Fabrice Richard
email: fabrice.richard@univ-fcomte.fr

Mylène Villars

Sébastien Thibaud

Keywords: indentation, viscoelastic, damage, human cartilage, osteoarthritis, identifiability, inverse method 2

come

Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation

Introduction

Articular cartilage (AC) is the thin layer (0.5-5.0 mm thick) of hydrated soft tissue covering the articulating bony ends in diarthrodial joints. It plays a vital role in the function of the musculoskeletal system by allowing almost frictionless motion to occur between the articular surfaces of a diarthrodial joint. Furthermore, it distributes the loads over a large contact area, thereby minimizing the contact stress, and dissipates the energy associated with dynamic loads [START_REF] Mow | Biomechanics of diarthrodial joints: a review of twenty years of progress[END_REF].

Normal articular cartilage

AC can be considered as a solid matrix, saturated with water and mobile ions. The solid matrix consists of cartilage cells (chondrocytes) embedded in an extracellular matrix. The major components of the extracellular matrix are collagen molecules and negatively charged proteoglycans. The mechanical properties of cartilage depend primarily on the properties of the extracellular matrix. Composition, maintenance and gradual turnover of the matrix depend on the biosynthetic activity of chondrocytes [START_REF] Maroudas | Biophysical chemistry of cartilaginous tissues with special reference to solute and fluid transport[END_REF].

Many studies have been conducted on the mechanical properties of cartilage [START_REF] Wilson | The role of computational models in the search for the mechanical behavior and damage mechanisms of articular cartilage[END_REF]. It is generally accepted that, for a correct description of the mechanical behavior of AC, an anisotropic, inhomogeneous and viscoporoelastic model should be used. Concurrent viscoelasticity and poroelasticity effects have been observed in articular cartilage and in poroviscoelastic model both types are included [START_REF] Disilvestro | A cross-validation of the biphasic poroviscoelastic model of articular cartilage in unconfined compression, indentation, and confined compression[END_REF]. Viscoelasticity results from the conformational change of macromolecules (fluid flow-independent), and poroelasticity results from the migration of small molecules (fluid flow-dependent). For articular cartilage the viscoelastic relaxation time (material-specific) is about 10 s and the poroelasticic relaxation time is above 1000 s in macro length scale. These values allow the classification of observations made with different lengths (radius of contact in an indentation test) and times [START_REF] Hu | Viscoelasticity and poroelasticity in elastomeric gels[END_REF].

The objective of this study is to propose a model of articular cartilage taking into account loading rate and degree of osteoarthritis that it can be easily used in predictive numerical simulations of the hip joint behavior during daily various activities [START_REF] Pustoc | Normal and osteoarthritic hip joint mechanical behaviour : a comparison study[END_REF]. For this application, the observation time is approximately in same order of magnitude as the viscoelastic relaxation time and much lower than the poroelastic relaxation time (viscoelasticity relaxing and poroelasticity unrelaxed). For this reason, only viscoelastic model is used in this study, unlike various recent studies that are in different observation time and/or length [START_REF] Gupta | A fiber reinforced poroelastic model of nanoindentation of porcine costal cartilage: a combined experimental and finite element approach[END_REF][START_REF] Keenan | New resource for the computation of cartilage biphasic material properties with the interpolant response surface method[END_REF][START_REF] Miller | Use of microindentation to characterize the mechanical properties of articular cartilage: comparison of biphasic material properties across length scales[END_REF].

Cartilage damage due to osteoarthritis

Osteoarthritis (OA) is the most common degenerative disease of human AC, especially in the population aged over 65 years [START_REF] Hamerman | Clinical implications of osteoarthritis and aging[END_REF]. It is characterized by an extracellular matrix damage and an important loss in tissue cellularity [START_REF] Vignon | Quantitative histological changes in osteoarthritic hip cartilage: morphometric analysis of 29 osteoarthritic and 26 normal human femoral heads[END_REF].

