P Azzi

R Desmorat

B Kolev

F Priziac

DISTANCE TO A CONSTITUTIVE TENSOR ISOTROPY STRATUM BY LASSERRE POLYNOMIAL OPTIMIZATION METHOD

Keywords: July 9, 2022. 2020 Mathematics Subject Classification. 90C23, 14P10, 90C22, 74B05, 74E10 polynomial optimization, Lasserre's method, semidefinite programming, distance to a symmetry class, cubic symmetry, elasticity, piezoelectricity, semialgebraic and real algebraic geometry

We give a detailed description of a polynomial optimization method allowing to solve a problem in continuum mechanics: the determination of the elasticity or the piezoelectricity tensor of a specific isotropy stratum the closest to a given experimental tensor, and the calculation of the distance to the given tensor from the considered isotropy stratum. We take advantage of the fact that the isotropy strata are semialgebraic sets to show that the method, developed by Lasserre and coworkers which consists in solving polynomial optimization problems with semialgebraic constraints, successfully applies.

Introduction

In mechanics, linear constitutive laws are described by the orbit space of a representation of the three-dimensional orthogonal group on the vector space of the considered constitutive tensors T [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF]13,[START_REF] Eringen | Electrodynamics of Continua , tomes I et II[END_REF]. This orbit space is endowed with a natural stratification by isotropy classes [𝐻], the strata Σ [𝐻] being the set of tensors with symmetry group conjugate to 𝐻.

The symmetry group of a measured (raw) tensor T 0 is in general trivial. However, in practice, appealing to Curie principle-the symmetries of the causes are to be found in the effectsa symmetry of a constitutive tensor is often expected by observing the micro-structure of a material [START_REF] Arts | A study of general anisotropic elasticity in rocks by wave propagation[END_REF][START_REF] François | Une nouvelle analyse des symétries d'un matériau élastique anisotrope. exemple d'utilisation à partir de mesures ultrasonores[END_REF][START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF]. For instance, the elasticity tensor of a single crystal alloy with cubic crystal network is expected to be cubic ([O], see figure 1), the piezoelectric tensor of an aluminum nitride (AlN) alloyed with rocksalt transition metal nitrides is expected to become cubic ([O -]) for a high chromium concentration [START_REF] Manna | Enhanced piezoelectric response of AlN via CrN alloying[END_REF]. The mechanical problem thus comes down to the computation of the distance 𝑑(T 0 , Σ [𝐻] ) of a raw constitutive tensor T 0 to a closed isotropy stratum Σ [𝐻] .

In linear elasticity, which involves a fourth-order tensor E, the distance to an isotropy stratum has been formulated as the minimization problem [START_REF] Gazis | The elastic tensor of given symmetry nearest to an anisotropic elastic tensor[END_REF][START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF][START_REF] Dellinger | Computing the optimal transversely isotropic approximation of a general elastic tensor[END_REF][START_REF] Moakher | The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry[END_REF][START_REF] Bucataru | Invariant properties for finding distance in space of elasticity tensors[END_REF] min

E∈Σ [𝐻]
‖E 0 -E‖, E = 𝜌 4 (𝑔)A, A ∈ Fix(𝐻), 𝑔 ∈ SO(3), with the natural parameterization by normal form A (a fixed point set for a representative symmetry group 𝐻) and rotation 𝑔. This problem has, however, many local minima and several global minima, making the determination of all the solutions numerically difficult.

In this paper, we formulate the computation of the distance to an isotropy stratum as a polynomial optimization problem. To do so, we make use of the property that the closed isotropy strata Σ [𝐻] are basic closed semialgebraic sets [START_REF] Abud | The geometry of orbit-space and natural minima of Higgs potentials[END_REF][START_REF] Abud | The geometry of spontaneous symmetry breaking[END_REF][START_REF] Procesi | Inequalities defining orbit spaces[END_REF][START_REF] Schwarz | The topology of algebraic quotients[END_REF]. For the fourth-order elasticity tensor, an explicit characterization of the closed strata Σ [𝐻] by polynomial equations and inequalities has recently been obtained, by means of polynomial covariants [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]Theorem 10.2]. Since such a result is not yet available in piezoelectricity, we have provided in theorem 7.2 a polynomial characterization of the cubic symmetry stratum Σ [O -] for the third-order piezoelectricity tensor.

We formulate the distance problems in question in such a way that we can apply a semialgebraic optimization method designed by Lasserre and coworkers [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF][START_REF] Lasserre | Moments, Positive Polynomials and Their Applications[END_REF][START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF][START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF][START_REF] Jeyakumar | On polynomial optimization over non-compact semi-algebraic sets[END_REF] to compute directly the global miminum of a polynomial function over polynomial constraints describing a basic closed semialgebraic set. This method consists in building a sequence of semidefinite programs whose optimal values converge to the desired minimum, under some hypothesis on the constraints. The benefit is that there exist efficient algorithms to solve numerically semidefinite programs, based on methods used in linear programming [START_REF] Dantzig | Linear programming and extensions[END_REF][START_REF] Luenberger | Linear and nonlinear programming[END_REF][START_REF] Vanderbei | Linear programming: foundations and extensions[END_REF], such as the ellipsoid method [START_REF] Grötschel | The ellipsoid method and its consequences in combinatorial optimization[END_REF] or the interior point method [START_REF] Nesterov | Interior-point polynomial algorithms in convex programming[END_REF][START_REF] Alizadeh | Interior point methods in semidefinite programming with applications to combinatorial optimization[END_REF][START_REF] Nesterov | Self-scaled barriers and interior-point methods for convex programming[END_REF][START_REF] Todd | Semidefinite optimization[END_REF][START_REF] Jarre | An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices[END_REF][START_REF] Benterki | A numerical feasible interior point method for linear semidefinite programs[END_REF][START_REF] Bochnak | Real Algebraic Geometry[END_REF][START_REF] Hertog | Interior point approach to linear, quadratic and convex programming: algorithms and complexity[END_REF]. The considered algorithm has been implemented by Lasserre and Henrion [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF] in a Matlab freeware, named GloptiPoly [START_REF] Henrion | GloptiPoly: global optimization over polynomials with Matlab and SeDuMi[END_REF], that aims to solve a sequence of relaxed semidefinite programs using SeDuMi (a Matlab toolbox for solving semidefinite programs created by Sturm [START_REF] Sturm | Primal-dual interior point approach to semidefinite programming[END_REF][START_REF] Sturm | Using SeDuMi 1.02, a matlab toolbox for optimization over symmetric cones[END_REF]). Moreover, this algorithm, when its stopping criterion is satisfied, extracts (approximated) minimizers for the considered minimized function.

Organization of the paper. The paper is organized as follows. In section 2, we recall basic material on isotropy classes and we pose the problem of the distance to an isotropy class. In section 3, we introduce polynomial optimization and its formulation to semidefinite programs. In section 3 and section 4, we describe the Lasserre and coworkers method for solving polynomial optimization problems with semialgebraic constraints as well as the corresponding algorithm, implemented as the software GloptiPoly. As a direct application, we deal with three examples of constitutive tensors. In section 5, we illustrate the method with the academic example of the distance of a symmetric second-order tensor to the transversely isotropic stratum Σ [O [START_REF] Abud | The geometry of spontaneous symmetry breaking[END_REF]] . In section 6, we compute the distance of an experimental elasticity (fourth-order) tensor of a Nickel-based single crystal superalloy to the cubic stratum Σ [O] , and consequently we extract the cubic elasticity tensor the closest to the experimental one. Finally, in section 7, we detail how polynomial optimization allows to find the cubic piezoelectricity (third-order) tensors, in Σ [O -] , the closest to raw tensors for wurtzite alloys.

All the tensorial components will be expressed with respect to an orthonormal basis. Hence, no distinction will be made between covariant and contravariant components. The notation q = (𝛿 𝑖𝑗 ) stands for the Euclidean metric tensor.

Isotropy classes and strata -Distance to an isotropy stratum

Let 𝐺 be a compact group and 𝜌 : 𝐺 → GL(V) be a continuous representation of 𝐺 on a finite dimensional real vector space V. Given 𝑣 𝑣 𝑣 ∈ V, its orbit is the subset of V defined by Orb(𝑣 𝑣 𝑣) := {𝜌(𝑔)𝑣 𝑣 𝑣; 𝑔 ∈ 𝐺} , and its symmetry group (or isotropy group) is defined as

𝐺 𝑣 𝑣 𝑣 := {𝑔 ∈ 𝐺; 𝜌(𝑔)𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣} .
The concept of symmetry group allows to define an equivalence relation on V, which is coarser than the relation "to be in the same orbit" and defined as follows: two vectors 𝑣 𝑣 𝑣 1 and 𝑣 𝑣 𝑣 2 have the same isotropy class (or same symmetry class in mechanics [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF][START_REF] Forte | Symmetry classes and harmonic decomposition for photoelasticity tensors[END_REF]) if they have conjugate symmetry groups. In the following, we shall denote by

[𝐻] := { 𝑔𝐻𝑔 -1 ; 𝑔 ∈ 𝐺 }
the conjugacy class of the subgroup 𝐻 of 𝐺. To each conjugacy class [𝐻], where 𝐻 is a closed subgroup of 𝐺, corresponds the subset of V defined by

Σ [𝐻] := {𝑣 𝑣 𝑣 ∈ V; [𝐺 𝑣 𝑣 𝑣 ] = [𝐻]} .
If this subset is not empty, [𝐻] is called an isotropy class and Σ [𝐻] is the isotropy stratum associated to [𝐻]. It is known (see [START_REF] Mostow | On a conjecture of montgomery[END_REF][START_REF] Bredon | Finiteness of number of orbit types[END_REF][START_REF] Mann | Finite orbit structure on locally compact manifolds[END_REF]) that there is only a finite number of isotropy classes for any finite dimensional representation of a compact group. The set of conjugacy classes [𝐻] of closed subgroups of a compact group is endowed with a partial order relation (reflexivity and transitivity are direct and true even if 𝐺 is not compact but anti-symmetry requires the compacity of 𝐺 [11, Proposition 1.9]), given by

[𝐻] ⪯ [𝐾] ⇐⇒ ∃𝑔 ∈ 𝐺, 𝑔𝐻𝑔 -1 ⊂ 𝐾.
Due to the order relation defined on the conjugacy classes, we define a closed stratum to be the set consisting of vectors having at least the symmetry [𝐻], denoted by Σ [𝐻] , and defined by

Σ [𝐻] = {𝑣 𝑣 𝑣 ∈ V; [𝐻] ⪯ [𝐺 𝑣 𝑣 𝑣 ]} = ⋃ [𝐻]⪯[𝐾] Σ [𝐾] .
The isotropy stratum Σ [𝐻] and the closed isotropy stratum Σ [𝐻] are semialgebraic sets [START_REF] Abud | The geometry of orbit-space and natural minima of Higgs potentials[END_REF][START_REF] Abud | The geometry of spontaneous symmetry breaking[END_REF][START_REF] Procesi | Inequalities defining orbit spaces[END_REF][START_REF] Schwarz | The topology of algebraic quotients[END_REF], i.e defined by polynomial equations and inequalities [START_REF] Coste | An Introduction to Semialgebraic Geometry[END_REF][START_REF] Bochnak | Real Algebraic Geometry[END_REF]. Actually, if 𝐺 is a subgroup of GL(V), we can give a direct proof of this fact. Indeed, if 𝐺 is a compact subgroup of GL(V), 𝐺 is a real algebraic set by [START_REF] Onishchik | Lie Groups and Algebraic Groups[END_REF]Chapter 3,paragraph 4,Theorem 5]. Notice that so is the subset 𝐺 𝑣 𝑣 𝑣 , if 𝑣 𝑣 𝑣 ∈ V, as it is described by polynomial equations in the coefficients of the matrices of the real algebraic set 𝐺. Now, since 𝐻 is a closed subgroup of 𝐺, 𝐻 is in particular a compact subgroup of GL(V) and then a real algebraic set as well. As a consequence, the closed isotropy stratum Σ [𝐻] = { 𝑣 𝑣 𝑣 ∈ V; ∃𝑔 ∈ 𝐺, ∀ℎ ∈ 𝐻, 𝑔ℎ𝑔 -1 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 } , and the isotropy stratum

