Let S = [b 1 , . . . , b N ] be a basis of A with deg(b i ) ≤ δ.
With J as above, let Syl S (J) be the transpose matrix of the map

N i=1 c i b i → J • N i=1 c i b i ∈ K[x] ∆ for c i ∈ K. Then [T r(b i b j )] N i,j=1 = Syl S (J) • X,
where X is the unique extension of the matrix M S (y) such that Mac ∆ (f ) • X = 0.

Once we compute the matrix of traces

R := [T r(b i b j )] N i,j=1 and the matrices R x k := [T r(x k b i b j )] N i,j=1 = Syl S (x k J) • X for k = 1, .
. . , m, we can use the results of [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] to compute a system of multiplication matrices for the (approximate) radical of I as follows: if R is a (numerical) maximal non-singular submatrix of R and Rx k is the submatrix of R x k with the same row and column indices as in R, then the solution M x k of the linear matrix equation

RM x k = Rx k
is an (approximate) multiplication matrix of x k for the (approximate) radical of I. See [START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] for the definition of (approximate) multiplication matrices. Note that a generating set for the radical √ I can be obtained directly from the definition of multiplication matrices, in particular, it corresponds to the rows of the matrices M x1 , . . . , M xm .

We also point out that in the s = m case these multiplication matrices M x k can be obtained even more simply using the nullspace of Mac ∆ (f ) and the Jacobian J of f , without computing the matrices of traces.

We also note here that in a follow up paper we will consider an extension of our present results which works also in the non-Gorenstein case to compute the matrices of traces. Furthermore, that paper will also extend our results to the affine complete intersection case using Bezout matrices.

Related Work

The motivation for this work was the papers [START_REF] Lasserre | A unified approach to computing real and complex zeros of zero-dimensional ideals[END_REF][START_REF] Lasserre | Semidefinite characterization and computation of zerodimensional real radical ideals[END_REF] where they use moment matrices to compute the radical of real and complex ideals. They present two versions of the method for the complex case: first, in [START_REF] Lasserre | Semidefinite characterization and computation of zerodimensional real radical ideals[END_REF] they double up the machinery for the real case to obtain the radical of the complex ideal. However, in [START_REF] Lasserre | A unified approach to computing real and complex zeros of zero-dimensional ideals[END_REF] they significantly simplify their method and show how to use moment matrices of maximal rank to compute the multiplication matrices of an ideal between I and its radical √ I. In particular, in the Gorenstein case they can compute the multiplication matrices of I. In fact, in [START_REF] Lasserre | A unified approach to computing real and complex zeros of zero-dimensional ideals[END_REF] they cite our previous work [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF] to compute the multiplication matrices of √ I from the multiplication matrices of I, but the method proposed in the present paper is much simpler and more direct.

Note that one can also obtain the multiplication matrices of I with respect to the basis S = [b 1 , . . . , b N ] by simply eliminating the terms not in S from x k b i using Mac δ+1 (f ). The advantage of computing multiplication matrices of the radical √ I is that it returns matrices which are always simultaneously diagonalizable, and possibly smaller than the multiplication matrices of I, hence easier to work with. Moreover, if S contains the monomials 1, x 1 , . . . , x m , one eigenvector computation yield directly the coordinates of the roots.

Computation of the radical of zero dimensional complex ideals is very well studied in the literature: methods most related to ours include [START_REF] González | The computation of the radical for a zero dimensional ideal in a polynomial ring through the determination of the trace for its quotient algebra[END_REF][START_REF] Becker | Radical computations of zero-dimensional ideals and real root counting[END_REF] where matrices of traces are used in order to find generators of the radical, and the matrices of traces are computed using Gröbner Bases; also, in [START_REF] Armendáriz | On the computation of the radical of polynomial complete intersection ideals[END_REF] they use the traces to give a bound for the degree of the generators of the radical and use linear solving methods from there; in [START_REF] González-Vega | Using symmetric functions to describe the solution set of a zerodimensional ideal[END_REF] they describe the computation of the radical using symmetric functions which are related to traces. One of the most commonly quoted method to compute radicals is to compute the projections I ∩ K[x i ] for each i = 1, . . . , m and then use univariate squarefree factorization (see for example [START_REF] Gianni | Gröbner bases and primary decomposition of polynomial ideals[END_REF][START_REF] Krick | Membership problem, representation problem and the computation of the radical for one-dimensional ideals[END_REF][START_REF] Cox | Using Algebraic Geometry[END_REF][START_REF] Greuel | A Singular introduction to commutative algebra[END_REF] ). The advantage of the latter is that it can be generalized for higher dimensional ideals (see for example [START_REF] Krick | An algorithm for the computation of the radical of an ideal in the ring of polynomials[END_REF]). We note here that an advantage of the method using matrices of traces is that it behaves stably under perturbation of the roots of the input system, as was proved in [START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF]. Other methods to compute the radical of zero dimensional ideals include [START_REF] Kobayashi | On radical zero-dimensional ideals[END_REF][START_REF] Gianni | Algebraic solution of systems of polynomial equations using Gröbner bases[END_REF][START_REF] Lakshman | On the complexity of computing a Gröbner basis for the radical of a zero-dimensional ideal[END_REF][START_REF] Lakshman | A single exponential bound on the complexity of computing Gröbner bases of zerodimensional ideals[END_REF][START_REF] Lakshman | On the complexity of zero-dimensional algebraic systems[END_REF][START_REF] Yokoyama | Solutions of systems of algebraic equations and linear maps on residue class rings[END_REF]. Applications of computing the radical include [START_REF] Heiß | On inverse systems and squarefree decomposition of zero-dimensional polynomial ideals[END_REF], where they show how to compute the multiplicity structure of the roots of I once the radical is computed.

Methods for computing the matrix of traces directly from the generating polynomials of I, without using multiplication matrices, include [START_REF] Díaz-Toca | An explicit description for the triangular decomposition of a zerodimensional ideal through trace computations[END_REF][START_REF] Briand | Multivariate Newton sums: Identities and generating functions[END_REF] where they use Newton Sums, [START_REF] Cardinal | Algebraic approach of residues and applications[END_REF][START_REF] Cattani | Computing multidimensional residues[END_REF][START_REF] Cattani | Residues and resultants[END_REF] where they use residues and [START_REF] Jeronimo | Rational formulas for traces in zero-dimensional algebras[END_REF] using resultants. Besides computing the radical of an ideal, matrices of traces have numerous applications mainly in real algebraic geometry [START_REF] Becker | Sums of squares and quadratic forms in real algebraic geometry[END_REF][START_REF] Pedersen | Counting real zeros in the multivariate case[END_REF][START_REF] Becker | Recent advances in real algebraic geometry and quadratic forms[END_REF], or in [START_REF] Rouiller | Solving zero-dimensional systems through the rational univariate representation[END_REF] where trace matrices are applied to find separating linear forms deterministically.

Moment Matrices and Matrices of Traces

Let f = [f 1 , . . . , f s ] be a system of polynomials of degrees d 1 ≥ • • • ≥ d s in K[x]
, where x = [x 1 , . . . , x m ] and K is an arbitrary algebraically closed field. Let I be the ideal generated by f 1 , . . . , f s in K[x] and define A := K[x]/I. We assume throughout the paper that A is a finite dimensional vector space over K and let A * denote the dual space of A.

Let us first recall the definition of a Gorenstein algebra (c.f. [START_REF] Kunz | Advanced lectures in Mathematics[END_REF][START_REF] Scheja | Über Spurfunktionen bei vollständigen Durschnitten[END_REF][START_REF] Elkadi | Introduction à la résolution des systèmes polynomiaux[END_REF][START_REF] Lasserre | A unified approach to computing real and complex zeros of zero-dimensional ideals[END_REF]). Note that these algebras are also referred to as Frobenius in the literature, see for example [START_REF] Becker | Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula[END_REF].