Several models were used to analyze the mechanical behavior of healthy and degenerated cartilage [START_REF] Korhonen | Fibril reinforced poroelastic model predicts specifically mechanical behavior of normal, proteoglycan depleted and collagen degraded articular cartilage[END_REF][START_REF] Disilvestro | Biphasic poroviscoelastic characteristics of proteoglycan-depleted articular cartilage: simulation of degeneration[END_REF][START_REF] Mäkelä | Structure-function relationships in osteoarthritic human hip joint articular cartilage[END_REF]. However, the initiation and progression of cartilage degeneration associated with experimental results on fresh normal human cartilage and osteoarthritis has so far not been studied and modeled. Understanding the mechanical behavior of damaged cartilage and associated parameters allows, through appropriate computational modeling, to better understand the beginning of some diseases such as osteoarthritis and efficiently analyze the pressure distribution in the joint.

Cartilage damage due to osteoarthritis is believed to be mechanically induced and the mechanical properties of cartilage tissue are strongly related to the OA stage of cartilage [START_REF] Kleemann | Altered cartilage mechanics and histology in knee osteoarthritis: relation to clinical assessment (ICRS Grade)[END_REF]. Nonetheless, the initial event that triggers the pathological process of cartilage degeneration is still unknown. In order to investigate the initiation and the evolution of damage due to osteoarthritis, a behavior model must be determined with a damage variable which represents OA level.

Indentation testing to characterize articular cartilage

Basically, instrumented indentation testing is a mechanical characterization process designed to determine the materials properties by putting an indenter in the surface of a material sample.

Indentation has been commonly used in testing the mechanical properties of articular cartilage at the macro, micro and nanoscales, and its loading conditions is considered more relevant to the physiological condition.

Elastic properties may be obtained from the measurements of the indentation load ( ) P t versus the penetration depth ( ) h t from the well-known equation for spherical indentation with the assumption of purely elastic solids [START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF]. It should be noted that unlike in simple tension or compression tests, in indentation test, the two material constants E (Young's modulus) and ν (Poisson's ratio) are coupled in the indentation load-deflection relations, and additional independent tests are generally needed to uncouple them [START_REF] Jin | Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters[END_REF]. It has been shown that human articular cartilage exhibits viscous behavior during indentation and that it makes the analysis of the load-penetration curve more difficult. Moreover, it should be noted that viscous phenomena introduce errors when traditional elastic-plastic analysis of indentation data is used and time-dependence is ignored [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]. Linear viscoelastic analytical solutions of indentation on a semi-infinite solid with a spherical-tip indenter developed by the method of functional equations are available for the load relaxation, creep tests [START_REF] Cheng | Spherical-tip indentation of viscoelastic material[END_REF] and harmonic loading [START_REF] Argatov | Accounting for the thickness effect in dynamic spherical indentation of a viscoelastic layer: Application to non-destructive testing of articular cartilage[END_REF], but the direct identification of the flow stress curve using an analytical model is impossible for more general loading and non linear behavior.

In this paper, an inverse method is used to identify the parameters of the constitutive law. The identification is based on optimization techniques to adjust material parameters in order that the calculated response obtained by numerical modeling (Finite Element Method) matches the measured one.

The aim of this study is to identify the viscoelastic parameters of normal and OA femoral head cartilage on four distinct areas using instrumented indentation testing coupled with an identification method by the way of a numerical approach. The experimental part of this study is based on the preparation of human cartilage samples (normal and OA) and indentation tests. The numerical part is based on the simulation of indentation experiments by the finite element method coupled with an optimization algorithm to minimize the error between the model and experiments. The identifiability analysis of the model parameters is also proposed.

Analysis results allow to propose a single phase viscoelastic model with an internal damage variable in the thermodynamic framework. This damaged model is a good representation of the viscoelastic behavior in the two states (normal and OA) and provides a prediction of the intermediate states.

Materials and methods

Cartilage sampling

Samples of normal human cartilage from femoral head were obtained during hip replacement surgery from 6 patients exclusively with femoral neck fracture and showed no other pathology.

Samples of OA human cartilage from femoral head were obtained during surgery from 4 patients with clinical and radiological features of advanced coxarthrosis (Table 1).