Σ [𝐻] = { 𝑣 𝑣 𝑣 ∈ V; ∃𝑔 ∈ 𝐺, ∀ℎ ∈ 𝐻, 𝑔ℎ𝑔 -1 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 and ∃𝑔 ′ ∈ 𝐺, ∀𝑘 ∈ 𝐺, 𝑘𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 ⇒ 𝑔 ′ 𝑘𝑔 ′ -1 ∈ 𝐻 } ,
are both described by first-order formulae (in the sense of [9, Definition 2.2.3]) so that the sets Σ [𝐻] and Σ [𝐻] are semialgebraic sets by [9, Proposition 2.2.4] (the latter cited result is an avatar of Tarski-Seidenberg theorem which is an angular stone of semialgebraic geometry). We shall introduce the distance of a vector 𝑣 𝑣 𝑣 0 ∈ V to the closed isotropy stratum Σ

[𝐻] (1) ∆(𝑣 𝑣 𝑣 0 , Σ [𝐻] ) 2 := min 𝑣 𝑣 𝑣∈Σ [𝐻] ‖𝑣 𝑣 𝑣 0 -𝑣 𝑣 𝑣‖ 2 ,
for some 𝐺-invariant norm ‖•‖. A minimizer will be denoted by 𝑣 𝑣 𝑣 * .

Examples of interest for the present work are provided by Continuum Mechanics, for which 𝐺 is either SO [START_REF] Alizadeh | Interior point methods in semidefinite programming with applications to combinatorial optimization[END_REF] or O(3), V is a space of tensors on R 3 , endowed with the invariant norm

‖T‖ := √ 𝑇 𝑖 1 ...𝑖𝑛 𝑇 𝑖 1 ...𝑖𝑛 ,
and the action on a tensor T is written (in an orthonormal basis)

(𝜌(𝑔)T) 𝑖 1 ...𝑖𝑛 := 𝑔 𝑖 1 𝑗 1 . . . 𝑔 𝑖𝑛 𝑗𝑛 𝑇 𝑗 1 ...𝑗𝑛 , T ∈ V, 𝑔 ∈ 𝐺.
Finally, the O(3)-subgroups will be denoted according to the notations in [START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF].

Example 2.1. In elasticity,

V = Ela = S 2 (S 2 (R 3 )) = { E ∈ ⊗ 4 R 3 , 𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑘𝑙𝑖𝑗 }
is the 21-dimensional vector space of elasticity tensors, and 𝐺 = SO(3). In that case there are exactly eight isotropy classes [START_REF] Abud | The geometry of orbit-space and natural minima of Higgs potentials[END_REF],

[Z 2 ], [D 2 ], [D 3 ], [D 4 ], [O(2)], [O], [SO(3)] [22]
, and the problem of the distance to an elasticity isotropy stratum has been investigated in [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF][START_REF] Dellinger | Computing the optimal transversely isotropic approximation of a general elastic tensor[END_REF][START_REF] Moakher | The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry[END_REF][START_REF] Kochetov | On obtaining effective transversely isotropic elasticity tensors[END_REF][START_REF] Kochetov | On obtaining effective orthotropic elasticity tensors[END_REF].

Example 2.2. In piezoelectricity,

V = Piez = { e ∈ ⊗ 3 R 3 , e 𝑖𝑗𝑘 = e 𝑖𝑘𝑗 }
is the 18-dimensional vector space of piezoelectricity tensors, and 𝐺 = O(3). In that case there are exactly 16 isotropy classes [START_REF] Abud | The geometry of orbit-space and natural minima of Higgs potentials[END_REF], [START_REF] Nye | Physical Properties of Crystals[END_REF][START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF][START_REF] Olive | Symmetry classes in piezoelectricity from second-order symmetries[END_REF], and the problem of the distance to a piezoelectricity isotropy stratum has been investigated in [START_REF] Zou | Symmetry types of the piezoelectric tensor and their identification[END_REF].

[Z 2 ], [Z 3 ], [D 𝑧 2 ], [D 𝑧 3 ], [Z - 2 ], [Z - 4 ], [D 2 ], [D 3 ], [D 𝑑 4 ], [D 𝑑 6 ], [SO(2)], [O(2)], [O(2) -], [O -], [O(3)] [
When polynomial equations and/or inequalities characterizing the semialgebraic set Σ [𝐻] are known (see [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]), the distance to an isotropy stratum problem (1) reduces to minimize a polynomial function (the quadratic function ∆( • , Σ [𝐻] ) 2 ) under polynomial constraints. In that case, we can solve the distance to an isotropy stratum problem using polynomial and semialgebraic optimization [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF][START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF][START_REF] Mevissen | Introduction to concepts and advances in polynomial optimization[END_REF][START_REF] Schweighofer | Optimization of polynomials on compact semialgebraic sets[END_REF], which allows to approximate numerically the global minimum of the function ∆( • , Σ [𝐻] ) 2 .

Semialgebraic optimization method

The problem of determining the constitutive tensor having a specific symmetry the closest to an experimental one can be viewed as an example of the problem of minimizing a polynomial function over polynomial constraints with 𝑔 1 , . . . , 𝑔 𝑚 ∈ R[𝑋] (see [START_REF] Coste | An Introduction to Semialgebraic Geometry[END_REF][START_REF] Bochnak | Real Algebraic Geometry[END_REF] for self-contained references on semialgebraic geometry). We now describe Lasserre's method [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF][START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF], that will allow us to solve numerically the problem of the distance from an experimental tensor to a closed stratum. The method consists in constructing a sequence of semidefinite programs whose optimal values form a nondecreasing sequence which converges to the optimum 𝑓 * .

In this section, apart from theorem 4.1 which is a refinement of [42, Theorem 6.2], there is no original statement: we give the essential steps and results of the approach for pedagogical reasons and to be self-contained. For more details on Lasserre's method and the involved mathematical results and background, we refer to [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF][START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF][START_REF] Schweighofer | Optimization of polynomials on compact semialgebraic sets[END_REF].

The first step of the method is to notice that the optimization problem (2) can be reformulated as This statement is a reformulation of [42, theorem 2.44] and is due to Putinar ([64]) and Jacobi-Prestel ( [START_REF] Jacobi | Distinguished representations of strictly positive polynomials[END_REF]). We recall the proof below but, first, we have to define the essential Archimedean hypothesis. [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF]). Suppose that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 describing 𝐾 satisfy the Archimedean hypothesis, and let

𝑝 ∈ R[𝑋]. If 𝑝(𝐾) ⊂]0; +∞[, then 𝑝 ∈ M(𝑔 1 , . . . , 𝑔 𝑚 ).
See also [66, section 2] and [44, section 3.7] for alternative proofs.

Proof of theorem 3.1. Let 𝑦 ∈ R N 𝑛 . The direct implication is actually true even if the polynomials 𝑔 1 , . . . , 𝑔 𝑚 do not satisfy the Archimedean hypothesis. Indeed, suppose that 𝑦 has a representing measure 𝜇 on 𝐾. Now, take any vector p = (𝑝 𝛼 ) 𝛼∈N 𝑛 of R N 𝑛 with finitely many nonzero coordinates and set 𝑝 := ∑ 𝛼 𝑝 𝛼 𝑋 𝛼 . If 𝑀 is any matrix, denote by 𝑀 𝑇 its transpose. If 𝑔 is any polynomial of {𝑔 0 , . . . , 𝑔 𝑚 }, we have then

p 𝑇 𝑀 (𝑔 • 𝑦) p = ∑ 𝛼,𝛽 𝑝 𝛼 𝑝 𝛽 ( ∑ 𝛾 𝑔 𝛾 ∫ R 𝑛 𝑥 𝑥 𝑥 𝛼+𝛽+𝛾 d𝜇(𝑥 𝑥 𝑥) ) = ∫ R 𝑛 𝑔(𝑥 𝑥 𝑥)𝑝(𝑥 𝑥 𝑥) 2 d𝜇(𝑥 𝑥 𝑥) ≥ 0.
By definition, for all 𝑥 𝑥 𝑥 ∈ 𝐾, 𝑔(𝑥 𝑥 𝑥) ≥ 0, and the support of 𝜇 is included in 𝐾.

Conversely, suppose that the matrices 𝑀 (𝑦), 𝑀 (𝑔 1 • 𝑦), . . . , 𝑀 (𝑔 𝑚 • 𝑦) of 𝑦 are positive semidefinite, and denote by 𝐿 𝑦 the linear mapping

𝑝 = ∑ 𝛼 𝑝 𝛼 𝑋 𝛼 ∈ R[𝑋] ↦ → ∑ 𝛼 𝑝 𝛼 𝑦 𝛼 ∈ R.
Let 𝑔 ∈ {𝑔 0 , . . . , 𝑔 𝑚 } and consider the symmetric bilinear form

R[𝑋] × R[𝑋] → R (𝑝, 𝑞) ↦ → 𝐿 𝑦 (𝑝𝑞𝑔) ,
which is represented by the localizing matrix 𝑀 (𝑔 • 𝑦) in the canonical basis of R[𝑋]. In particular, for every polynomial 𝑝, if p denotes the vector (𝑝 𝛼 ) 𝛼∈N 𝑛 , we have

𝐿 𝑦 ( 𝑝 2 𝑔 ) = p 𝑇 𝑀 (𝑔 • 𝑦) p ≥
∈ R[𝑋] such that 𝑝(𝐾) ⊂]0; +∞[. If 𝑝 ∈ R[𝑋] satisfies 𝑝(𝐾) ⊂ [0; +∞[ then,
for any positive real number 𝜖, the polynomial 𝑝 + 𝜖 has positive values on 𝐾 so that 𝐿 𝑦 (𝑝) + 𝜖 = 𝐿 𝑦 (𝑝 + 𝜖) ≥ 0, and therefore 𝐿 𝑦 (𝑝) ≥ 0. We can then apply Haviland's theorem ( [START_REF] Haviland | On the momentum problem for distribution functions in more than one dimension[END_REF], see also [50, section 3.2] and Theorem 4.15 and section 4.6 of the up-to-date version of [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF]) to the mapping 𝐿 𝑦 : there exists a measure 𝜇 on R 𝑛 with support in 𝐾 such that

𝐿 𝑦 (𝑝) = ∫ R 𝑛 𝑝(𝑥 𝑥 𝑥)d𝜇(𝑥 𝑥 𝑥) for all 𝑝 ∈ R[𝑋].
In particular, for all 𝛼 ∈ N 𝑛 , we have

𝑦 𝛼 = 𝐿 𝑦 (𝑋 𝛼 ) = ∫ R 𝑛 𝑥 𝑥 𝑥 𝛼 d𝜇(𝑥 𝑥 𝑥)
i.e., 𝑦 is the moment sequence of the measure 𝜇.

From now on, we assume that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis so that we can write [START_REF] Arts | A study of general anisotropic elasticity in rocks by wave propagation[END_REF] 𝑓 * = inf 𝑦∈R N 𝑛 {⟨𝑓, 𝑦⟩; 𝑦 0 = 1, 𝑀 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 = 0, . . . , 𝑚} , where 𝑦 0 := 𝑦 (0,...,0) .