Definition 1 A finite dimensional K-algebra A is Gorenstein (over K) if there exists a nondegenerate K-bilinear form B(x, y) on A such that B(ab, c) = B(a, bc) for every a, b, c ∈ A.
Note that this is equivalent to the fact that A and A * are isomorphic as A modules. It is also equivalent to the existence of a K-linear function Λ : A → K such that the bilinear form B(a, b) := Λ(ab) is nondegenerate on A.

Assumption 2 Throughout the paper we assume that A is Gorenstein. Furthermore, we also assume that we have a bound δ > 0 such that

N := dim K K[x] δ / f 1 , . . . , f s δ = dim K K[x] d / f 1 , . . . , f s d (1)
for all d ≥ δ and that

N = dim A. (2)
Here

f 1 , . . . , f s d := i f i p i : deg(p i ) ≤ d -d i . (3)
We fix S = [b 1 , . . . , b N ] a monomial basis for A such that deg(b i ) ≤ δ for all i = 1, . . . , N . Let D be the maximum degree of the monomials in S. Thus D ≤ δ.

We have the following theorem giving bounds for δ in the case when f has finitely many projective roots.

Theorem 3 Let f = [f 1 , . . . , f s ] be a system of polynomials of degrees d 1 ≥ • • • ≥ d s in K[x]
. Assume that f 1 , . . . , f s has finitely many projective common roots in P m K . Assume further that f 1 , f 2 , . . . , f s have no common roots at infinity. Then: 1) and ( 2) are satisfied. Furthermore, in this case A is always Gorenstein.

1. If s = m then for δ := m i=1 (d i -1) conditions (

Proof.

For the first assertion let f h be the homogenization of f using a new variable x m+1 . Using our assumption that f h has finitely many roots in P m K and s = m, one can see that (f h ) is a regular sequence in R :

= K[x 1 , . . . , x m , x m+1 ].
Define the graded ring B := R/ f h . Following the approach and notation in [START_REF] Stanley | Combinatorics and commutative algebra[END_REF], we can now calculate the Hilbert series of B, defined by H(B, λ) = d H B (d)λ d , where H B is the Hilbert function of B. We have

H(R, λ) = H(B, λ) (1 -λ d1 ) • • • (1 -λ dm ) ,
and using the simple fact that

H(R, λ) = 1 (1 -λ) m+1 we obtain that H(B, λ) = (1 + λ + • • • + λ d1-1 ) • • • (1 + λ + • • • + λ dm-1 ) (1 -λ) = g(λ)(1 + λ + . . .), where g(λ) = (1 + λ + • • • + λ d1-1 ) • • • (1 + λ + • • • + λ dm-1 ).
This implies that the Hilbert function

H B (δ) = H B (δ + 1) = H B (δ + 2) = . . .

Note that dehomogenization induces a linear isomorphism

B d → K[x] d / f 1 , . . . , f s d
, where B d stands for the degree d homogeneous part of B. From this, using that there are no common roots at infinity, we infer that for 1) and ( 2). Note that the common value N = H B (δ) is the sum of the coefficients of g, which is

d ≥ δ dim K K[x] d / f 1 , . . . , f s d = dim K A = N , which implies (
g(1) = m i=1 d i .
To prove that A is Gorenstein, we cite [START_REF] Elkadi | Introduction à la résolution des systèmes polynomiaux[END_REF]Proposition 8.25,p. 221] where it is proved that if f 1 , . . . , f m is an affine complete intersection then the Bezoutian B 1,f1,...,fm defines an isomorphism between A * and A.

To prove the second assertion we note that [33, Theorem 3.3] implies that

dim K B δ = dim K B δ+1 = . . . .
From here we obtain (1) and (2) as in the Case 1.

Remark 4 Note that in general

I d = f 1 , . . . , f s d
, where I d is the set of elements of I with degree at most d and f 1 , . . . , f s d was defined in (3). This can happen when the system has a root at infinity, for example, if

f 1 = x + 1, f 2 = x then I 0 = span K (1) but f 1 , f 2
0 = {0} However, using the homogenization f h 1 , . . . , f h s , the degree d part of the homogenized ideal is always equal to the space spanned by the multiples of f h 1 , . . . , f h s of degree d. The above example also demonstrates that dim A is not always the same as dim K

[x] d / f 1 , . . . , f s d for large enough d, because above dim A = 0 but dim K[x, y] d / f 1 , f 2 d = 1 for all d ≥ 0.
Next we will define Sylvester and Macaulay type resultant matrices for f 1 , . . . f s .

Definition 5 Define

∆ := max(δ, 2D + 1)

where δ and D are defined in Assumption 2.

Let Syl ∆ (f ) be the transpose matrix of the linear map

i K[x] ∆-di -→ K[x] ∆ (4)
(g 1 , . . . , g s ) → s i=1

f i g i
written in the monomial bases. So, in our notation, Syl ∆ (f ) will have rows which correspond to all polynomials f i x α of degree at most ∆.

Let Mac ∆ (f ) be a row submatrix of Syl ∆ (f ) of maximal size with linearly independent rows.

Remark 6

In the case where s = m, for generic f we can directly construct Mac ∆ (f ) by taking the restriction of the map (4) to

m i=1 S i (∆) -→ K[x] ∆ where S i (∆) = span{x α : |α| ≤ ∆ -d i , ∀j < i, α j < d j }.
Here Mac ∆ (f ) is a submatrix of the classical Macaulay matrix of the homogenization of f and some f h m+1 , where f h m+1 is any homogeneous polynomial of degree ∆-δ: we only take the rows corresponding to the polynomials in f . Since the Macaulay matrix is generically non-singular, Mac ∆ (f ) will also be generically full rank.

Note that with our assumption that f 1 , . . . , f m has finitely many projective roots, we have that Mac ∆ (f ) has column corank N := m i=1 d i . Since ∆ ≥ δ, by Assumption 2 the corank of Mac ∆ (f ) = N , where N is the dimension of A. Also, we can assume that the elements of the basis S of A are monomials of degree at most δ, and that the first columns of Mac ∆ (f ) correspond to the basis S of A.

Fix an element

y = [y α : α ∈ N m , |α| ≤ ∆] T of the nullspace Null(Mac ∆ (f )), i.e. Mac ∆ (f ) • y = 0.
Definition 7 Let S be the basis of A as above, consisting of monomials of degree at most D. Using y we can define Λ y ∈ A * by Λ y (g) := x α ∈S y α g α , where g = x α ∈S g α x α ∈ A. Note that every Λ ∈ A * can be defined as Λ y for some y ∈ Null(Mac ∆ (f )) or more generally with an element of K[x] * which vanishes on the ideal I. Define the moment matrix M S (y) to be the N × N matrix given by

M S (y) = [y α+β ] α,β ,
where α and β run through the exponents of the monomials in S. Note that M S is only a submatrix of the usual notion of moment matrices in the literature, see for example [START_REF] Curto | Solution of the truncated complex moment problem for flat data[END_REF]. For p ∈ A, we define the linear function p • Λ ∈ A * as p • Λ(g) := Λ(pg) for all g ∈ A.

Remark 8 If one considers a linear function Λ on A, such that the bilinear form (x, y) → Λ(xy) is nondegenerate on A, then the moment matrix corresponding to this Λ will be the one whose (i, j)-th entry is just Λ(b i b j ). Moreover, for g, h ∈ A

Λ y (gh) = coeff S (g) T • M S (y) • coeff S (h)
where coeff S (p) denotes the vector of coefficients of p ∈ A in the basis S.

The following proposition is a simple corollary of [31, Prop 3.3 and Cor. 3.1].

Proposition 9 Let y be a random element of the vector space Null(Mac ∆ (f )). With high probability, M S (y) is non-singular.

Remark 10 Using the above proposition, one can detect whether the algebra A is not Gorenstein with high probability by simply computing the rank of M S (y) for (perhaps several) random elements y in Null(Mac ∆ (f )).