These specimens were removed immediately after surgery from four study areas (superior, inferior, posterior and anterior) delimited by the surgeon at the end of the operation. The samples were then stored at 4°C in a solution of Phosphate Buffered Saline (PBS) until the mechanical tests that did not exceed 24 hours after collection. Samples have a thickness of 8 mm, a length of 25 mm and a width of 15 mm (subchondral bone and a small thickness of cancellous bone were preserved). For OA femoral head, the samples were carried out in areas where enough cartilage remained.

A total of 29 samples were analyzed (24 normal and 5 OA). To validate the method, other samples with the same preparation method were also used and some results are presented to illustrate the mechanical behavior of the material and the relevance of the identification process.

Indentation

Experimental testing

The tests were performed by using an instrumented micro-indenter system developed at DMA/FEMTO-ST institute that is able to measure the indentation load P and the penetration depth h in various mechanical modes. The principle of this testing method is presented in Fig. 1.

A spherical and impermeable stainless steel indenter (radius 1.5 mm R =

) was used during the experiments (Fig. 2).

Three consecutive measures spaced by 2 mm were performed on each sample (random measurements between indentation). Tests were conducted in load-unload configuration with a depth of less than 250 µm and a rate of about 5 µm/s. Load-unload cycle analysis can be used to access to viscoelastic properties [START_REF] Qasmi | Viscoelastic mechanical properties determined by nanoindentation tests and its numerical modelling of polypropylene modified by He+ particle implantation and e(-) irradiation[END_REF] and it seems a realistic test for the main application of the model. The load rate (5 μm/s) fully activates viscoelastic phenomena and was chosen to keep experimental observation time much lower than poroelastic relaxation time for an indentation depth about 10% of the cartilage thickness.

To validate model, other tests have been also carried out on other samples: load-unload (rate from 1 µm/s to 40 µm/s), creep and relaxation.

Viscoelastic constitutive law

Articular cartilage can be characterized as an isotropic viscoelastic material with no interstitial fluid flow during short time response. Isotropic viscoelastic constitutive law is based on the idea that the dilatational components of the stress tensor kk σ which involves the volumetric compression, and the deviatoric part ij s which only accounts for shape modifications, are decoupled. Thus, the constitutive law can be written as

(1 ) (1 ) ij ij ij ij E H E E H s s e e η ν ν η + + = + + + & & (1) (1 2 ) (1 2 ) kk kk kk kk E H E E H σ σ ε ε η ν ν η + + = + - - & & (2) with 1 1 , 3 3 ij ij ij kk ij ij ij kk s e σ δ σ ε δ ε = - = - (3)
In these equations, E is the instantaneous Young's modulus and H a tension modulus connected to the relaxed Young's modulus / (

) E EH E H ∞ = +
. ν is the Poisson' ratio and η the viscosity coefficient of the model. ij δ is the second order unity tensor, ij s and ij e represent the stress and strain deviatoric tensors. ij σ and ij ε are the stress and the strain tensors respectively. The above equations correspond to the standard linear solid model as shown in Fig. 3.

It should be noted here that the Poisson ratio is assumed to be constant. This assumption is validated experimentally from indentation test [START_REF] Jin | Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters[END_REF]. It also notices that in tension, Eq. ( 1) and ( 2) are summarized by the following expression

11 11 11 11 1 E E σ σ ε ε τ τ ∞ + = + & & (4) Where / ( ) E EH E H ∞ = + and / ( ) E H τ η = +
are, respectively, the relaxed tension modulus and the relaxation constant.

The identification of the four parameters model

[ , , , ] T E H ν η = θ
is required. The identification of these material parameters was performed for each sample by using an inverse method from experimental results and numerical modeling of the indentation test. An identifiability analysis was performed to determine uncertainties on the obtained parameters.

Numerical testing

The indentation test was modeled in a 2D axisymmetric representation with the ANSYS® finite element software (Fig. 4). The indenter is considered as a rigid semi-circular arc. Cartilage thickness h c is adapted to samples (from 1 mm to 3.2 mm).