Lasserre's method to solve the optimization problem (4) consists, then, in relaxing this infinitedimensional problem into a sequence of finite-dimensional problems which are semidefinite programs. Semidefinite programs, or SDP's, are optimization problems over finite positive semidefinite symmetric matrices which generalize linear programs, and for which there exist efficient algorithms of numerical resolution. SDP-solving algorithms include methods inspired by the ones used in linear programming, such as interior point methods (see for instance the references [START_REF] Alizadeh | Interior point methods in semidefinite programming with applications to combinatorial optimization[END_REF][START_REF] Vandenberghe | Semidefinite programming[END_REF][START_REF] Sturm | Primal-dual interior point approach to semidefinite programming[END_REF][START_REF] Todd | Semidefinite optimization[END_REF][START_REF] Freund | Introduction to semidefinite programming (sdp)[END_REF][START_REF] Wolkowicz | Handbook of Semidefinite Programming[END_REF]).

Below, we follow Lasserre's notations in [42, section 6.1

.1]. First, if 𝑘 ∈ N, let Λ(𝑘) := {(𝛼 1 , . . . , 𝛼 𝑛 ) ∈ N 𝑛 ; 𝛼 1 + • • • + 𝛼 𝑛 ≤ 𝑘} and, if 𝑦 ∈ R Λ(2𝑘) and 𝑘 ′ ∈ N satisfies 𝑘 ′ ≤ 𝑘, set 𝑀 𝑘 ′ (𝑦) := (𝑦 𝛼+𝛽 ) 𝛼,𝛽∈Λ(𝑘 ′ ) . If 𝑔 ∈ R[𝑋], set 𝑔 • 𝑦 := ⎛ ⎝ ∑ 𝛽∈N 𝑛 𝑔 𝛽 𝑦 𝛼+𝛽 ⎞ ⎠ 𝛼∈Λ(𝑘) ∈ R Λ(𝑘) .
Finally, for 𝑖 ∈ {0, . . . , 𝑛}, denote

𝑣 𝑖 := ⌈ deg(𝑔 𝑖 ) 2

⌉

(notice that 𝑣 0 = 0) and let 𝑑 0 be the integer max (⌈

deg(𝑓 ) 2

⌉

, 𝑣 1 , . . . , 𝑣 𝑚 ) . For 𝑑 any integer such that 𝑑 ≥ 𝑑 0 , we then consider the optimization problem ( 6)

𝜌 𝑑 = inf 𝑦∈R Λ(2𝑑) {⟨𝑓, 𝑦⟩; 𝑦 0 = 1, 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 = 0, . . . , 𝑚} ,
relaxed from [START_REF] Arts | A study of general anisotropic elasticity in rocks by wave propagation[END_REF].

For a given 𝑑 ≥ 𝑑 0 , the optimization problem ( 6) is a semidefinite program (and can then be numerically solved using SDP solvers). Indeed, for all 𝑦 ∈ R Λ(2𝑑) such that 𝑦 0 = 1 and all 𝑖 ∈ {0, . . . , 𝑚}, we can write

𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) = 𝐴 0 𝑖 + ∑ 𝛼∈Λ(2𝑑)∖{0} 𝑦 𝛼 𝐴 𝛼 𝑖
where, for all 𝛼 ∈ Λ(2𝑑), 𝐴 𝛼 𝑖 is a symmetric square matrix of size Λ(𝑑 -𝑣 𝑖 ) (see also [66, section 5]).

The following theorem of Lasserre ([42, Theorem 6.2], see also [START_REF] Schweighofer | Optimization of polynomials on compact semialgebraic sets[END_REF]Theorem 1.5]) asserts that the sequence of optima (𝜌 𝑑 ) 𝑑≥𝑑 0 converges to 𝑓 * : Theorem 3.4 (Lasserre). The sequence (𝜌 𝑑 ) 𝑑≥𝑑 0 is a nondecreasing sequence that converges to 𝑓 * .

Proof. Let 𝑑 be an integer such that 𝑑 ≥ 𝑑 0 and denote by 𝐹 𝑑 the set of vectors

𝑦 ∈ R Λ(2𝑑) such that 𝑦 0 = 1 and 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0 for all 𝑖 ∈ {0, . . . , 𝑚}. The set {⟨𝑓, 𝑦⟩; 𝑦 ∈ 𝐹 𝑑+1 } is included in the set {⟨𝑓, 𝑦⟩; 𝑦 ∈ 𝐹 𝑑 }. Indeed, if 𝑦 ∈ 𝐹 𝑑+1 and if we denote by 𝑦 the truncation (𝑦 𝛼 ) 𝛼∈Λ(2𝑑) of 𝑦, we have 𝑦 0 = 𝑦 0 = 1 and, for 𝑖 ∈ {0, . . . , 𝑚}, 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0 (because 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) is a principal submatrix of the positive semidefinite matrix 𝑀 𝑑+1-𝑣 𝑖 (𝑔 𝑖 • 𝑦)), as well as ⟨𝑓, 𝑦⟩ = ⟨𝑓, 𝑦⟩ since deg 𝑓 ≤ 2𝑑. As a consequence, 𝜌 𝑑 ≤ 𝜌 𝑑+1 .
We then show that the nondecreasing sequence (𝜌 𝑑 ) 𝑑≥𝑑 0 is bounded by 𝑓 * . Consider the formulation (5) of our optimization problem and denote by 𝐹 the set of sequences 𝑦 ∈ R N 𝑛 such that 𝑦 0 = 1 and 𝑀 (𝑔 𝑖 • 𝑦) ⪰ 0 for all 𝑖 ∈ {0, . . . , 𝑚}. Let 𝑦 be in 𝐹 and let 𝑦 := (𝑦 𝛼 ) 𝛼∈Λ(2𝑑) be the truncation of 𝑦. Again, we have 𝑦 0 = 1, 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 ∈ {0, . . . , 𝑚}, and ⟨𝑓, 𝑦⟩ = ⟨𝑓, 𝑦⟩, so that 𝜌 𝑑 ≤ 𝑓 * . Therefore, the sequence (𝜌 𝑑 ) 𝑑≥𝑑 0 converges.

The last step is to show that 𝑓 * is actually the limit of (𝜌 𝑑 ) 𝑑≥𝑑 0 . If 𝜖 is a positive real number, one can show that there exists 𝑑 ≥ 𝑑 0 such that 𝑓 * -𝜖 ≤ 𝜌 𝑑 ≤ 𝑓 * : the interested reader is invited to refer to [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF]Theorem 6.2] or [START_REF] Schweighofer | Optimization of polynomials on compact semialgebraic sets[END_REF]Theorem 1.5]. The proof involves the dual SDP associated to [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF], together with Putinar's Positivstellensatz 3.3.

Lasserre's algorithm -GloptiPoly

The principle of Lasserre's algorithm to solve problem (2) is to numerically compute the sequence of optima (𝜌 𝑑 ) 𝑑≥𝑑 0 (which by theorem 3.4 converges to 𝑓 * ) using SDP solvers at each step. In order to complete this approach, one has to define a stopping criterion for the algorithm. In [42, section 6.1], Lasserre chooses a sufficient condition in terms of ranks of moment matrices, a condition which is motivated by the theorem below. The result we show is actually a slight generalization of [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF]Theorem 6.6], that we decided to state in order to take into account the fact that a SDP solver, when applied to the SDP [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF], only provides, at best, a numerical approximation of the optimum 𝜌 𝑑 . Let 𝜖 be a nonnegative real number, 𝑑 be an integer such that 𝑑 ≥ 𝑑 0 and denote 𝑣 := max(𝑣 1 , . . . , 𝑣 𝑚 ).

Theorem 4.1. Let 𝑦 ∈ 𝐹 𝑑 (we defined 𝐹 𝑑 in the proof of theorem 3.4) such that 𝜌 𝑑 ≤ ⟨𝑓, 𝑦⟩ ≤ 𝜌 𝑑 + 𝜖. If rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦) then 𝑓 * ≤ ⟨𝑓, 𝑦⟩ ≤ 𝜌 𝑑 + 𝜖 ≤ 𝑓 * + 𝜖.
Moreover, if we denote 𝑠 := rank 𝑀 𝑑 (𝑦), there exist at least 𝑠 points 𝑥 𝑥 𝑥 of 𝐾 such that 𝑓 * ≤ 𝑓 (𝑥 𝑥 𝑥) ≤ 𝑓 * + 𝜖.

In other words, if an optimal solution 𝑦, up to a fixed precision 𝜖, of the SDP (6) satisfies the above rank condition on its moment matrix, then ⟨𝑓, 𝑦⟩ is an approximation of 𝑓 * up to precision 𝜖. Furthermore, there exist at least rank 𝑀 𝑑 (𝑦) points of 𝐾 which are global minimizers of 𝑓 up to precision 𝜖. Remark 4.2.

(1) For 𝜖 = 0, we recover Theorem 6.6 in [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF].

(2) The SDP solver used in the algorithm of Lasserre implemented in the freeware Glop-tiPoly 3 computes an element 𝑦 of 𝐹 𝑑 which is an approximation of an optimal solution of ( 6) and such that the rank 𝑟 of 𝑀 𝑑 (𝑦) is maximal among the ranks of moment matrices of elements of 𝐹 𝑑 . GloptiPoly then checks if the numerical rank of the principal submatrix 𝑀 𝑑-𝑣 (𝑦) of 𝑀 𝑑 (𝑦) is equal to 𝑟. The numerical rank of a matrix 𝑀 is, roughly speaking, the number of singular values of 𝑀 which are greater than a fixed precision, and the numerical rank of 𝑀 is not greater than its rank. As a consequence, if the numerical rank of 𝑀 𝑑-𝑣 (𝑦) is (at least) 𝑟, we have the inequalities 𝑟 ≤ rank 𝑀 𝑑-𝑣 (𝑦) ≤ rank 𝑀 𝑑 (𝑦) = 𝑟 so that rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦) and the stopping criterion of theorem 4.1 applies. More details about these questions can be found in [34,

𝑦 0 = 1, we have 1 = 𝑦 0 = ∫ R 𝑛 𝜇(𝑥 𝑥 𝑥) = ∑ 𝑠 𝑖=1 𝜆 𝑖 . Then ⟨𝑓, 𝑦⟩ = ∑ 𝛼∈Λ(2𝑑) 𝑓 𝛼 𝑦 𝛼 = ∫ R 𝑛 ∑ 𝛼∈Λ(2𝑑) 𝑓 𝛼 𝑥 𝑥 𝑥 𝛼 d𝜇(𝑥 𝑥 𝑥) = 𝑠 ∑ 𝑖=1 𝜆 𝑖 𝑓 (𝑥 𝑥 𝑥 𝑖 ) ≥ 𝑠 ∑ 𝑖=1 𝜆 𝑖 𝑓 * = 𝑓 * , so that 𝑓 * + 𝜖 ≥ 𝜌 𝑑 + 𝜖 ≥ ⟨𝑓, 𝑦⟩ ≥ 𝑓 * .
Finally suppose that there exists 𝑖 ∈ {1, . . . , 𝑠} such that 𝑓 (𝑥 𝑥 𝑥 𝑖 ) > 𝑓 * + 𝜖. This implies that

𝑠 ∑ 𝑗=1 𝜆 𝑗 𝑓 (𝑥 𝑥 𝑥 𝑗 ) > ⎛ ⎝ 𝑠 ∑ 𝑗=1 𝜆 𝑗 𝑓 * ⎞ ⎠ + 𝜖 = 𝑓 * + 𝜖,
which is not true according to the above inequalities. As a consequence, for all 𝑖 ∈ {1, . . . , 𝑠}, 𝑓 * ≤ 𝑓 (𝑥 𝑥 𝑥 𝑖 ) ≤ 𝑓 * + 𝜖.