Remark 11 By [31, Theorem 2.6 and Lemma 3.2] one can extend y to ỹ ∈ K N m such that the infinite moment matrix M(ỹ) := [ỹ α+β ] α,β∈N m has the same rank as M S (y) and the columns of M(ỹ) vanish on all the elements of the ideal I.

Next we define a basis dual to S = [b 1 , . . . , b N ] with respect to the moment matrix M S (y). Using this dual basis we also define a polynomial J which is in some sense a generalization of the Jacobian of a well-constrained polynomial system.

Definition 12

From now on we fix y ∈ Null(Mac ∆ (f )) such that M S (y) is invertible and we will denote by Λ the corresponding element Λ y ∈ A * . We define Also note that the notion of generalized Jacobian was also introduced in [START_REF] Becker | Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula[END_REF]. Its name come from the fact that if s = m and if Λ is the so called residue (c.f. [START_REF] Elkadi | Introduction à la résolution des systèmes polynomiaux[END_REF]), then

M -1 S (y) =: [c ij ] N i,
N i=1 b i b * i = J is the Jacobian of f 1 , . . . , f m .
We now recall the definition of the multiplication matrices and the matrix of traces as presented in [START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF]. The next results relate the multiplication by J matrix to the matrix of traces R.

Proposition 15 Let M J be the multiplication matrix of J with respect to the basis S. We then have that

M J = [T r(b i b * j )] N i,j=1 .
Proof. Let Λ ∈ A * be as in Definition 12. For any h ∈ A we have that

h = N j=1 Λ(hb j )b * j = N j=1 Λ(hb * j )b j ⇒ hb i = N j=1 Λ(hb * j b i )b j = M h [i, j] = Λ(hb * j b i ) ⇒ T r(h) = N i=1 Λ(hb * i b i ) = Λ(h N i=1 b * i b i ). Since J = N i=1 b * i b i in A, we have T r(h) = Λ(hJ). Therefore M J [i, j] = Λ(Jb * j b i ) = T r(b * j b i ) = T r(b i b * j )
Corollary 16

M J • M S (y) = [T r(b i b j )] N i,j=1 = R, or equivalently J • Λ = T r in A * . Proof. The coefficients of b * i in the basis S = [b 1 , . . . , b N ] are the columns of M -1 S (y), which implies that M J = [T r(b i b * j )] N i,j=1 = [T r(b i b j )] N i,j=1 • M -1 S (y). Therefore we have that M J • M S (y) = [T r(b i b j )] N i,j=1 .
Finally, we prove that the matrix of traces R can be computed directly from the Sylvester matrix of f 1 , . . . , f s and J, without using the multiplication matrix M J . First we need a lemma.

Theorem 19

Syl S (J)

•

MS(y) RS(y) = [T r(bibj)] N i,j=1
Proof. Since the j-th column of the matrix

MS(y) RS(y)
represents the values of b j • Λ on all the monomials of degree less than or equal to ∆, and the i-th row of Syl S (J) is the coefficient matrix of b i J, we have

Syl S (J) • MS(y) RS(y) = [(bj • Λ)(biJ)] N i,j=1 = [Λ(Jbibj)] N i,j=1 = [T r(bibj )] N i,j=1 .
We can now describe the algorithm to compute a set of multiplication matrices M xi , i = 1, . . . , m of the radical √ I of I with respect to a basis of K[x]/ √ I. To prove that the algorithm below is correct we need the following result from [START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF]Proposition 8.3] which is the consequence of the fact that the kernel of the matrix of traces corresponds to the radical of A: Proposition 20 Let R be a maximal non-singular submatrix of the matrix of traces R. Let r be the rank of R, and T := [b i1 , . . . , b ir ] be the monomials corresponding to the columns of R. Then T is a basis of the algebra K[x]/ √ I and for each k = 1, . . . , m, the solution M x k of the linear matrix equation

RM x k = Rx k is the multiplication matrix of x k for √ I with respect to T . Here Rx k is the r × r submatrix of [T r(x k b i b j )] N i,j=1
with the same row and column indices as in R.

Algorithm 21 Input: f = [f 1 , . . . , f s ] ∈ K[x] of

Compute

R = [T r(bibj )] N i,j=1 = Syl S (J) • MS (y) RS(y)
and

Rx k :=[T r(x k bibj )] N i,j=1 = Syl S (x k J) • MS(y)

RS(y)

for k = 1, . . . , m.

9. Compute R, a maximal non-singular submatrix of R. Let r be the rank of R, and T := [b i1 , . . . , b ir ] be the monomials corresponding to the columns of R.

10. For each k = 1, . . . , m solve the linear matrix equation RM x k = Rx k , where Rx k is the submatrix of R x k with the same row and column indices as in R.

Remark 22 Since the bound given in Theorem 3 might be too high, it seems reasonable to design the algorithm in an iterative fashion, similarly to the algorithms in [START_REF] Lasserre | A unified approach to computing real and complex zeros of zero-dimensional ideals[END_REF][START_REF] Lasserre | Semidefinite characterization and computation of zerodimensional real radical ideals[END_REF][START_REF] Zhi | Solving nonlinear polynomial systems via symbolic-numeric elimination method[END_REF], in order to avoid nullspace computations for large matrices. The bottleneck of our algorithm is doing computations with Mac ∆ (f ), since its size exponentially increases as ∆ increases. .

Even though this way we will not get matrices of traces, a system of multiplication matrices of the radical √ I can still be recovered: if Q denotes a maximal non-singular submatrix of Syl S (J) • X, and Qx k is the submatrix of Syl S (x k J) • X with the same row and column indices as in Q, then the solution M x k of the linear matrix equation QM x k = Qx k gives the same multiplication matrix of √ I w.r.t. the same basis T as the above Algorithm.

Remark 24 As M x k is the matrix of multiplication by x k modulo the radical ideal √ I, its eigenvectors are (up to a non-zero scalar) the interpolation polynomials at the roots of I. Similarly the eigenvectors of the transposed matrix M t

x k are (up to a non-zero scalar) the evaluation at the roots ζ of I (see [START_REF] Mourrain | Computing isolated polynomial roots by matrix methods[END_REF][START_REF] Elkadi | Introduction à la résolution des systèmes polynomiaux[END_REF] for more details).

Examples

In this section we present three examples. Each of them has three polynomials in two variables. The first one is a system which has roots with multiplicities, the second one is a system which has clusters of roots, and the third one is a system obtained by perturbing the coefficients of the first one. For each of them we compute the Macaulay matrix Mac ∆ (f ), the vector y in its nullspace, the moment matrix M S (y), the polynomial J, the matrix of traces R and the (approximate) multiplication matrices of the (approximate) radical, following Algorithm 21.

The exact system: f has common roots (-1, 3) of multiplicity 3 and (2, 2) of multiplicity 2.

f =              3x 2 1 + 18x1x2 -48x1 + 21x
The system with clusters: f has two clusters: (-1, 3), (-0.9, 3), (-1.01, 3.1) and (2, 2), (1.9, 2) each of radius 10 -1 .

f =              3x 2
The perturbed system: is obtained from f by a random perturbation of size 10 -3 . This system has no common roots. We set δ = 6, D = 2 and ∆ = 6. The Sylvester matrices in all three cases were size 28 × 28 and in the first two cases they had rank 23 while in the last case it was full rank. In the first two cases the fact that the corank is 5 indicates that there are 5 solutions, counting multiplicities. For these cases we computed a basis S := [1, x 1 , x 2 , x 1 x 2 , x 2 1 ] for the factor algebra by taking maximum rank submatrices of the Macaulay matrices. In the third case, we simply erased the columns of the Macaulay matrix corresponding to the monomials in S. From here, we chose random elements in the nullspaces of the (cropped) Macaulay matrices to compute the moment matrices: The polynomials J, computed from the moment matrices are:

f =              3x 2 1 + 18x1x2 -48x1 + 21.001x 2
        1 0 0 0 0 0 0 0 - 6
J = 5 - 3 10 x1 - 26 15 x2 - 1 30 x1x2 - 1 5 x 2 1 J = 5 + 0.916x1 -1.952x2 -0.636x1x2 -0.106x 2 1 Ĵ = 4.999 -0.306x1 -1.733x2 -0.030x1x2 -0.200x 2 1 .
After computing the matrices Syl S (J) and R S (y), we obtain the matrices of traces: The first matrix R has rank 2, while R and R have rank 5. In the first case we follow steps 9 and 10 of Algorithm 21 to obtain the multiplication matrices of the radical with respect to its basis T = [1, x 1 ]: with respective eigenvalues [2, -1] and [START_REF] Becker | Sums of squares and quadratic forms in real algebraic geometry[END_REF][START_REF] Becker | Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula[END_REF].