No friction between indenter/cartilage was considered. Thicknesses and mechanical behavior of subchondral h sb and cancellous bones h cb were taken into account in the model and considered as elastic and isotropic ( 2000 MPa, 0.3, 600 MPa, 0.2

sb sb cb cb E E ν ν = = = =
). The low sensitivity of these parameters on the loading-penetration curve (P-h) was tested.

Substrat's inferior part was embedded. The contact algorithm between rigid indenter and cartilage was the augmented Lagrangian method. The considered material behavior is a viscoelastic model detailed in the previous section. An implicit method with non-linear geometric consideration is used with higher order 2-D elements. The mesh density is defined in order that the smallest elements are localized in the contact region. A convergence analysis has been performed in order to avoid any mesh influence.

Inverse method for parametric identification

A least-squares error minimization approach

The identification of material parameters can be seen as an inverse formulation. In this study, the main idea is to simulate the performed experiment (load-unload indentation with a constant rate displacement), by adapting the 4 material parameters

1 2 3 4 [ , , , ] [ , , , ] T T E H θ θ θ θ ν η = = θ in order
to numerically obtain the same results as the experimental ones (Fig. 5). This approach consists of an optimization problem where the objective is to minimize the gap between the experimental ( ) E P t and the numerical results ( , ) N P t θ , whatever the time variable t belonging to the period of the experimental test. The gap (or the residuals) between the experimental and the numerical results can be defined as

[ ] 2 1 [0, ] 1 ( ) ( ) ( , ) 2 max ( ) N N E i N i i E t t f P t P t N P t = ∈ = - ∑ θ θ (5)
In previous equation, the th i data points corresponding to the acquisition times i t . The minimization problem can be written formally:

[ , ]

[ , , , ] argmin ( )

T E H f ν η -+ * ∈ = = θ θ θ θ θ (6)
Where * θ is the solution of the identification problem.

- θ and + θ are the minimum and maximum limits on the material parameters. The minimization problem was solved for each cartilage sample with the optimization software MIC2M [START_REF] Richard | MIC2M Software, Modélisation et Identification du Comportement Mécanique des Matériaux[END_REF] by considering an algorithm based on the Levenberg-Marquardt method. From numerical modeling of the indentation test and experimental results, this software was able to write the functional (5) to be minimized.

Identifiability analysis

Samples preparation follows the procedure describes in section 2.1. Parameters were identified from a loading-unloading experimental indentation test with a rate displacement 5 µm/s h ≈ & following the procedure describes in the previous section. The solid lines in Fig. 6 Fig. 6 shows a good correlation between simulation and experiment, but the information contained in the available experimental data must be evaluated more precisely to determine if they are suitable for a reliable parameter estimation, mainly due to the uncertainty of the Poisson's ratio.

All techniques of identifiability analysis are based on local sensitivity functions. These functions quantify the relation between the outputs and the parameters of the model. In this study, the sensitivity function is mathematically defined by ( , ) ( , )

j N i ij j N i P t S P t θ θ * * * ∂ = ∂ θ θ θ (7)
The sensitivity ranking of the th j parameter is done by using the relation (adjusted for each sample) is equal to 0.2 while the others are negligible ( , , ,...

sb cb sb h h E ).
The poor identifiability of the model parameters can be due to a small sensitivity of the model results to a parameter, or by a linear approximation dependence of sensitivity functions on the results with respect to the parameters. It is evaluated through an indicator which is a measure of the shape of the confidence region

max min / E J λ λ = (9)
where max λ and min λ are respectively the largest and the smallest eigenvalue of the Fisher's Information Matrix T = F SS, defined from normalized sensitivities (7) of the set of considered parameters. According to [START_REF] Gujarati | Basic Econometrics, 4th Edition[END_REF], a low value of E J <100 indicates a low collinearity (good identifiability) and a high value ( E J >1000) indicates a very high collinearity (poor identifiability).