Proof of theorem 4.3. We point out the essential steps of the reasoning, referring to [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF] for the detailed proofs. We have

rank 𝑀 𝑑-𝑣 (𝑦) ≤ rank 𝑀 𝑑-𝑣+1 (𝑦) ≤ • • • ≤ rank 𝑀 𝑑-1 (𝑦) ≤ rank 𝑀 𝑑 (𝑦)
and suppose that rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦): we obtain that rank 𝑀 𝑑-1 (𝑦) = rank 𝑀 𝑑 (𝑦). We can then recursively apply the Flat Extension Theorem 5.14 of [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF] (originally due to Curto and Fialkow in [START_REF] Curto | Solution of the truncated complex moment problem for flat data[END_REF]) to assert the existence of a sequence ̃ 𝑦 of R N 𝑛 such that, for all 𝛼 ∈ Λ(2𝑑),

 𝑦 𝛼 = 𝑦 𝛼 and, for all 𝑘 ∈ N satisfying 𝑘 ≥ 𝑑 -𝑣, rank 𝑀 𝑘 (̃ 𝑦) = rank 𝑀 𝑑 (𝑦) = 𝑠.
In particular, for all 𝑘 ≥ 𝑑, since the principal submatrix 𝑀 𝑑 (𝑦) = 𝑀 𝑑 (̃ 𝑦) of 𝑀 𝑘 (̃ 𝑦) is positive semidefinite (because 𝑦 is in 𝐹 𝑑 ) and rank 𝑀 𝑘 (̃ 𝑦) = rank 𝑀 𝑑 (̃ 𝑦), the symmetric matrix 𝑀 𝑘 (̃ 𝑦) is also positive semidefinite (see [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF]Definition 1.1]). In other words, the (infinite) moment matrix 𝑀 (̃ 𝑦) is positive semidefinite. Since, furthermore, rank 𝑀 (̃ 𝑦) = 𝑠, by [44, Theorem 5.1 (i)], there is a 𝑠-atomic measure 𝜇 representing ̃ 𝑦, and then 𝑦, with support the finite real algebraic set 𝑉 (𝐼) := {𝑥 𝑥 𝑥 ∈ R 𝑛 ; for all 𝑝 ∈ 𝐼, 𝑝(𝑥 𝑥 𝑥) = 0} where 𝐼 := {𝑝 ∈ R[𝑋]; 𝑀 𝑝 = 0} (the proof of Theorem 5.1 (i) of [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF] involves real algebraic geometry).

The last step is then to prove that this support is included in 𝐾. Write 𝑉 (𝐼) = {𝑥 𝑥 𝑥 We finally present the algorithm implemented by Lasserre and Henrion in the Matlab freeware GloptiPoly 3 to numerically solve polynomial optimization problems. For details on GloptiPoly and its use, see [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF] and Appendix B of [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF].

In order to solve a SDP relaxation 𝜌 𝑑 , 𝑑 ≥ 𝑑 0 , GloptiPoly 3 uses by default the SDP solver SeDuMi of Sturm [START_REF] Sturm | Using SeDuMi 1.02, a matlab toolbox for optimization over symmetric cones[END_REF]. Other SDP solvers can also be used as long as they are interfaced through Yalmip [START_REF] Lofberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF] (see section 5.9 of [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF]).

The inputs of GloptiPoly are

• the variables 𝑋 1 , . . . , 𝑋 𝑛 , • the polynomial 𝑓 ∈ R[𝑋 1 , . . . , 𝑋 𝑛 ],
• the polynomials 𝑔 𝑗 ∈ R[𝑋 1 , . . . , 𝑋 𝑛 ], 𝑗 ∈ {1, . . . , 𝑚}, satisfying the Archimedean hypothesis, • a maximal relaxation order 𝑑 max ≥ 𝑑 0 . The first output is a status number 𝜉 ∈ {-1, 0, 1}:

• 𝜉 = -1 means that the consider SDP solver could not solve numerically any of the relaxations 𝜌 𝑑 , 𝑑 ∈ {𝑑 0 , . . . , 𝑑 max }; • 𝜉 = 0 means that the solver numerically solved (that is up to a prescribed precision 𝜖) at least one of the relaxations 𝜌 𝑑 , 𝑑 ∈ {𝑑 0 , . . . , 𝑑 max }, but at each such success either no optimal solution was provided by the solver, either the rank stopping criterion of Theorem 4.1 was not satisfied by the (approximated up to precision 𝜖) obtained optimal solution 𝑦 𝑑 . In that case, the algorithm also outputs the last computed (and then greatest) optimal value 𝜌 𝑑 which is (up to precision 𝜖) a lower bound for 𝑓 * ; • 𝜉 = 1 means that the rank stopping criterion of Theorem 4.1 has been satisfied by an optimal solution 𝑦 𝑑 of a solved relaxation 𝜌 𝑑 , 𝑑 ∈ {𝑑 0 , . . . , 𝑑 max }. In that case, the algorithm also outputs 𝜌 𝑑 which is then an approximation of 𝑓 * up to the prescribed precision 𝜖. Lasserre's algorithm for polynomial optimization is, in pseudo code, the following (see [ (1) When the rank condition is satisfied, we can also ask GloptiPoly to extract minimizers up to precision 𝜖 (in the sense of Theorem 4.1), which involves the algorithm described in [42, section 6.1.2].

(2) If the output 𝜉 is 0 or -1, one can increase 𝑑 max to try to obtain an approximation (or a better lower bound) of 𝑓 * at a higher relaxation order. (3) There is no complexity known for Lasserre's method. Actually, we do not know if there is a maximal relaxation degree 𝑑 max , dependent on the inputs of the problem, which would ensure the rank stopping criterion to be satisfied at some ordre 𝑑 ≤ 𝑑 max . However, what makes this method advantageous is that it benefits from the interesting complexity of SDP solvers to solve semidefinite programs (see for instance [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF] A.1.2).

We conclude this part by the following remark: the convergence of Lasserre's polynomial optimization method, described in the previous sections, takes place when the constraint set is a semialgebraic compact set satisfying the Archimedean property 3.2. Nevertheless, when the Archimedean condition is not satisfied but the polynomial function 𝑓 is coercive, Jeyakumar-Lasserre-Li in [START_REF] Jeyakumar | On polynomial optimization over non-compact semi-algebraic sets[END_REF] provide a way to consider the optimization problem (2) as a problem with constraints satisfying the Archimedean condition : Lemma 4.5 (Jeyakumar-Lasserre-Li). Suppose that the polynomial function 𝑓 : R 𝑛 → R associated to 𝑓 ∈ R[𝑋 1 , . . . , 𝑋 𝑛 ] is coercive, and let 𝑐 > 0 and 𝑦 𝑦 𝑦 ∈ 𝐾 such that 𝑐 > 𝑓 (𝑦 𝑦 𝑦). Then the quadratic module M(𝑔 1 , . . . , 𝑔 𝑚 , 𝑐 -𝑓 ) associated to the semialgebraic set In other words, even if the polynomials 𝑔 1 , . . . , 𝑔 𝑚 do not satisfy themselves the Archimedean hypothesis, provided that 𝑓 is coercive, we can place ourselves in the range of application of Lasserre's method by adding the inequality 𝑐 -𝑓 ≥ 0 to the constraints 𝑔 1 ≥ 0, . . . , 𝑔 𝑚 ≥ 0. Remark 4.6. In [42, theorem 6.5], Lasserre states some classical conditions (known as the Karush-Kuhn-Tucker (KKT) conditions [42, section 7.1]), already encountered in nonlinear programming, to ensure the finite convergence of the hierarchy of the semidefinite relaxations [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF]. These conditions are initially a certificate for global optimality [42, theorem 7.4 and 7.5] and hold generically for a polynomial optimization problem [42, theorem 7.6].

 𝐾 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0, 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0} is Archimedean (in particular, ̃ 𝐾 is compact). Furthermore, 𝑓 * = inf 𝑥 𝑥 𝑥∈𝐾 𝑓 (𝑥 𝑥 𝑥) = inf

Distance to the transversely isotropic stratum of the symmetric second-order tensor

Let V = S 2 (R 3 ) be the vector space of symmetric second-order tensors, endowed with the natural action 𝜌 2 (𝑔)a = 𝑔 a 𝑔 𝑇 , for a ∈ S 2 (R 3 ), 𝑔 ∈ 𝐺 = SO(3). Let q = (𝛿 𝑖𝑗 ) be the Euclidean metric,

a ′ = a - 1 3 tr(a) q,
be the traceless part of a, 𝜀 𝜀 𝜀 be the Levi-Civita tensor. We denote by ( ) 𝑠 the total symmetrization of a tensor. The generalized cross-product of symmetric tensors is defined as [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF] (7)

a × b := -(a • 𝜀 𝜀 𝜀 • b) 𝑠 , (𝑖.𝑒., (a × b) 𝑖𝑗𝑘 := -(𝑎 𝑖𝑙 𝜀 𝑙𝑗𝑠 𝑏 𝑠𝑘 ) 𝑠 ) .
For a, b ∈ S 2 (R 3 ), it is a totally symmetric third-order tensor with 10 independent components. There are three isotropy classes for the symmetric second-order tensors (S 2 (R 3 ), SO(3)): 3)] (isotropy, characterized by the linear equation a ′ = 0), if a has three equal eigenvalues. We illustrate through this first example the accuracy of Lasserre's polynomial optimization method to compute the distance to an isotropy stratum. We shall obtain by this way the distance ∆(a 0 , Σ [O(2)] ) of the orthotropic second-order tensor

• [D 2 ] (
a 0 = ⎛ ⎝ -7 4 -4 4
5 -2 -4 -2 5 ⎞ ⎠ and compare the numerical results obtained with the algebraic solution derived in [START_REF] Antonelli | Distance to plane elasticity orthotropy by Euler-Lagrange method[END_REF]: where 𝐾 = { a; a 2 × a = 0 } . Then, in order to properly apply the algorithm described in section 4, we need to ensure the Archimedean property (definition 3.2), and for that we use lemma 4.5. Therefore, to the 10 equations a 2 × a = 0, we add the inequality 𝑐 -‖a 0 -b‖ 2 ≥ 0 where we take

(8) ∆(a 0 , Σ [O(2)] ) 2 = ‖a 0 -a ** ‖ 2 = 18, where (9)
b = ⎛ ⎝ 1 0 0 0 1 0 0 0 -2 ⎞ ⎠ ∈ 𝐾,
and choose accordingly 𝑐 = 300. GloptiPoly then computes the approximation [START_REF] Bredon | Finiteness of number of orbit types[END_REF] ‖a

0 -a * ‖ 2 of the minimum ∆(a 0 , Σ [O(2)] ) 2 = min a∈ ̃ 𝐾 ‖a 0 -a‖ 2 , where (11) ̃ 𝐾 = { a; a 2 × a = 0, 𝑐 -‖a 0 -a‖ 2 ≥ 0 } .
The optimal result computed in 1. is close to the exact solution ( 8)-( 9), with the constraints accurately satisfied:

max 𝑖 |𝑔 𝑖 (a * )| ‖a 0 ‖ 3 = max 𝑝,𝑞,𝑟 |(a * 2 × a * ) 𝑝𝑞𝑟 | ‖a 0 ‖ 3 = 5.696 10 -9 .
For different values 𝑐 ∈ [202, 450], one gets 18.000007 ≤ ∆(a 0 , Σ [O(2)] ) 2 ≤ 18.00006, with the GloptiPoly convergence obtained for the first degree of relaxation 𝑑 = 𝑑 0 = 2. The value chosen for 𝑐 affects the numerical solution. In fact, by increasing 𝑐 we get closer to the true minimum (=18), but the convergence is lost for 𝑐 ≥ 500 (with a GloptiPoly status 𝜉 = 0 at the first relaxation).