      5
For the second case we use the method described in [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] to compute the approximate multiplication matrices of the approximate radical of the clusters. Using Gaussian Elimination with complete pivoting, we found that the almost vanishing pivot elements were of the order of magnitude of 10 -1 which clearly indicated the numerical rank. Using the submatrices obtained from the complete pivoting algorithm we got the following approximate multiplication matrices of the approximate radical with respect to the basis T = [x 1 x 2 , x 2 ]: distance from the centers of gravity of the clusters, as was shown in [START_REF] Janovitz-Freireich | Approximate radical of ideals with clusters of roots[END_REF][START_REF] Janovitz-Freireich | Approximate radical for clusters: a global approach using gaussian elimination or svd[END_REF] (recall that the radius of the clusters was 10 -1 ).

In the third case, the numerical rank was not easy to determine using either SVD or complete pivoting. However, when we assume that the numerical rank of R is 2, and we cut the matrix R using the output of the complete pivoting algorithm, then we obtain the multiplication matrices with respect to the basis T = [x 1 x 2 , x 2 ]: The norm of the commutator of these matrices is 0.010 and their eigenvalues are respectively [1.997, -0.987] and [1.999, 2.993] (recall that the perturbation of the polynomials was of size 10 -3 ).

Conclusion

In this paper we gave an algorithm to compute matrices of traces and the radical of an ideal I which has finitely many projective common roots, none of them at infinity and its factor algebra is Gorenstein. A follow-up paper will consider an extension of the above algorithm which also works in the non-Gorenstein case and for systems which have roots at infinity, as well as an alternative method using Bezout matrices for the affine complete intersection case to compute the radical √ I.

2 .

 2 If s > m then for δ := m+1 i=1 d i -m conditions (1) and (2) are satisfied.

  j=1 . Let b * i := N j=1 c ji b j . Then [b * 1 , . . . , b * N ] corresponds to the columns of the inverse matrix M -1 S (y) and they also form a basis for A. Note that we have Λ(b i b * j ) = 1, if i = j, and 0 otherwise. Define the generalized Jacobian by J := N i=1 b i b * i mod I (5) expressed in the basis S = [b 1 , . . . , b N ] of A. Remark 13 Note that since N i=1 b i b * i has degree at most 2D, and ∆ > 2D, we can use Mac ∆ (f ) to find its reduced form, which is J. Because of this reduction, we have that deg(J) ≤ D ≤ δ.

Definition 14

 14 Let p ∈ A. The multiplication matrix M p is the transpose of the matrix of the multiplication map M p : A -→ A g → pg written in the basis S. The matrix of traces is the N × N symmetric matrix: R = [T r(b i b j )] N i,j=1 where T r(pq) := T r(M pq ), M pq is the multiplication matrix of pq as an element in A in terms of the basis S = [b 1 , . . . , b N ] and T r indicates the trace of a matrix.

degrees d 1 2 . 3 . 4 .

 1234 , . . . , d s generating an ideal I and δ > 0 such that they satisfy the conditions in Assumption 2. An optional input is D ≤ δ, which by default is set to be δ. Output: A basis T for the factor algebra K[x]/ √ I and a set of multiplication matrices {M xi |i = 1, . . . , m} of √ I with respect to the basis T . 1. Compute Mac ∆ (f ) for ∆ := max(2D + 1, δ) Compute a basis S of K[x] ∆ / f ∆ such that the polynomials in S have degrees at most D. Let S = [b 1 , . . . , b N ]. Compute a random combination y of the elements of a basis of N ull(Mac ∆ (f )). Compute the moment matrix M S (y) defined in Definition 7 and R S (y) defined in Lemma 17. 5. Compute M -1 S (y) and the basis [b * 1 , . . . , b * N ] defined in Definition 12. 6. Compute J = N i=1 b i b * i mod I using Mac ∆ (f ). 7. Compute Syl S (J) and Syl S (x k J) for k = 1, . . . , m defined in Definition 18.

Remark 23

 23 Note that if s = m then we can use the conventional Jacobian of f 1 , . . . , f m in the place of J, and any |Mon ≤ (∆)| × |S| matrix X such that it has full rank and Mac ∆ (f ) • X = 0 in the place of

  The vector which represents this evaluation at ζ in the dual space A * is the vector of values of [b 1 , . . . , b N ] at ζ. To obtain these vectors, we solve the generalized eigenvalue problem ( Rt x k -z Rt ) w = 0 and compute v = Rt w. The vectors v will be of the form [b 1 (ζ), . . . , b N (ζ)] for ζ a root of I. If b 1 = 1, b 2 = x 1 , . . . , b m+1 = x m , we can read directly the coordinates of ζ from this vector.

  The norm of the commutator of these matrices is 0.002 and their eigenvalues are respectively[1.949, -0.972] and [2.001, 3.036]. Note that the corresponding roots [1.949, 2.001] and [-0.972, 3.036] are within 10 -2

	0.976 1.895	1 -4.623 × 10 -7	and	2.346 -0.671	-0.354 2.691	.

Research supported by NSF grant CCR-0347506. Research partially supported by the ANR GECKO.

Research supported in part by OTKA grant NK63066.

Proof. By our assumption that the first columns of Mac ∆ (f ) correspond to S we have

, where the columns of B are indexed by the monomials in S. Note here that by Assumption 2 the rows of Mac ∆ (f ) span I ∆ , and the monomials in S span the factor space K[x] ∆ /I ∆ . These together imply that the (square

which implies that Define Syl S (P ) to be the matrix with rows corresponding to the coefficients of the polynomials (b 1 P ), . . . , (b N P ) in the monomial basis Mon ≤ (∆) (we use here that deg(b

Furthermore, we assume that the monomials corresponding to the columns of Syl S (P ) are in the same order as the monomials corresponding to the columns of Mac ∆ (f ).


Mohamed Zoubair Saidi

El Chaouki

Andreea Moujahid

Nadia Pasc

Clara Canilho

Alain Delgado-Sanchez

Vanessa Celzard

Richard Fierro

Tarik Kouitat-Njiwa

Chafik

Chaouki El Moujahid

Andreea Pasc
email: andreea.pasc@univ-lorraine.fr

Nadia Canilho

Clara Delgado-Sanchez

Alain Celzard

Vanessa Fierro

Richard Kouitat-Njiwa

Tarik Chafik

Enhanced tribological properties of wind turbine engine oil formulated with flower-shaped MoS2 nano-additives

Keywords: MoS 2 nano-additives, Hydrothermal synthesis, Surface hydrophobization, Wind turbine lubricant, Formulation, Friction/Wear modifiers