Remark 5.1. The transversely isotropic closed stratum Σ [O( 2)] can also be characterized by a single scalar equation of degree 6, ( 12)

𝑔(a) = 12‖a 2 × a‖ 2 = ( tr(a ′ 2 ) ) 3 -6 ( tr(a ′3 ) ) 2 = 0,
with a ′ , the traceless part of a. However, there is no finite convergence of the associated relaxation problem, since grad a 𝑔(a) = 6𝑔(a) tr(a ′ 2 ) a ′ = 0 when 𝑔(a) = 0. In particular, the independence of the gradients of the constraint functions at the minimum (first order KKT sufficient condition mentioned in [42, theorem 6.5, theorem 7.2], see remark 4.6) is not satisfied.

The present example illustrates the strong dependence of the GloptiPoly convergence issue on the characterization of the isotropy classes. Indeed, convergence is obtained for the covariant characterization a 2 × a = 0, but not for the invariant characterization [START_REF] Bucataru | Invariant properties for finding distance in space of elasticity tensors[END_REF].

Distance to cubic elasticity isotropy stratum

In this section, we compute the distance of an experimental elasticity tensor E 0 to the cubic isotropy stratum Σ [O] , and determine the associated minimizer E * . The distance to an isotropy stratum problem has been widely addressed in the Continuum Mechanics literature, by solving it in terms of an unknown rotation (either parameterized by Euler angles [START_REF] François | Une nouvelle analyse des symétries d'un matériau élastique anisotrope. exemple d'utilisation à partir de mesures ultrasonores[END_REF][START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF], or by a unit quaternion [START_REF] Dellinger | Computing the optimal transversely isotropic approximation of a general elastic tensor[END_REF][START_REF] Kochetov | On obtaining effective transversely isotropic elasticity tensors[END_REF][START_REF] Kochetov | On obtaining effective orthotropic elasticity tensors[END_REF]). Here, we use the characterization of the (cubic) isotropy stratum by means of at most quadratic covariants in order to formulate such a distance problem as a quadratic polynomial optimization problem. This makes us able to apply Lasserre's method, and to show that using GloptiPoly allows to compute an accurate solution of this non trivial example.

6.1. Formulation of the distance problem as a polynomial optimization problem. Let

V = Ela = { E ∈ ⊗ 4 R 3 , 𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑘𝑙𝑖𝑗 } (dim Ela = 21)
be the set of elasticity tensors E : S 2 (R 3 ) → S 2 (R 3 ), introduced in example 2.

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
The vector space Ela decomposes into a direct sum of SO(3)-irreducible subspaces (so-called harmonic decomposition [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Spencer | A note on the decomposition of tensors into traceless symmetric tensors[END_REF])

Ela = H 0 (R 3 ) ⊕ H 0 (R 3 ) ⊕ H 2 (R 3 ) ⊕ H 2 (R 3 ) ⊕ H 4 (R 3 )
, where H 𝑛 (R 3 ) denotes the space of harmonic tensors of order 𝑛 (dim H 𝑛 (R 3 ) = 2𝑛 + 1). Letting

d = tr 12 E, v = tr 13 E,
the harmonic decomposition of E ∈ Ela can be expressed as (see Appendix A for explicit formulas)

E = (𝛼, 𝛽, d ′ , v ′ , H), with 𝛼 = tr d, 𝛽 = tr v ∈ H 0 (R 3 ) the scalar (isotropic) components of E, with d ′ , v ′ ∈ H 2 (R 3 )
its second-order harmonic components (the traceless parts of d and v), and H ∈ H 4 (R 3 ) its fourth-order harmonic component. The squared Euclidean norm of E is then ( 14)

‖E‖ 2 = 5𝛼 2 + 4𝛽 2 + 2 21 ‖d ′ + 2v ′ ‖ 2 + 4 3 ‖d ′ -v ′ ‖ 2 + ‖H‖ 2 .
We consider the triclinic experimental elasticity tensor E 0 representing the Nickel-based aeronautics single crystal superalloy (of CMSX-4 type), measured in [START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF]. In Voigt notation: [START_REF] Curto | Solution of the truncated complex moment problem for flat data[END_REF] [

E 0 ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 243
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ GPa.
This material has an expected symmetry, namely the cubic symmetry [O], deduced from its cubic microstructure (see figure 1). We then aim a computing of ( 16)

∆(E 0 , Σ [O] ) 2 = min E∈Σ O ‖E 0 -E‖ 2 .
This optimization problem has 21 variables 𝐸 𝑖𝑗𝑘𝑙 . To set it as a polynomial optimization problem, we take advantage of the fact that the cubic elasticity stratum is an algebraic set, characterized by explicit polynomial equations.

Theorem 6.1 (Olive et al [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]). Let E = (𝛼, 𝛽, d ′ , v ′ , H) ∈ Ela be an elasticity tensor,

d 2 = H . . . H (i.e., (d 2 ) 𝑖𝑗 = 𝐻 𝑖𝑝𝑞𝑟 𝐻 𝑝𝑞𝑟𝑗 ) and d ′ 2 = d 2 -1 3 tr(d 2 ) q be second-order covariants of E. Then E ∈ Σ [O] (is at least cubic) if and only if v ′ = d ′ = 0 and d ′ 2 = 0, and E ∈ Σ [O] (is cubic) if and only if furthermore H ̸ = 0.
We have then the following result.

Theorem 6.2. Let E = (𝛼, 𝛽, d ′ , v ′ , H) and E 0 = (𝛼 0 , 𝛽 0 , d ′ 0 , v ′ 0 , H 0 )
be two elasticity tensors. The 21-dimensional minimization problem (16) is equivalent to the 9-dimensional polynomial optimization problem min

d ′ 2 =0
‖H 0 -H‖ 2 , with E = (𝛼 0 , 𝛽 0 , 0, 0, H), and

O ‖E 0 -E‖ 2 = 2 21 ‖d ′ 0 + 2v ′ 0 ‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 ‖ 2 + min d ′ 2 =0 ‖H 0 -H‖ 2 . (17) min E∈Σ
Proof. The squared distance function in ( 16) is

‖E 0 -E‖ 2 = ‖(𝛼 0 -𝛼, 𝛽 0 -𝛽, d ′ 0 -d ′ , v ′ 0 -v ′ , H 0 -H)‖ 2 .
It can be expressed as

‖E 0 -E‖ 2 = 5(𝛼 0 -𝛼) 2 +4(𝛽 0 -𝛽) 2 + 2 21 ‖d ′ 0 +2v ′ 0 -(d ′ +2v ′ )‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 -(d ′ -v ′ )‖ 2 +‖H 0 -H‖ 2 ,
by using [START_REF] Coste | An Introduction to Semialgebraic Geometry[END_REF]. By theorem 6.1, taking 𝛼 = 𝛼 0 , 𝛽 = 𝛽 0 , d ′ = 0 and v ′ = 0, we get [START_REF] Dantzig | Linear programming and extensions[END_REF], and the polynomial optimization problem ( 16) is reduced to the following problem in only 9 variables (the components of

H ∈ H 4 (R 3 ), dim H 4 (R 3 ) = 9) instead of 21, min d ′ 2 =0
‖H 0 -H‖ 2 .

Remark 6.

3. An elasticity tensor E ∈ Ela corresponds to a quadratic elastic energy density, which must be positive semidefinite. This condition can be characterized using SO(3)-invariant polynomial inequalities on E, and thus added to the set of polynomial constraints, if necessary, using the following fact. Given a symmetric real 𝑛 × 𝑛 matrix 𝐴, we get

𝐴 is positive semidefinite ⇐⇒ ∀𝑖 ∈ {1, . . . , 𝑛} , 𝜎 𝑖 ≥ 0,
where 𝜎 1 , . . . , 𝜎 𝑛 denote the elementary symmetric polynomials in the eigenvalues 𝜆 𝑖 of 𝐴. Indeed, if 𝜆 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}, then 𝜎 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}. Conversely, assume that 𝜎 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}. Then, the polynomial

𝑝 := (𝑋 + 𝜆 1 )(𝑋 + 𝜆 2 ) . . . (𝑋 + 𝜆 𝑛 ) = 𝑋 𝑛 + 𝜎 1 𝑋 𝑛-1 + . . . + 𝜎 𝑛-1 𝑋 + 𝜎 𝑛 , satisfies 𝑝(𝑥) ≥ 𝑥 𝑛 > 0, ∀𝑥 > 0.
Hence, the (real) roots of 𝑝, namely -𝜆 1 , . . . , -𝜆 𝑛 , belong to ] -∞, 0], and thus 𝜆 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}.

In practice, an experimental tensor E 0 is most often measured as semidefinite and the tensor the closest to E 0 computed as semidefinite, so, here we do not add the semidefiniteness constraint E ≥ 0 to our optimization problem.

6.2.

Resolution by Lasserre's method. A fourth order harmonic tensor H ∈ H 4 (R 3 ) is represented by the following real matrix (in Voigt notation) [START_REF] Dellinger | Computing the optimal transversely isotropic approximation of a general elastic tensor[END_REF] [H] =

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
In practice we set

𝑥 𝑥 𝑥 = (Λ 1 , Λ 2 , Λ 3 , 𝑋 1 , 𝑋 2 , 𝑌 1 , 𝑌 2 , 𝑍 1 , 𝑍 2 )
, and GloptiPoly computes the approximation

(19) ‖H 0 -H * ‖ 2 of the minimum ∆(H 0 , Σ [O] ) 2 = min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥), where 𝑓 (𝑥 𝑥 𝑥) = ‖H 0 -H‖ 2 and  𝐾 = { 𝑥 𝑥 𝑥; d ′ 2 = 0, 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0 } ,
with 𝑐 = 58000 > 𝑓 (0) to ensure the Archimedean property on the set of constraints. The five quadratic scalar equations (d ′ 2 ) 𝑖𝑗 = 0 are detailed in Appendix B. For E 0 given by ( 15), we have

𝑓 (𝑥 𝑥 𝑥) = 540Λ 2 + 620Λ 3 -88𝑋 1 + 668Λ 1 -264𝑍 1 -264𝑍 2 -456𝑋 2 + 8𝑍 2 2 + 8Λ 2 1 + 8Λ 2 2 + 8Λ 2 3 + 8𝑌 2 2 + 16𝑋 2 1 + 8𝑋 2 2 + 16𝑌 2 1 + 16𝑍 2 1 -392𝑌 1 -808𝑌 2 + 8𝑍 1 𝑍 2 + 2Λ 1 Λ 2 + 2Λ 2 Λ 3 + 8𝑋 1 𝑋 2 + 8𝑌 1 𝑌 2 + 2Λ 3 Λ 1 + 2026042 35 .
We obtain the result at the first relaxation order 𝑑 = 𝑑 0 = 1 with GloptiPoly status 𝜉 = +1 and value min

𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥) ≈ 𝑓 (𝑥 𝑥 𝑥 * ) = 2530.474727 GPa 2 ,
The computation time is of 0.9 seconds. The computed minimizer is By [START_REF] Dellinger | Computing the optimal transversely isotropic approximation of a general elastic tensor[END_REF], it corresponds to the fourth-order harmonic tensor H * solution of [START_REF] Hertog | Interior point approach to linear, quadratic and convex programming: algorithms and complexity[END_REF]. We get, by theorem 6.2, E * = (𝛼 0 , 𝛽 0 , 0, 0, H * ), i.e.,

𝑥 𝑥 𝑥 * = (-
E * = 1 15 (𝛼 0 + 2𝛽 0 ) q ⊙ q + 1 6 (𝛼 0 -𝛽 0 ) q ⊗ (2,2)
q + H * , with ⊙ the symmetric tensor product (see Appendix A). The elasticity tensor E * is cubic (and not isotropic) since H * ̸ = 0. Finally, the computed cubic tensor

E * ∈ Σ [O] the closest to E 0 is, in Voigt notation, [E * ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 240
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ GPa. It corresponds to ∆(E 0 , Σ [O] ) ≈ ‖E 0 -E * ‖ = 74.
131148 GPa and to the relative distance to cubic symmetry ‖E 0 -E * ‖ ‖E 0 ‖ = 0.103910, slightly better than the solution computed in [START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF] using a parameterization by Euler angles together with a simplex minimization method. Note that the constraint d ′ 2 = 0 is satisfied accurately, since

d ′ 2 ‖H 0 ‖ 2 = 10 -6 ⎛ ⎝ -4.097
-2.8 10 -6 -4.9 10 -6 -2.8 10 -6 -4.455 4.2 10 -6 -4.9 10 -6 4.2 10 -6 -8.552

⎞ ⎠ ≈ 0.
One can choose other values for 𝑐 satisfying 𝑐 > 𝑓 (𝑦 𝑦 𝑦) for some 𝑦 𝑦 𝑦 ∈ 𝐾. The GloptiPoly solution varies slightly as 𝑐 runs the interval [58000, 61000], with a computation time of 0.9 seconds for 𝑐 = 58000, of 0.8 seconds for 𝑐 = 60000, and of 0.1 seconds for 𝑐 = 61000. Outside from this narrow interval, the GloptiPoly convergence is lost (Gloptipoly status 𝜉 = 0). Remark 6.4. The computation time is lower for this quadratic optimization problem (with 9 variables) than for the degree 3 polynomial optimization problem of section 5 (with 6 variables).

Distance to cubic piezoelectricity isotropy stratum

In this final section, we apply Lasserre's polynomial optimization method to compute the distance ∆(e 0 , Σ [O -] ) of a raw piezoelectricity third-order tensor 1 e 0 to the cubic piezoelectricity stratum Σ [O -] . This problem seems to have never been addressed before. It is important for the design of dielectric materials, since for instance the piezolectricity behavior strongly depends on the crystal primitive cell symmetry. 7.1. Formulation of the distance problem as a polynomial optimization problem. According to the three-dimensional piezoelectricity framework [START_REF] Eringen | Electrodynamics of Continua , tomes I et II[END_REF][START_REF] Geymonat | Symmetry classes of piezoelectric solids[END_REF], we denote by The vector space Piez decomposes into a direct sum of O(3)-irreducible subspaces (so-called harmonic decomposition [START_REF] Spencer | A note on the decomposition of tensors into traceless symmetric tensors[END_REF])

V = Piez = { e ∈ ⊗ 3 R 3 ,
Piez = H 1 (R 3 ) ⊕ H 1 (R 3 ) ⊕ H 2♯ (R 3 ) ⊕ H 3 (R 3
). The notation H 𝑛 (R 3 ) still refers to the vector space of 𝑛-th order harmonic tensors endowed with the standard O(3)-representation 𝜌 𝑛 , while H 𝑛♯ (R 3 ) refers to the same vector space endowed with the twisted O(3)-representation ρ𝑛 , such that ρ𝑛 (𝑔) = (det 𝑔) 𝜌 𝑛 (𝑔). One has e = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a, h)

with 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 ∈ H 1 (R 3 ), a ∈ H 2♯ (R 3 ) and h ∈ H 3 (R 3 ).
Let ⊙ be the symmetric tensor product and e 𝑠 ∈ S 3 (R 3 ) denote the totally symmetric part of e (of components (e 𝑠 ) 𝑖𝑗𝑘 = 1 3 (e 𝑖𝑗𝑘 + e 𝑗𝑖𝑘 + e 𝑘𝑗𝑖 )). Any piezoelectricity tensor e ∈ Piez can be decomposed as the sum e = g + h where [START_REF] Dixmier | Le nombre minimum d'invariants fondamentaux pour les formes binaires de degré 7[END_REF] h := e 𝑠 -3 5 q ⊙ tr(e 𝑠 ) ∈ H 3 (R 3 ), is the leading harmonic part of e, and g := eh = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a), is orthogonal to h (i.e., ⟨g, h⟩ = 𝑔 𝑖𝑗𝑘 ℎ 𝑖𝑗𝑘 = 0).

Remark 7.1. The third-order tensors g = g(e) and h = h(e) are linear covariants of e.

The squared Euclidean norm of e is then (21) ‖e‖ 2 = e 𝑖𝑗𝑘 e 𝑖𝑗𝑘 = ‖g‖ 2 + ‖h‖ 2 .

We will first consider the following raw (triclinic) piezoelectricity tensor e 0 for pure wurtzite AlN (aluminum nitride, 𝑥 = 0), of Voigt representation, [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF] [e 0 ] = ⎛ ⎝ 0 0 -0.0505 -0.0394 -0.2854 -0.0637 -0.0637 -0.0042 0.0332 -0.2818 -0.0058 0.0185 -0.5807 -0.5822 1.4607 0.0022 0.0002 0.0043

⎞ ⎠ C/m 2 ,
in Coulomb per square meter, computed by Density Functional Theory (DFT), using abinitio simulations, by Manna and coworkers [49, Fig. 3]. We will also consider wurtzite alloys Cr 𝑥 Al 1-𝑥 N and the associated raw piezoelectricity tensors e 𝑥 0 (given in the Appendix D for chromium concentrations 0 ≤ 𝑥 ≤ 0.25). Note that pure rocksalt CrN corresponds to a Crconcentration 𝑥 = 1, and that the value 𝑥 = 0.25 is the so-called wurzite to rocksalt phase transition point [START_REF] Mayrhofer | Structure, elastic properties and phase stability of Cr1-𝑥Al𝑥n[END_REF].

We aim at computing by polynomial optimization [START_REF] Forte | Symmetry classes and harmonic decomposition for photoelasticity tensors[END_REF] ∆(e 0 , Σ With the same proof as for theorem 6.2, we have the following result.

Theorem 7.3. Let e = g + h and e 0 = g 0 + h 0 be two piezoelectricity tensors, with h and h 0 their leading harmonic parts. The 15-dimensional minimization problem [START_REF] Forte | Symmetry classes and harmonic decomposition for photoelasticity tensors[END_REF] is equivalent to the 7-dimensional polynomial optimization problem

min d ′ 2 =0
‖h 0 -h‖ 2 , with e = h, and min

e∈Σ O - ‖e 0 -e‖ 2 = ‖g 0 ‖ 2 + min d ′ 2 =0
‖h 0 -h‖ 2 .

7.2.

Resolution by Lasserre's method. A third order harmonic tensor h ∈ H 3 (R

(E 0 , Σ [O -] ) ≈ ‖e 0 -e * ‖ is in C/m 2 ).
Remark 7.5. The computation times are of the same order of magnitude as for the cubic elasticity case. For this quadratic optimization problem (with 7 variables) as well, they are lower than for the 6 variables but degree 3 optimization problem of section 5.

Conclusion

Some isotropy strata of tensorial representations of the orthogonal group are explicitly characterized by polynomial covariants. We have taken advantage of this fact to formulate the computation of the distance to these strata as a polynomial optimization problem. We have used the property that the isotropy classes for the representation of SO(3) on the vector space of elasticity tensors are in general semialgebraic. The present work shows the interest of the characterization of the isotropy classes by means of polynomial covariants, rather than by means of invariants. In particular, the covariant characterization of the cubic piezoelectricity symmetry stratum (theorem 7.2), which is the cornerstone of our methodology, is a new result.

We have then recalled Lasserre's method to solve polynomial optimization problems under semialgebraic constraints. Under the so-called Archimedean hypothesis, this approach consists in writing the initial problem as an infinite semidefinite program from which is constructed a sequence of relaxed semidefinite programs that converges to the desired global minimum. We have presented the corresponding algorithm implemented in the freeware GloptiPoly, in particular its stopping criterion.

We have applied this polynomial optimization method to compute the cubic tensor the closest to a raw (measured) constitutive tensor, both in continuum mechanics elasticity and piezoelectricity. We have considered the following examples

• of an elasticity tensor measured by François and coworkers [START_REF] François | Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements[END_REF] for an aeronautics Nickelbased single crystal superalloy, • of nine piezoelectricity tensors computed for wurtzite alloys using Density Functional Theory (DFT) and ab-initio simulations, by Manna and coworkers [START_REF] Manna | Enhanced piezoelectric response of AlN via CrN alloying[END_REF].

In both cases, we took advantage of the distance being a coercive polynomial function to adapt the constraints so that they can satisfy the Archimedean condition, in order to ensure the convergence of the method to the desired minimum.

Appendix A. Explicit harmonic decomposition of an elasticity tensor An elasticity tensor E ∈ Ela admits the following explicit harmonic decomposition [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF]: with d ′ = d -1 3 tr(d)q and v ′ = v -1 3 tr(v)q respectively the traceless parts of d = tr 12 E and v = tr 13 E.

(26) E = E 𝑖𝑠𝑜 + q ⊗ (4) a + q ⊗ (2,2) b + H.
In [START_REF] Freund | Introduction to semidefinite programming (sdp)[END_REF], q is the Euclidean canonical bilinear 2-form represented by the components (𝛿 𝑖𝑗 ) in any orthonormal basis and the tensor products ⊗ We have, for Euclidean norm,

‖E‖ 2 = 5𝛼 2 + 4𝛽 2 + 2 21 ‖d ′ + 2v ′ ‖ 2 + 4 3 ‖d ′ -v ′ ‖ 2 + ‖H‖ 2 ,
Using ( 28)-( 29) we obtain for the experimental elasticity tensor E 0 (given by ( 15)) the harmonic decomposition Here, we will formulate alternative integrity bases for both R[H 3 (R 3 )] SO (3) and R[H 3 (R 3 )] O (3) , which happen to be more useful in order to characterize the cubic symmetry class in H 3 (R 3 ) for O(3). These will be used to prove theorem 7.2.

E 0 = (𝛼 0 , 𝛽 0 , d ′ 0 , v ′ 0 , H
Theorem C.1. Let h ∈ H 3 (R 3 ) be an harmonic third-order tensor, d 2 = h : h, and (1) A minimal integrity basis of R[H 3 (R 3 )] SO (3) is constituted by the five invariants (2) A minimal integrity basis of R[H 3 (R For such a tensor we get 𝐼 2 (h 0 ) = 6𝛿 2 , 𝐼 4 (h 0 ) = 0, 𝐼 6 (h 0 ) = 0, 𝐼 10 (h 0 ) = 0.

𝑣
Therefore, since 𝐼 2 (h) = ‖h‖ 2 ≥ 0, we can find a real number 𝛿 such that 6𝛿 2 = 𝐼 2 (h), and thus an at least cubic tensor h 0 such that 𝐼 2 (h 0 ) = 𝐼 2 (h), 𝐼 4 (h 0 ) = 𝐼 4 (h), 𝐼 6 (h 0 ) = 𝐼 6 (h), 𝐼 10 (h 0 ) = 𝐼 10 (h).

But an integrity basis for a real representation of a compact group separate the orbits [2, Appendix C]. Hence, h and h 0 are necessarily in the same orbit, which means that h = 𝜌 3 (𝑔)h 0 , for some 𝑔 ∈ O(3).

(Perla Azzi) CNRS, Sorbonne université, IMJ -Institut de mathématiques Jussieu, 75005, Paris, France & Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS -Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France

Email address, Perla Azzi: perla.azzi@ens-paris-saclay.fr (Rodrigue Desmorat) Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Email address: rodrigue.desmorat@ens-paris-saclay.fr (Boris Kolev) Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Email address: boris.kolev@ens-paris-saclay.fr (Fabien Priziac) Univ Bretagne Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France Email address: fabien.priziac@univ-ubs.fr

( 2 )

 2 𝑓 * = inf {𝑓 (𝑥 𝑥 𝑥); 𝑥 𝑥 𝑥 ∈ 𝐾} , where 𝑓 ∈ R[𝑋] := R[𝑋 1 , . . . , 𝑋 𝑛 ], 𝑥 𝑥 𝑥 = (𝑥 1 , . . . , 𝑥 𝑛 ) ∈ R 𝑛 and 𝐾 is a basic closed semialgebraic set 𝐾 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0} ,

(

  The numerical problem is first reduced to the following polynomial optimization problem min a∈𝐾 ‖a 0 -a‖ 2 ,

  2 seconds on a standard PC,∆(a 0 , Σ [O(2)] ) 2 ≈ 18.000007, a ** ≈ a * =

Figure 1 .

 1 Figure 1. Cubic microstructure of CMSX-4 Ni-based single crystal superalloy [51].

Remark 7 . 4 .

 74 We take 𝑐 = 3 > 𝑓 (0) to ensure the Archimedean property, but in the present case the convergence status 𝜉 does not seem to depend on 𝑐. Dropping the condition 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0 in K also leads to an accurate computed optimum. We obtain the result min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥) ≈ 𝑓 (𝑥 𝑥 𝑥 * ) = 1.060855 C 2 /m 4 at the first GloptiPoly relaxation 𝑑 = 𝑑 0 = 1 (with convergence status 𝜉 = +1 and for a computation time of 0.8 seconds). The components of the computed minimizer h * are (in C/m 2 ): ℎ 111 = -0.075476, ℎ 112 = -0.426450, ℎ 122 = 0.088998, ℎ 123 = -0.005937, ℎ 222 = 0.412070, ℎ 223 = -0.308913, ℎ 333 = 0.609783. By theorem 7.3, the computed cubic tensor e * ∈ Σ [O -] the closest to e 0 is simply e * = h * . In Voigt notation, [e * ] = ⎛ ⎝ -0.075476 0.088998 -0.013521 -0.005937 -0.300870 -0.426450 -0.426450 0.412070 0.0143797 -0.308913 -0.005937 0.088998 -0.300870 -0.308913 0.609783 0.014379 -0.013521 -0.005937 ⎞ ⎠ C/m 2

  + 2v ′ ), b = 2(d ′ -v ′ ).

6 (

 6 second-order tensors a, b, are defined as follows:(a ⊗ (4) b) 𝑖𝑗𝑘𝑙 = (a ⊙ b) 𝑖𝑗𝑘𝑙 = 1 𝑎 𝑖𝑗 𝑏 𝑘𝑙 + 𝑏 𝑖𝑗 𝑎 𝑘𝑙 + 𝑎 𝑖𝑘 𝑏 𝑗𝑙 + 𝑏 𝑖𝑘 𝑎 𝑗𝑙 + 𝑎 𝑖𝑙 𝑏 𝑗𝑘 + 𝑏 𝑖𝑙 𝑎 𝑗𝑘 )𝑏 𝑘𝑙 + 2𝑏 𝑖𝑗 𝑎 𝑘𝑙 -𝑎 𝑖𝑘 𝑏 𝑗𝑙 -𝑎 𝑖𝑙 𝑏 𝑗𝑘 -𝑏 𝑖𝑘 𝑎 𝑗𝑙 -𝑏 𝑖𝑙 𝑎 𝑗𝑘 ).

  0 -𝛽 0 ) q ⊗

  𝑣 𝑣 3 := h : d ′ 2 , 𝑣 𝑣 𝑣 5 := d ′ 2 • 𝑣 𝑣 𝑣 3 , 𝑣 𝑣 𝑣 7 := d ′ 2 • 𝑣 𝑣 𝑣 5 where d ′ 2 = d 2 -

𝐼 2 :

 2 = tr d 2 = ‖h‖ 2 , 𝐼 4 := tr(d ′ 2 2 ) = ‖d ′ 2 ‖ 2 , 𝐼 6 := ‖𝑣 𝑣 𝑣 3 ‖ 2 , 𝐼 10 := ‖d ′ 2 × 𝑣 𝑣 𝑣 3 ‖ 2 ,𝐼 15 := det(𝑣 𝑣 𝑣 3 , 𝑣 𝑣 𝑣 5 , 𝑣 𝑣 𝑣 7 ).

Proof of theorem 7 . 2 .

 72 Let e = g + h be a piezoelectricity tensor,g = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a) ∈ H 1 (R 3 ) ⊕ H 1 (R 3 ) ⊕ H 2♯ (R 3 ), h ∈ H 3 (R 3 ). If e ∈ Σ [O -] , then g = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a) ∈ Σ [O -] vanishes since an element in H 1 (R 3 ) or H 2♯ (R 3 ) with at least cubic symmetry ([O -]) is necessarily isotropic. For the same reason d ′ 2 (h) = 0. Conversely, if g = 0, then e = h ∈ H 3 (R 3 ) is harmonic, and it suffices to show that h ∈ Σ [O -] (is at least cubic). Since we assume furthermore d ′ 2 = 0, we have𝐼 2 (h) = ‖h‖ 2 ≥ 0, 𝐼 4 (h) = 0, 𝐼 6 (h) = 0, 𝐼 10 (h) = 0.Now an harmonic tensor in H 3 (R 3 ) which is fixed by O -is written (in Voigt notation (24)) as h

  𝐾, if 𝛿 𝑥 𝑥 𝑥 denotes the probability Dirac measure on R 𝑛 at 𝑥 𝑥 𝑥, we have 𝑓 (𝑥 𝑥 𝑥) = ∫ 𝑓 d𝛿 𝑥 𝑥 𝑥 and, conversely, if 𝜇 is a probability measure on R 𝑛 with support in 𝐾, Now, if (𝑦 𝛼 ) 𝛼∈N 𝑛 is a sequence of real numbers, denote by 𝑀 (𝑦) the infinite symmetric matrix (𝑦 𝛼+𝛽 ) 𝛼,𝛽∈N 𝑛 , (called the moment matrix associated to 𝑦) and, if 𝑔 = ∑ 𝛽∈N 𝑛 𝑔 𝛽 𝑋 𝛽 , set 𝑛 , ⟨𝑓, 𝑦⟩ := ∑ 𝛼 𝑓 𝛼 𝑦 𝛼 , and 𝑔 0 := 1. Here, if 𝑀 is a finite or infinite matrix with real coefficients, 𝑀 ⪰ 0 means that 𝑀 is positive semidefinite (an infinite symmetric matrix is called positive semidefinite if all its principal submatrices are positive semidefinite). The formulation (4) is a direct consequence of the following solution of the moment problem on 𝐾. Theorem 3.1 (Putinar, Jacobi-Prestel). Suppose that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 describing 𝐾 satisfy the Archimedean hypothesis. Then, for all 𝑦 ∈ R N 𝑛 , 𝑦 has a representative measure on 𝐾 (i.e. there exists a finite Borel measure 𝜇 on R 𝑛 with support in 𝐾 such that, for any 𝛼 ∈ N 𝑛 , 𝑦 𝛼 = ∫ R 𝑛 𝑥 𝑥 𝑥 𝛼 d𝜇(𝑥 𝑥 𝑥)) if and only if the moment matrices 𝑀 (𝑦), 𝑀 (𝑔 1 •𝑦), . . . , 𝑀 (𝑔 𝑚 •𝑦) are positive semidefinite.

	⎛	⎞
	3) Indeed, for 𝑥 𝑥 𝑥 ∈ ∫ 𝑓 * = inf {∫ R 𝑛 𝑓 d𝜇; 𝜇 probability measure on R 𝑛 with support in 𝐾 𝑓 (𝑥 𝑥 𝑥)d𝜇(𝑥 𝑥 𝑥) ≥ ∫ ⎝ ∑ 𝛽∈N 𝑛 𝑔 𝛽 𝑦 𝛼+𝛽 ⎠ 𝛼∈N 𝑛 ∈ R N 𝑛 . Under an hypothesis called the Archimedean hypothesis, we can write the optimization problem } . (3) as (4) 𝑓 * = inf 𝑦∈R N 𝑛 { ⟨𝑓, 𝑦⟩; 𝑦 (0,...,0) = 1, 𝑀 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 = 0, . . . , 𝑚 } 𝑓 𝑔 • 𝑦 := where, if 𝑦 ∈ R N
	R 𝑛	R 𝑛

* d𝜇(𝑥 𝑥 𝑥) = 𝑓 * .

  𝑗 ). We say that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis (or that M(𝑔 1 , . . . , 𝑔 𝑚 ) is an Archimedean module) if there exists a positive integer 𝑁 such that Notice that if 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis, then 𝐾 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0} is necessarily compact. The crucial point is that, if 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis, then we have access to Putinar's Positivstellensatz.

	Definition 3.2. Consider the R-module
		{	𝑚	}
		M(𝑔 1 , . . . , 𝑔 𝑚 ) :=	𝜎 0 +	∑	𝜎 𝑖 𝑔 𝑖 ; 𝜎 0 , . . . , 𝜎 𝑚 sum of squares	⊂ R[𝑋]
				𝑖=1
	(a polynomial 𝑝 ∈ R[𝑋] is a sum of squares if there exist polynomials 𝑝 1 , . . . , 𝑝 𝑁 ∈ R[𝑋] such
	that 𝑝 =	∑ 𝑁 𝑗=1 𝑝 2
				𝑛
			𝑁 -	∑	𝑋 2 𝑖 ∈ M(𝑔 1 , . . . , 𝑔 𝑚 ).
				𝑖=1
	We refer to [66, Theorem 1.1] (a result due to Schmüdgen) for a list of properties equivalent
	to the Archimedean hypothesis. Theorem 3.3 (Putinar

  0 and, consequently, the linear mapping 𝐿 𝑦 has nonnegative values on M(𝑔 1 , . . . , 𝑔 𝑚 ). By Putinar's Positivstellensatz 3.3, this implies that 𝐿 𝑦 has nonnegative values on any polynomial 𝑝

  sections 4.4.1 and 4.4.2].(3) In [42, section 6.1.2] is described the algorithm, implemented in GloptiPoly, which extract (approximated) global minimizers of 𝑓 when the rank condition is satisfied.Theorem 4.1 is a consequence of the following one whose sketch of proof is postponed below. For any 𝑟 ∈ N ∖ {0}, a Borel measure 𝜇 on R 𝑛 is said to be 𝑟-atomic if there exist 𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑟 ∈ R 𝑛 and positive real numbers 𝜆 1 , . . . , 𝜆 𝑟 such that 𝜇 = ∑ 𝑟 𝑖=1 𝜆 𝑖 𝛿 𝑥 𝑥 𝑥 𝑖 . Theorem 4.3 (Curto-Fialkow[START_REF] Curto | The truncated complex 𝑘-moment problem[END_REF], Laurent[START_REF] Laurent | Revisiting two theorems of Curto and Fialkow on moment matrices[END_REF]). Let 𝑦 ∈ 𝐹 𝑑 . If rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦), then 𝑦 can be represented by a 𝑠-atomic measure, where 𝑠 := rank 𝑀 𝑑 (𝑦), whose support is included in 𝐾.Proof of theorem 4.1. We adapt the proof of[42, theorem 6.6]. Suppose that rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦). Then, by theorem 4.3, 𝑦 has a 𝑠-atomic representing measure 𝜇 with support included in 𝐾 : there exist 𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑠 ∈ 𝐾 and 𝜆 1 , . . . , 𝜆 𝑠 ∈]0; +∞[ such that 𝜇 = ∑ 𝑠 𝑖=1 𝜆 𝑖 𝛿 𝑥 𝑥 𝑥 𝑖 . In particular, since