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Improving energy efficiency and sustainability in modern industry is one of the major global challenges facing the world today. Indeed, energy losses caused by excessive wear and friction in mechanical systems are estimated to account for more than 25% of the total energy loss worldwide [START_REF] Luo | Green laser irradiationstimulated fullerene-like MoS 2 nanospheres for tribological applications[END_REF][START_REF] Holmberg | Influence of tribology on global energy consumption, costs and emissions[END_REF]. In the wind energy sector, maintenance costs account for about 30% of the total cost of the energy produced [START_REF] Shafiee | Maintenance optimization and inspection planning of wind energy assets: Models, methods and strategies[END_REF]. In particular, frequent critical failures and extensive maintenance work in wind turbines, mainly on gearboxes and bearing components, account for about 13% of total maintenance costs [START_REF] Peeters | Vibration-based bearing fault detection for operations and maintenance cost reduction in wind energy[END_REF][START_REF] Graca | Failure and oil analysis in wind turbine gearboxes[END_REF]. The origin of these failures is related to severe and unstable operating conditions such as high loads, extreme temperature, etc., which cause irreversible cumulative damage to mechanical components (micro-pitting, cracking, false brinelling and fretting corrosion, gear scuffing, …) and thus increase energy losses through heat generation [START_REF] Nutakor | An experimental characterization of the friction coefficient of a wind turbine gearbox lubricant[END_REF]. Therefore, one of the most suitable and effective ways to overcome these failures and increase both economic and long-term benefits is the development of a new generation of lubricants that can ensure a long life and high efficiency of machine components and, consequently, a reduced maintenance time. The latest developments in lubrication technology have led to the design of new generations of lube oils based on advanced nanoparticles as lubricant additives (extreme pressure (EP), anti-wear (AW) and friction modifier (FM)). These advanced nano-additives are designed to react chemically with metal surfaces, forming protective film layers that reduce asperity contact as well as the coefficient of sliding friction. This prevents severe wear and seizure, overcoming the failure of vital engineering components and increasing the performance of mechanical systems under complex operating conditions, as well as increasing productivity by reducing energy consumption and increasing the reliability of contacting components [START_REF] Spear | 2D-Nanomaterials for controlling friction and wear at interfaces[END_REF][START_REF] Rasheed | Graphene based nanofluids and nanolubricants -Review of recent developments[END_REF][START_REF] Liu | Research progress in application of 2D materials in liquid-phase lubrication system[END_REF].

Compared to conventional lubricant additives, nanoparticles exhibit many advantages [START_REF] Spikes | Friction modifier additives[END_REF], such as the multiple possibilities of film formation on many different types of surfaces, durability and low volatility to withstand high temperature conditions. Furthermore, nanomaterials have good tribological properties and excellent thermal conductivity. Their use as additives would lead to highly conductive nanofluids and thus limit overheating by homogenizing the temperature, which would be of high interest in extreme temperature conditions. Thereby, thanks to the development of nanoscience and nanotechnologies, several studies have been carried out in order to examine the lubricating efficiency of different types of nanomaterials such as metals [START_REF] Padgurskas | Tribological properties of lubricant additives of Fe, Cu and Co nanoparticles[END_REF], metal oxides [START_REF] Ali | Improving the tribological characteristics of piston ring assembly in automotive engines using Al 2 O 3 and TiO 2 nanomaterials as nano-lubricant additives[END_REF], metal dichalcogenides [START_REF] Liu | Synthesis and Tribological Properties of Tubular NbS 2 and TaS 2 Nanostructures[END_REF][START_REF] Irving | On the lubricity of transition metal dichalcogenides: an ab initio study[END_REF], borates [START_REF] Saffari | Tribological properties of waterbased drilling fluids with borate nanoparticles as lubricant additives[END_REF] and so on, which has considerably widened the number of potential candidates to replace traditional additives. Among these advanced nanomaterials, molybdenum disulfide (MoS 2 ) is emerging as one of the most efficient anti-friction and extreme-pressure lubrication additives thanks to its special, graphite-like lubricating quality. The latter is based on its layered structure and low van der Waals force between layers allowing easy sliding, while its highly polarized sulfur atoms promote interaction between the particles and the metal surface, offering a great capacity to reduce friction and wear between opposing contact [START_REF] Wu | Surface modification of MoS 2 nanosheets as effective lubricant additives for reducing friction and wear in Poly-α-olefin[END_REF][START_REF] Chen | Mechanism of antiwear property under high pressure of synthetic oil-soluble ultrathin mos 2 sheets as lubricant additives[END_REF][START_REF] Kalin | Mechanisms and improvements in the friction and wear behavior using MoS 2 nanotubes as potential oil additives[END_REF].

In addition, the tribological response of MoS 2 can be tuned by various intrinsic parameters such as morphology, size, crystal structure and concentration on their operating mechanism [START_REF] Hu | Tribological properties of 3 types of MoS 2 additives in different base greases[END_REF][START_REF] Rabaso | Boundary lubrication: Influence of the size and structure of inorganic fullerene-like MoS 2 nanoparticles on friction and wear reduction[END_REF]. Tubular and spherical nanoparticles are considered the most suitable morphologies because of their high rolling and exfoliation capacities between sliding surfaces [START_REF] Tomala | Synergisms and antagonisms between MoS 2 nanotubes and representative oil additives under various contact conditions[END_REF]. The size and concentration of nanoparticles in lubricating oils also have a great influence both on tribological performance and colloidal stability; large particles accelerate the sedimentation rate, while high concentration favors the collision and agglomeration of particles and, consequently, accelerate their sedimentation by gravitational effect [START_REF] Wu | Preparation and tribological properties of chemically decorated MoS 2 nanosheets with oleic diethanolamide[END_REF][START_REF] Koshy | Evaluation of the tribological and thermo-physical properties of coconut oil added with MoS 2 nanoparticles at elevated temperatures[END_REF]. Furthermore, nanoparticles are characterized by their high surface energy due to their high surface-tovolume ratio, which increases their ability to attract each other and agglomerate, and thus lose their lubricating properties.

In addition to size reduction, chemical modification of the surface of nanoparticles by dispersants or surfactants is considered an essential tool to improve colloidal stability by providing repulsive energy, which increases the distance between suspended nanoparticles and reduces their tendency to agglomerate. Furthermore, surface modification can generate a lipophilic layer around the surface of the nanoparticles, which increases their compatibility with oils and therefore slows down their sedimentation rate and increases their colloidal stability.

Gulzar et al. [START_REF] Gulzar | Improving the AW/EP ability of chemically modified palm oil by adding CuO and MoS 2 nanoparticles[END_REF] showed that the colloidal stability of lubricants containing CuO and MoS 2 nanoparticles modified with 1wt.% of oleic acid was achieved for more than 70h at room temperature. Yang et al. [START_REF] Yang | Facile synthesis of stable colloidal MoS 2 nanoparticles for combined tumor therapy[END_REF] indicated that the use of MoS 2 nanoparticles coated with PolyVinyl Pyrrolidone (PVP) produces a homogenous and stable system with negligible fluctuations for 48h in water, RPMI 1640 medium and PBS. Liu et al. [START_REF] Liu | Fabrication of coral-like MoS 2 and its application in improving the tribological performance of liquid paraffin[END_REF] found that the use of CTAB-modified coral-like MoS 2 enhances the dispersion and stability of nanoparticles in liquid paraffin and also gives better results in improving lubrication performance due to its better oxidation resistance, better dispersion stability and easier exfoliation in liquid paraffin.

Our group [START_REF] Saidi | Effect of morphology and hydrophobization of MoS 2 microparticles on the stability of poly-aolefins lubricants[END_REF] investigated the effect of the morphology (1-5 m platelets vs 600-800 nm spheres) of MoS 2 particles on their stability in poly--olefins oils (PAO) of different viscosities. The results revealed that the increase in oil viscosity and hydrophobization degree of the particles (related to the length of the hydrocarbon chain of the dispersant) produces extremely stable formulations in the long term. In addition, closer stability results were obtained for both the platelet-like particles and submicron spherical particles due to the drag forces applied by the lubricating medium on the surface of the particles, which were found to be greater for the platelet shape than for the spherical shape. Wu et al. [START_REF] Wu | Fabrication and tribological properties of oil-soluble MoS 2 nanosheets decorated by oleic diethanolamide borate[END_REF] indicated that the dispersion stability of hydrothermally produced MoS 2 nanosheets in oil could be significantly enhanced with oleic diethanolamide borate (ODAB). Furthermore, the average friction coefficient, the average wear scar diameter and the performance at extreme pressure of ODAB-MoS 2 -based oil were improved by approximately 27.9%, 22.9% and 17.4%, respectively, compared to the base oil. Kumari et al [START_REF] Kumari | Structural defects-mediated grafting of alkylamine on fewlayer MoS 2 and its potential for enhancement of tribological properties[END_REF] successfully functionalized the surface of MoS 2 nanosheets with octadecylamine (ODA) and demonstrated that sulfur vacancies and structural defects on the surface of MoS 2 nanosheets facilitate the grafting of ODA and ensure good dispersibility in mineral oil, resulting in a significant reduction in friction (48%) and wear (44%).