  1 , . . . , 𝑥 𝑥 𝑥 𝑠 } and let 𝜆 1 , . . . , 𝜆 𝑠 ∈]0; +∞[ such that 𝜇 = ∑ 𝑠 𝑖=1 𝜆 𝑖 𝛿 𝑥 𝑥 𝑥 𝑖 . By Lemma 5.6 of [44], there exist 𝑝 1 , . . . , 𝑝 𝑠 ∈ R[𝑋] of degree at most 𝑑 -𝑣 such that, for all 𝑖, 𝑗 ∈ {1, . . . , 𝑠}, 𝑝 𝑖 (𝑥 𝑥 𝑥 𝑗 ) = 𝛿 𝑖𝑗 (see also [44, Lemma 2.3]). For all 𝑖 ∈ {1, . . . , 𝑠} and 𝑗 ∈ {1, . . . , 𝑚}, we then have, because 𝑦 ∈ 𝐹 𝑑 , 0 ≤ 𝑝 𝑇 𝑖 𝑀 𝑑-𝑣 (𝑔 𝑗 • 𝑦)𝑝 𝑖 = ∫ R 𝑛 𝑔 𝑗 (𝑥 𝑥 𝑥)𝑝 𝑖 (𝑥 𝑥 𝑥) 2 d𝜇(𝑥 𝑥 𝑥) = 𝑘 𝑔 𝑗 (𝑥 𝑥 𝑥 𝑘 )𝑝 𝑖 (𝑥 𝑥 𝑥 𝑘 ) 2 = 𝜆 𝑖 𝑔 𝑗 (𝑥 𝑥 𝑥 𝑖 ) (see the proof of proposition 3.1 above for the first equality) and, since 𝜆 𝑖 > 0, 𝑔 𝑗 (𝑥 𝑥 𝑥 𝑖 ) ≥ 0. As a consequence, for all 𝑖 ∈ {1, . . . , 𝑠}, 𝑥 𝑥 𝑥 𝑖 ∈ {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0} = 𝐾.

𝑠 ∑ 𝑘=1 𝜆

  𝐸 1111 𝐸 1122 𝐸 1133 𝐸 1123 𝐸 1113 𝐸 1112 𝐸 1122 𝐸 2222 𝐸 2233 𝐸 2223 𝐸 1223 𝐸 1222 𝐸 1133 𝐸 2233 𝐸 3333 𝐸 2333 𝐸 1333 𝐸 1233 𝐸 1123 𝐸 2223 𝐸 2333 𝐸 2323 𝐸 2331 𝐸 2312 𝐸 1113 𝐸 1223 𝐸 1333 𝐸 2331 𝐸 1313 𝐸 3112 𝐸 1112 𝐸 1222 𝐸 1233 𝐸 2312 𝐸 3112 𝐸 1212

			⎛
	(13)	[E] =	⎜ ⎜ ⎜ ⎜ ⎜ ⎜
			⎝

1, and 𝐺 = SO(3). An elasticity tensor E ∈ Ela can be represented by a 6 × 6 symmetric matrix, in Voigt notation,

  36.401489, -20.227012, -38.908985, -6.396664, 27.780748, -2.277546, 44.251364, -4.557344, 21.161507).

  e 𝑖𝑗𝑘 = e 𝑖𝑘𝑗 } (dim Piez = 18), the vector space of piezoelectricity tensors e : S 2 (R 3 ) → R 3 (see example 2.2), and set 𝐺 = O(3). A piezoelectricity tensor e ∈ Piez can be represented by a 3 × 6 matrix, in so-called Voigt 122 e 133 e 123 e 113 e 112 e 211 e 222 e 233 e 223 e 213 e 212 e 311 e 322 e 333 e 323 e 313 e 312

	representation,
		⎛ e 111 e ⎞
	[e] =	⎝	⎠ .

  [O -] ) 2 = min e∈Σ O - ‖e 0 -e‖ 2 , and e * ∈ Σ [O -] the closest to e 0 . In order to succeed, we first have to characterize the cubic piezoelectricity stratum Σ [O -] by polynomial equations (a proof of the following theorem is provided in Appendix C).Theorem 7.2. Let e = g + h ∈ Piez be a piezoelectricity tensor, with h ∈ H 3 (R 3 ) its leading harmonic part, let d 2 = h : h (i.e., (d 2 ) 𝑖𝑗 = ℎ 𝑖𝑘𝑙 ℎ 𝑘𝑙𝑗 ) , and d ′ 2 = d 2 -1 3 tr(d 2 ) q be second-order covariants of e. Then e ∈ Σ [O -] (is at least cubic) if and only if g = 0 and d ′ 2 = 0, and e ∈ Σ [O -] (is cubic) if and only if furthermore h ̸ = 0.

  3 ) has seven independent components and is represented by the following real matrix (in Voigt notation) 111 , ℎ 112 , ℎ 122 , ℎ 123 , ℎ 222 , ℎ 223 , ℎ 333 ).

			⎛	ℎ 111	ℎ 122 -ℎ 111 -ℎ 122			ℎ 123	-ℎ 223 -ℎ 333 ℎ 112	⎞
	(24)	[h] =	⎝	ℎ 112	ℎ 222 -ℎ 112 -ℎ 222			ℎ 223	ℎ 123	ℎ 122	⎠
				-ℎ 223 -ℎ 333 ℎ 223	ℎ 333	-𝐻 112 -ℎ 222 -ℎ 111 -ℎ 122 ℎ 123
	The traceless second order tensor d ′ 2 has five independent components (d ′ 2 ) 𝑖𝑗 detailed in Appen-
	dix B.
	We set
	𝑥 𝑥 𝑥 = (ℎ GloptiPoly computes the approximation
	(25)				‖h 0 -h  𝐾 = { 𝑥 𝑥 𝑥; d ′ 2 = 0, 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0	}	,	𝑐 = 3.
	For e 0 given by (22), we have (in C 2 /m 4 )
		𝑓 (𝑥 𝑥 𝑥) = 6ℎ 111 ℎ 122 + 6ℎ 223 ℎ 333 + 6ℎ 112 ℎ 222 + 4ℎ 2 111 + 6ℎ 2 112 + 6ℎ 2 122 + 4ℎ 2 222
				+ 6ℎ 2 223 + 4ℎ 2 333 + 6ℎ 2 123 -0.1002ℎ 111 -0.1742ℎ 122 + 0.1636ℎ 123
				-0.0114ℎ 223 -5.2244ℎ 333 + 0.4574ℎ 112 + 0.0836ℎ 222 + 2.7367.

* ‖ 2 of the minimum ∆(e 0 , Σ [O -] ) 2 = min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥), where 𝑓 (𝑥 𝑥 𝑥) = ‖h 0 -h‖ 2 and

Table 1 .

 1 The distance and the relative distance to cubic piezoelectricity are finally∆(e 0 , Σ [O -] ) ≈ ‖e 0 -e * ‖ = 1.214681 C/m 2 , ‖e 0 -e * ‖ ‖e 0 ‖ = 0.684256.The results obtained for the raw piezoelectricity tensors e 𝑥 0 given in the Appendix D for wurzite Cr 𝑥 Al 1-𝑥 N, with different chromium concentrations, are summarized in Table1. Results for the raw piezoelectricity tensors of[START_REF] Manna | Enhanced piezoelectric response of AlN via CrN alloying[END_REF] for different Crconcentrations 𝑥 (the distance ∆

	𝑥	∆(E 0 , Σ [O -] )	‖e 0 -e * ‖ ‖e 0 ‖	Computation time (s)
	0 (AlN)	1.214681	0.684256	0.7
	0.035	1.307327	0.715295	0.7
	0.07	1.364909	0.729065	0.8
	0.10	1.541726	0.785604	0.6
	0.13	1.542293	0.758240	1.0
	0.16	1.665883	0.793355	0.6
	0.19	1.852505	0.813719	0.7
	0.225	1.877377	0.781094	1.2
	0.255	1.944763	0.752770	0.7

  3 )] O(3) is constituted by the four invariants 𝐼 2 , 𝐼 4 , 𝐼 6 , and 𝐼 10 .Proof. To prove the theorem, it is enough to show that Smith and Bao's invariants can be expressed as polynomials of 𝐼 2 , 𝐼 4 , 𝐼 6 , 𝐼 10 , 𝐼 15 , since, then, this set will be generating and moreover the cardinal of a minimal integrity basis of homogeneous invariants does not depend on the choice of a particular basis[START_REF] Dixmier | Le nombre minimum d'invariants fondamentaux pour les formes binaires de degré 7[END_REF]. Indeed, one can check that

	𝐾 4 = 𝐼 4 +	1 3	𝐼 2	2 ,
	𝐾 10 = -	4 3	𝐼 10 -	1 27	𝐼 2	3 𝐼 4 +	1 9	𝐼 2	2 𝐼 6 +	2 9	𝐼 2 𝐼 4	2 +	2 3	𝐼 4 𝐼 6 ,

𝐾 15 = 2𝐼 15 ,

which achieves the proof.

relating induced polarization in a dielectric material to the strain tensor.

The authors were partially supported by CNRS Projet 80-Prime GAMM (Géométrie algébrique complexe/réelle et mécanique des matériaux).

2

H the fourth-order harmonic tensor given by [START_REF] Dellinger | Computing the optimal transversely isotropic approximation of a general elastic tensor[END_REF], are: Smith and Bao [START_REF] Smith | Isotropic invariants of traceless symmetric tensors of orders three and four[END_REF] have derived a minimal integrity basis of five invariants for the algebra R[H 3 (R 3 )] SO (3) , of polynomial SO(3)-invariants of the third-order harmonic tensors h ∈ H 3 (R 3 ). These five invariants (equations (2.3) and (2.4) in [START_REF] Smith | Isotropic invariants of traceless symmetric tensors of orders three and four[END_REF]) can be recast in a more intrinsic form as

where d 2 = h : h and 𝑣 𝑣 𝑣 3 := h : d ′ 2 . In [START_REF] Olive | Isotropic invariants of a completely symmetric third-order tensor[END_REF], Olive and Auffray have used these results to deduce that a minimal integrity basis for the algebra R[H 3 (R 3 )] O (3) , of polynomial O(3)-invariants of h ∈ H 3 (R 3 ) consists of the four invariants 𝐼 2 , 𝐾 4 , 𝐼 6 , and 𝐾 10 .

Appendix D. Raw piezoelectricity tensors for wurtzite

The raw piezoelectricity tensors e 𝑥 0 considered in section 7 correspond to the mean values computed in [START_REF] Manna | Enhanced piezoelectric response of AlN via CrN alloying[END_REF] for wurtzite Cr 𝑥 Al 1-𝑥 N, with 𝑥 the chromium concentration (in C/m 2 ),