Our group also investigated [START_REF] Saidi | Improved tribological properties, thermal and colloidal stability of poly-α-olefins based lubricants with hydrophobic MoS 2 submicron additives[END_REF] the colloidal stability and the tribological behavior of submicron solvothermal MoS 2 particles dispersed in poly-α-olefins (PAO) base oils as a function of the hydrophobization degree and viscosity of the oil and demonstrated that formulations containing 1wt.% of alkylsilane-modified MoS 2 nanoparticles remained highly stable (Turbiscan Stability Index < 1), even at high temperature (80°C). The tribological properties of the formulated poly--olefins lubricants also increased with the degree of hydrophobization of the MoS 2 particles.

Herein, we describe an efficient mixed hydrothermal method for the preparation of MoS 2 nanoparticles, with emphasis on the effect of the solvent to water ratio on the growth mechanism and the structural properties of the resulting particles. In order to examine their effect on friction and wear parameters in the presence of other common additives, the tribological potential of MoS 2 nanoparticles was evaluated in fully formulated commercial lubricating oil used for the lubrication of gearboxes in wind turbines. The results indicate excellent tribological performance and the presence of other additives did not alter the lubricating potential of these nanoparticles. Surface hydrophobization of these nanoparticles with alkylsilane groups provides high dispersibility and homogeneity in the oil system and further enhances the lubricity of the prepared formulations.

Experimental section

Syntheses of MoS 2 nanoparticles

MoS 2 nanoparticles were synthesized by a single-step hydrothermal approach using a Teflonlined stainless steel autoclave filled with a mixture of water and ethylene glycol up to 60% of its total volume (35 mL). In a typical procedure, 0.276 g of sodium molybdate (Na 2 MoO 4 .2H 2 O), 0.106 g of sulfur powder (S) and 0.075 g of manganese dichloride were added to the autoclave filled with 21 mL of ethylene glycol -water mixture and maintained at 190°C for 24h. After natural cooling, the black product was separated from the solution by centrifugation, washed several times with distilled water and ethanol, and finally dried under vacuum at 60°C for 3h. Four volume ratios (R EG / water ) of ethylene glycol (EG) to water (W) in the mixture were used as follows: 1:1, 1:2, 1:3 and 1:4 and the syntheses were labelled EG1W1, EG1W2, EG1W3 and EG1W4 respectively.

Formulation of MoS 2 -based nano-lubricants

In this study, fully formulated commercial oils used for the lubrication of wind turbine engines, are used as a continuous phase for the dispersion of 0.1 wt.% of MoS 2 nanoadditives. 1 wt.% of alkyl trichlorosilane derivatives [octadecyltrichlorosilane (ODTS, C 18 H 37 SiCl 3 ) and octyltrichlorosilane (OTS, C 8 H 17 SiCl 3 )] were added to enhance the colloidal stability of the prepared formulations. The formulations, with and without alkylsilane dispersants, were sonicated for 15 min using a 200W Sonopuls HD 2200 device used at 45% of its nominal power, then magnetically stirred for 2h at room temperature.

Characterization methods

X-ray diffraction (XRD) was performed using a Bruker D8 ADVANCE ECO diffractometer with Cu/Ka radiation ( = 0.154 nm) and using a scanning rate of 0.04° s -1 . The morphology and structure of the MoS 2 samples were observed by scanning electron microscopy (SEM, Hirox SH-4000M) and transmission electron microscopy (TEM, JEOL 2100F). FTIR analysis was performed on a Jasco 410 spectrometer at a spectral resolution of 4 cm -1 . XPS analyses were recorded using a KRATOS Axis Ultra X-ray photoelectron spectrometer (Kratos Analytical, Manchester, UK) equipped with a monochromatic Al K X-ray source (h = 1486.6 eV) operated at 150 W. The size distribution of the particles was measured by DLS on a Malvern Zetasizer 300HSA operating at an angle of 90°. The stability of the suspensions was determined using an optical analyzer (TURBISCAN LAB from Formulaction, L'Union, France), based on a light-scattering detection method. Tribological tests were performed using a CSM tribometer, with a pin-on-disk configuration. The ball used is in chrome steel type 100C6 with a diameter of 6 mm and the disk used is made of 304L stainless steel with a diameter of 30 mm and a height of 5 mm. The disks were "mirror polished" using finer silicon carbide abrasive disks in grades 400, 800 and 1200, in order to achieve the same finish and surface roughness for all samples. The topography of surfaces after the friction tests was studied using a smart WLI-extended optical 3D surface measurement system.

Results and discussion

Nanoparticles synthesis and characterization

The synthesis of the MoS 2 nanoparticles was carried out using a one-step hydrothermal method with a variable ethylene glycol to water volume ratio. All the products obtained were characterized by XRD, SEM and FTIR analysis, in order to determine the effect of ethylene glycol to water volume ratio on the intrinsic property of the products obtained.

The crystal structure and phase purity of the obtained samples were investigated by XRD. Fig. 1 displays the XRD patterns of the as-synthesized products as well as that of hexagonal 2H-MoS 2 (JCPD 37-1492). The XRD patterns of prepared samples (Fig. 1 (A-D)) are very similar and reveal the presence of broad diffraction peaks, characteristic of amorphous and disordered structure of MoS 2 materials. All diffraction patterns show two broad peaks towards 32° -33° and 57° -58°, corresponding to the reflections of the (100) and (110) diffraction planes and indicating that the four prepared samples have the same atomic arrangement as that of 2H-MoS 2 along the basal planes [START_REF] Li | Facile hydrothermal synthesis of MoS 2 nano-sheets with controllable structures and enhanced catalytic performance for anthracene hydrogenation[END_REF]. However, the absence of the (002) plane in the XRD pattern of sample EG1W1 indicates that the stacked layered structure of MoS 2 did not occur using an EG-water volume ratio of 1:1. Nonetheless, the appearance of a new broad peak centered at about 10° can be observed from the 1:2 volume ratio (sample EG1W2), and becomes more intense when the volume of ethylene glycol is reduced relative to water (Fig. 1.

(B-D)). This peak could be assigned to the (002) plane of 2H-MoS 2 shifted to lower angles [START_REF] Zhang | Template-free fabrication of hierarchical MoS 2 /MoO 2 nanostructures as efficient catalysts for hydrogen production[END_REF] and could corresponds to a periodic sequence of MoS 2 stacking layers along the c-axis.

The shift of the (002) plane towards lower Bragg angles has been attributed to the interplanar expansion caused by molecular defects or strains due to layer curvature, as reported by other researchers [START_REF] Dunn | The origin of an anomalous, low 2θ peak in x-ray diffraction spectra of MoS 2 films grown by ion beam assisted deposition[END_REF][START_REF] Wu | Surfactant-assisted fabrication of MoS 2 nanospheres[END_REF][START_REF] Ma | Biopolymer-assisted hydrothermal synthesis of flower-like MoS 2 microspheres and their supercapacitive properties[END_REF]. The SEM images of the as-prepared products are shown in Fig. 2. The variation in the volume ratio of ethylene glycol to water, R EG / water , initially shows a great influence on the size of the product obtained. Spherical nanoparticles with a relatively narrow size distribution and an average size of about 100-150 nm were obtained exclusively with an R EG /H2O of 1:4 (Fig. 2 (D)). By increasing the volume ratio of ethylene glycol, the size of the product obtained increases to more than 1 µm for particles resulting from an R EG / water of 1:1(Fig. 2 (A)). In addition to the impact on particle size, R EG / water has a strong effect on the morphology of the MoS 2 particles and on their agglomeration rate. For example, almost spherical particles with a diameter of 300 nm were obtained with the R EG / water of 1:3 (Fig. 2 (C)), while a R EG / water of 1:2 lead to highly agglomerated particles of irregular shape and inhomogeneous size (Fig. 2 (B)). However, a R EG / water of 1:1 produced irregular, agglomerated spheres with diameters greater than 1 µm, together with tubular and amorphous clusters (Fig. 2 (A)). TEM images (Fig. 3) reveal the stacked lamellar structure of MoS 2 nanoparticles (sample EG1W4) and confirm their flower-shaped morphology with an interlayer spacing of 0.64 nm, which is in good agreement with the theoretical interplanar distance of the (002) plane for hexagonal MoS 2 (0.614 nm).

Fig. 3. TEM images of the MoS 2 nanoparticles (sample EG1W4).

To fully understand the mechanism of MoS 2 formation under these synthesis conditions, the effect of each reagent must be determined. It has been reported that EG, in addition to its use as a solvent, can also act as a (weak) reducing and stabilizing agent at the same time [START_REF] Yue | Ethylene glycol: properties, synthesis, and applications[END_REF],

while elemental sulfur and sodium molybdate are used as precursors of sulfur and molybdenum, respectively. The mechanism of MoS 2 formation begins with the reaction of elemental sulfur with EG to form H 2 S by the following reactions [START_REF] Wang | Low-Temperature synthesis of near-monodisperse globular MoS 2 nanoparticles with sulphur powders[END_REF]:

2 HOCH 2 -CH 2 OH ⇄ 2 CH 3 CHO + 2 H 2 O (1) 2 CH 3 CHO + S ⇄ CH 3 CO-COCH 3 + H 2 S (2)
The H 2 S formed in the reaction medium causes the gradual reduction of Na 2 MoO 4 into Mo 4+ ions, which, in the presence of subcritical water, are transformed into MoS 3 nuclei that are then decomposed into MoS 2 , thermodynamically more stable [START_REF] Geng | Pure and stable metallic phase molybdenum disulfide nanosheets for hydrogen evolution reaction[END_REF]. Then, agglomeration and coalescence phenomena start spontaneously, until the MoS 2 nanoparticles reach their maximum size at which the particles are sufficiently stable [START_REF] Polte | Fundamental growth principles of colloidal metal nanoparticlesa new perspective[END_REF]. Thus, the increase in the R EG /H2O , could result in an increase of MoS 2 nuclei and thus to a higher nucleation rate and a continuous growth of the particles. On the contrary, a decrease in R EG /H2O could lead to reduced attraction forces between the nuclei during the growth process, thus creating an electrostatic barrier on the surface of the particles [START_REF] Polte | Fundamental growth principles of colloidal metal nanoparticlesa new perspective[END_REF][START_REF] Akiya | Roles of water for chemical reactions in high-temperature water[END_REF]. This barrier is added to that created by the ions Mn 2+ and Cl -coming from MnCl 2 , which considerably increases the repulsion forces during the agglomeration and coalescence process and minimizes the dipole-dipole attractions between the particles [START_REF] Ramasamy | Dopant induced diameter tuning of Mndoped CdTe nanorods in aqueous solution[END_REF]. These two electrostatic barriers lead to the formation of particles of smaller size.

All materials were characterized by FTIR and show similar spectra. As an example, Fig. 4 (A) shows the spectrum of EG1W4 nanoparticles. The first low intensity band observed around 465 cm -1 is assigned to the Mo-S bond [START_REF] Kumari | Octadecanethiol-grafted molybdenum disulfide nanosheets as oil-dispersible additive for reduction of friction and wear[END_REF], while those appearing at 664 cm -1 and 1048 cm -1 correspond to S 2 2-and sulfate species [START_REF] Afanasiev | Genesis of vesicle-like and tubular morphologies in inorganic precipitates: amorphous Mo oxysulfides[END_REF], respectively. A broad band at 801 cm -1 is also observed and may be due to the symmetrical and asymmetrical stretching vibrations of the cis dioxo MoO 2 2+ group [START_REF] Yuldashev | Synthesis and study by IR and UV methods of spectral analysis of a complex of Mo(VI) with quercetin[END_REF]. Other bands observed around 1624 cm -1 , 3220 cm -1 and 3425 cm -1 can be assigned to the water adsorbed on the surface of these nanoparticles [START_REF] Gerakines | The infrared band strengths of H 2 O, CO and CO 2 in laboratory simulations of astrophysical ice mixtures[END_REF]. Further insights in the chemical composition and atomic valence states of the as-synthesized MoS 2 nanoparticles were obtained from XPS analysis. Fig. 4(B) displays the Mo 3d spectrum with four peaks, two of which are predominant at 232.14 eV and 229.02 eV that can be assigned to Mo 3d3/2 and Mo 3d5/2 doublets of Mo 4+ , respectively [START_REF] Xu | Photoanode Current of large-area MoS 2 ultrathin nanosheets with vertically mesh-shaped structure on indium tin oxide[END_REF]. The peak at 235.14

eV can be attributed to Mo 3d5/2 of Mo 6+ , which implies the presence of an oxide state resulting from the oxidation of MoS 2 nanoparticles in air [START_REF] Mao | Morphologycontrolled synthesis and lithium storage properties of SnO 2 @C@MoS 2 hollow nanospheres with petaloid and granular MoS 2 nanosheets as the external layer in different solvents[END_REF]. Finally, a weak peak at 226.13

eV is ascribed to S 2s. As for the S 2p spectrum (Fig. 4(C)), the two mains doublets appeared at binding energies of 161.63eV and 163.04 eV and are due to S 2p3/2 and S2 p½ doublet of S 2-. The peak at 168.78eV indicates the existence of SO 4 2-species and confirms the partial oxidation of MoS 2 surface in contact with air [START_REF] Hou | Aging mechanism of MoS 2 nanosheets confined in N-doped mesoporous carbon spheres for sodium-ion batteries[END_REF]. The binding energies of Mo 3d and S 2p are in good agreement with the reported values for MoS 2 particles [START_REF] Liu | Flexible amorphous MoS 2 nanoflakes / N-doped carbon microtubes /reduced graphite oxide composite paper as binder free anode for full cell lithium ion batteries[END_REF][START_REF] Yi | The synthesis of two-dimensional MoS 2 nanosheets with enhanced tribological properties as oil additives[END_REF][START_REF] Kumari | Structural defects-mediated grafting of alkylamine on fewlayer MoS 2 and its potential for enhancement of tribological properties[END_REF].

Surface modification and nano-lubricant preparation

The surface modification of the MoS 2 nanoparticles was conducted in the commercial gearbox oil of a wind turbine. 0.1 wt.% of the prepared MoS 2 nanoparticles with and without alkyltrichlorosilane were added to the oil. The surface modification of MoS 2 nanoparticles was first investigated by FTIR analysis. The infrared spectra of alkylsilane-modified MoS 2 nanoparticles (Fig. 4 (A)) indicate the appearance of two new bands located at 2847 and 3923 cm -1 for both functionalized materials, attributed to the stretching vibration of methylene groups from the alkylsilane dispersants [START_REF] Saidi | Effect of morphology and hydrophobization of MoS 2 microparticles on the stability of poly-aolefins lubricants[END_REF]. In addition, the band observed at 1452 cm -1 could be assigned to the Si-C vibration, whereas those appearing at about 1001 and 761 cm -1

were attributed to Si-O and Mo-O-Si vibrations, respectively [START_REF] Saidi | Improved tribological properties, thermal and colloidal stability of poly-α-olefins based lubricants with hydrophobic MoS 2 submicron additives[END_REF]. Compared to the infrared spectra of unmodified MoS 2 nanoparticles, the presence of these news bands confirmed the chemisorption of alkyltrichlorosilane molecules on the surface of MoS 2 nanoparticles.

The prepared formulations, with modified and unmodified nanoparticles, were left at rest and visually inspected daily for 20 days, and then monthly, in order to follow their colloidal stability as a function of time (Fig. 5). Visual monitoring of the sedimentation process indicated that all formulations remained stable, and no changes were visually observed even after 1 year after their preparation. In order to demonstrate the effect of the dispersants used on the colloidal stability and agglomeration tendency of the MoS 2 nanoparticles dispersed in commercial oil used in wind turbines, the hydrodynamic size and the particle size distribution were investigated by dynamic light scattering (Fig. 6). However, the high viscosity of commercial oil prevents the movement of the particles, and therefore the measurement of their hydrodynamic size by the DLS technique. Thus, a 10-fold dilution was made in a less viscous base oil (PAO4). The results obtained are shown in Fig. 6 and indicate the power of alkylsilane moieties in reducing the agglomeration rate of blended oil/nano-MoS 2 systems.

The addition of 1 wt.% of dispersants resulted in a finer, monodisperse particle size distribution. Furthermore, the MoS 2 -ODTS formulation presents the smallest average size and the most homogenous particle distribution compared to bare MoS 2 and MoS 2 -OTS formulations, confirming the effect of the length of the hydrocarbon chain (hydrophobization rate) in improving colloidal stability.

Tribological behavior of formulations based on MoS 2

The friction-reducing and anti-wear properties of MoS 2 nanoparticles dispersed in wind turbine engine oil, with and without dispersants, were evaluated using a pin-on-disk laboratory tribometer under different loads (1N and 10N, corresponding respectively to about 450 MPa and 980 MPa mean contact pressure). It has already been reported that for certain types of anti-wear and friction-reducing additives, thermal activation is required to trigger their lubricating power [START_REF] Zhang | Template-free fabrication of hierarchical MoS 2 /MoO 2 nanostructures as efficient catalysts for hydrogen production[END_REF][START_REF] Dunn | The origin of an anomalous, low 2θ peak in x-ray diffraction spectra of MoS 2 films grown by ion beam assisted deposition[END_REF]. This may explain the instability of the friction behavior observed at the beginning of the test for wind turbine oil, and its stability after the 200 cycles. Conversely, the use of MoS 2 nanoparticles does not require any time for the formation of the tribofilm, but their strong aggregation tendency leads to the formation of large particles, which limits their passage to the contact surfaces. The reduction in the number of particles between the moving surfaces leads to the formation of a thin tribofilm that degrades rapidly. This can be confirmed by a simple comparison of the friction behavior observed at both 1N and 10N of bare MoS 2 nanoparticles with that of their analogs functionalized by alkylsilane groups. The functionalization of the surface of these nanoparticles leads to an improvement of their stability by reducing their agglomeration tendency, enabling them to exhibit again the best friction-reducing properties, with average values not exceeding 0.1 for the entire duration of the test.

Friction reduction abilities

Anti-wear properties

The effectiveness of a lubricant depends not only on its ability to reduce friction, but also on its ability to decrease wear rates and minimize surface damage between contacting components. To further explore the effect of MoS 2 nano-additives on contact sliding and surface damage, the surface topography was evaluated using a 3D optical surface measurement system. The analysis of the wear track surface (Fig. 10) indicates, in the case of pristine engine oils, the presence of grooves, scars and debris on the contact track. The wear depth obtained at a load of 10N is approximately 6.57 µm (Fig. 11). However, the addition of 0.1 wt.% of bare MoS 2 nanoparticles shows a slight reduction in wear depth of about 4.3% compared to the original oil. While the addition of nanoparticles modified with OTS and ODTS leads to significant reductions of up to 95.44% and 97.87 %, respectively. In addition, the comparison of 3D images of the wear tracks obtained with the different formulations confirms the negative effect of nanoparticles agglomeration on their performance and tribological efficiency as lubricant additives. They also show that the improvement in dispersibility and stability of these nano-additives dispersed in lubricant oils is a key and determining factor of their tribological power, leading not only to the improvement of their friction-reducing properties but also to a significant decrease in both the depth and width of wear.

Conclusion

In summary, the present study revealed that the variation in the ethylene glycol to water volume ratio subjected to hydrothermal treatment has a strong influence on the morphology and size of the MoS 2 nanoparticles obtained. Indeed, flower-shaped MoS 2 nanoparticles with an average size of about 150 nm have been obtained with a EG / W volume ratio of 1:4. By increasing the EG fraction, the particle size increases, and its morphology becomes less homogeneous and the rate of aggregation more important.

Alkyltrichlorosilane derivatives have proven their effectiveness as dispersing agents, leading to excellent dispersibility and long-term stability with TSI values below 0.5 in a fully formulated wind turbine oil. Subsequently, the evaluation of the tribological behavior of formulations with and without modified and unmodified MoS 2 nano-additives first shows that the excessive tendency of bare MoS 2 nanoparticles to agglomerate leads to the formation of large particles, acting as abrasive bodies between the sliding surfaces, and thus leading to the deterioration of their lubricity. On the other hand, improving the dispersibility and the stability of these nano-additives in lubricating oils has proven to be a key and determining factor in improving their tribological efficiency, leading not only to an improvement in their friction-reducing properties, but also to a significant decrease in depth and width of wear.

More importantly, MoS 2 nanoparticles modified with octadecyl moieties have been most

effective in reducing the friction and wear coefficients when added to wind turbine engine oil under laboratory conditions. This resulted in a reduction of more than 30% in the friction coefficient and up to 97% in the maximum wear depth.

Fig. 1 .

 1 Fig. 1. XRD pattern of MoS 2 particles obtained from different ethylene glycol to water volume ratios. A) 1:1 (sample EG1W1), B) 1:2 (sample EG1W2), C) 1:3 (sample EG1W3), D) 1:4 (sample EG1W4) vs 2H-MoS 2 .

Fig. 2 .

 2 Fig. 2. SEM images of the as-synthesized materials obtained from different ethylene glycol to water volume ratios. A) 1:1 (sample EG1W1), B) 1:2 (sample EG1W2), C) 1:3 (sample EG1W3), D) 1:4 (sample EG1W4).

Fig. 4 .

 4 Fig. 4. (A) FTIR spectra of MoS 2 nanoparticles with and without alkyl-silane derivatives, and XPS spectra of Mo 3d (B), and S 2p (C).

Fig. 5 .

 5 Fig. 5. Images of blended wind turbine oil and MoS 2 nanoparticles with and without alkylsilane dispersants after 20 days of preparation.

Fig. 6 .

 6 Fig. 6. Average particle size and particle size distribution of modified and unmodified MoS 2 nanoparticles.

Fig. 7 .

 7 Fig. 7. Turbiscan stability index as function of time.

Fig. 8

 8 Fig.8shows the anti-friction behavior of the different formulations at a load of 1N. The

Fig. 8 .

 8 Fig. 8. Friction-reducing behavior (A) and average Friction coefficient (B) of wind turbine oil under a load of 1N.

Fig. 9 .

 9 Fig. 9. Same as Fig. 8, but under 10N.

Fig. 10 .

 10 Fig. 10. Optical micrographs, pseudo-color view, 3D view and profile curve of wear track: (a, a', a'') Engine oil, (b, b', b'') Engine oil/MoS 2 , (c, c', c'') Engine oil/ MoS 2 -OTS, (d, d', d'') Engine oil/MoS 2 -ODTS.

Fig. 11 .

 11 Fig. 11. Maximum scar depth obtained with different formulations in wind turbine oil.

Acknowledgements

The authors thank the institute IRESEN/Morocco (Project Innowind13 Nanolubricant) for its financial support and for the scholarship awarded to M.Z. Saidi. The authors also thank S.

Haloui (Innovation and Development Center, University of Abdelmalek Essaadi) for performing X-ray measurements and Ovidiu Ersen and Dris Ihiawakrim (IPCMS, University of Strasbourg) for performing TEM analysis.