Table 2 shows a moderate correlation between the parameters for 4 k = ( 300 In the indentation test, it is well known that the elastic modulus is correlated with the Poisson ratio through the indentation modulus and variability of results may be limited by the following assumption that the value of Poisson's ratio can vary from 0 to 0.5. To estimate the error on the parameters obtained, an identification was performed with 0 ν = and the resulting parameters set is equal to: 12.1 MPa E * = , 2.7 MPa H * = , 112.7 MPa.s η * = . This solution leads to an error on the cost function (5) equivalent to the first identification results. It can be estimated that the error parameter is less than 30%. In practice, the Poisson's ratio is not set. It will be observed that this uncertainty is lower than inter and intra individual variability. Nevertheless for almost samples (normal and OA), the algorithm converges to a value of about 0.47 ν = . This result is closed to the results obtained by [START_REF] Jin | Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters[END_REF]. They consider different indenter radius to access to this value.

E J ≈ ).

Statistical analysis

For normal and OA cartilage, a variance analysis was performed to compare each viscoelastic parameters between the different areas. To compare each viscoelastic parameters between the normal and OA cartilage, a t-test was performed. For statistical tests, a significance level of 0.05 was used.

Results and discussion

Identification Results

A total of 29 samples were analyzed (24 normal and 5 OA). The different results obtained by using the identification procedure defined in section 2.3. are summarized in Table 3 andTable 4. No significant differences could be put in evidence in mean values of the viscoelastic parameters E , H , and η between the different areas (p>0.75). However, a significant difference was observed for Poisson's ratio (p<0.05).

Osteoarthritis cartilage

5 samples were analyzed from OA femoral heads. These parameters values obtained are close to the results proposed by Juras et al. (2011) for OA human femoral condyles (knee cartilage). Values of E , E ∞ and τ ranged respectively from 2.5 to 10.7 MPa (mean: 4 MPa), 0.1 to 2.9 MPa (mean: 0.8 MPa), and 3.8 to 22.6 s (mean: 11.0 s).

No significant differences could be put in evidence in mean values of viscoelastic parameters E , H , and η between the different areas (p>0.36). However, a significant difference was observed for Poisson's ratio (p<0.05).

Comparison normal/OA

The mean values of viscoelastic parameters E , H and η were significantly lower for OA cartilage than normal (p=0.002, 0.025, 0.013 respectively). However, no difference for ν was observed (p = 0.953).

The E and H moduli are about 3.5 times lower for the OA cartilage than the normal cartilage but it is interesting to note that the / E E ∞ ratio does not change with osteoarthritis. On the viscosity coefficient η and the relaxation constant τ , they are respectively six and twice lower for the OA cartilage than the normal cartilage, but the uncertainty in these values is quite important, due to high inter-individual and intra-individual variability.

The means values of cartilage thickness differed between the inferior area (1.61 mm) and superior area (2.97 mm). These results are in agreement with the literature [START_REF] Kurrat | The thickness of the cartilage in the hip joint[END_REF]. No significant correlation between thickness and the identified viscoelastic parameters is observed.

A viscoelastic model including degradation

In this section, an isotropic phenomenological viscoelastic model used to provide an accurate prediction of the mechanical response of human normal and OA cartilage is presented.

In the context of thermodynamics with internal variables [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF] for an isothermal process, the continuum damage mechanics concepts developed by the Lemaître's school [START_REF] Lemaitre | Engineering damage mechanics[END_REF] are considered. Starting from the early proposition of [START_REF] Kachanov | Time of the Rupture Process under Creep Conditions[END_REF] and [START_REF] Rabotnov | Creep Problems in Structural Members[END_REF], a scalar damage variable D which characterizes the degree of material degradation is introduced

1 D = - σ σ % (10)
where σ is the Cauchy stress tensor and % σ the effective stress tensor. The damage variable D may be interpreted as an indirect measure of degree of osteoarthritis and it is assumed that

0 1 ≤ ≤ ≤ c D D
, where c D is the critical damage parameter defining the local failure, here defined as the degree of osteoarthritis corresponding to advanced coxarthrosis. By applying the elastic strain equivalence hypothesis [START_REF] Lemaitre | Théorie mécanique de l'endommagement isotrope appliqué à la fatigue des métaux[END_REF], the strain behavior of damaged model material is represented by the constitutive equation of virgin material which stress is simply replaced by the effective stress such as:

1 1

: :

e - - = = ε C σ C σ % % (11)
where e ε is the elastic strain and C the elastic constitutive tensor for the normal cartilage which can be written in matrix notation as a function of the bulk modulus K and the shear modulus

G [ ] 4 /3 2 /3 2 /3 0 0 0 2 /3 4 /3 2 /3 0 0 0 2 /3 2 /3 4 /3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K G K G K G K G K G K G K G K G K G G G G + - - ⎡ ⎤ ⎢ ⎥ - + - ⎢ ⎥ ⎢ ⎥ - - + = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ C (12)
with and 3(1 2 ) 2( 1)

E E K G ν ν = = - + (13)
E is the virgin (undamaged) instantaneous modulus (Young's modulus). % C is the elastic constitutive tensor for the damaged material which can be written by using ( 10) and (11) as follows

(1 )

D = - C C % (14)
Assuming that the Poisson's ratio is unaffected by the damage process, the damaged bulk and shear moduli can be written as a function of the instantaneous damaged Young's modulus E % : , 3(1 2 ) 2( 1)

E E K G ν ν = = - + % % % % (15)
The strain partition assumption is used. The total strain is then partitioned in an elastic part e ε (instantaneous reversible strain) and a viscous part v ε (time-dependent reversible strain):

= + e v ε ε ε (16)
The state variables can be divided in observable variables and internal variables as defined in Table 6. The α and X tensors are associated to viscous phenomena.

The standard generalized framework [START_REF] Halphen | Sur les matériaux standards généralisées[END_REF] assumes that the mechanical behavior can be obtained when two potentials are defined: a free energy density ψ and a dissipation potential Ω. By applying the hypothesis that elasticity and viscous behaviors are uncoupled, the free energy is proposed as the following expression ( )

1 ( , , ) 1 : : : : 2 e e e H D D E ρψ ⎛ ⎞ = - + ⎜ ⎟ ⎝ ⎠ ε α ε C ε α C α ( 17
)
Where ρ is the mass density. State laws can be defined by:

( )

1 : ψ ρ ∂ = = - ∂ e e D σ C ε ε (18) ( ) 1 : H D E ψ ρ ∂ = = - ∂ X Cα α (19) 1 : : : : 2 e e H Y D E ψ ρ ∂ ⎛ ⎞ = =- + ⎜ ⎟ ∂ ⎝ ⎠ ε C ε α C α (20)
Based on the experimental results, the following dissipation potential is introduced

1 ( ):( ) ( ) 2 Y η Ω = - -+ ϒ σ X σ X % % % % (21)
The potential ϒ can be integrated into the evolution of creep damage and fatigue. Based on this potential Ω , the evolution of internal variables is expressed as

2 (1 ) v D η η ∂Ω - - = = = ∂ - σ X σ X ε σ & % (22) v ∂Ω = - = ∂ α ε X & & (23) D Y Y ∂Ω ∂ϒ = - = - ∂ ∂ & (24)
From a rheological point of view the model proposed here can be represented by a standard linear solid model with damage and damaged parameters can be defined using the above equations:

( ) ( ) ( ) 2 1 ; 1 ; 1 E D E H D H D η η = - = - = - % % % (25)
(1 ) and ( 1)

D E E D τ τ ∞ ∞ = - = - % % % % (26)

Conclusions

By using an inverse method coupled with a finite element method of the indentation test, the effects of osteoarthritis (OA) on the viscoelastic behavior of the human hip femoral cartilage is quantified.

In the normal and OA cartilage, no significant differences could be put in evidence between the different areas and the results showed a relative homogeneity of the viscoelastic parameters within each femoral head. However, it reveals a high intra and inter-individual variability.

Mean values of viscoelastic parameters were significantly lower for the OA cartilage than the normal: E and H moduli are about 3.5 times lower for OA cartilage than normal cartilage and the viscosity coefficient η is about six times lower.

The proposed isotropic phenomenological viscoelastic model provides an accurate prediction of the mechanical response of the normal and OA cartilages with advanced coxarthrosis.

Relationships between the viscoelastic material parameters E , H , η and the internal variable of damage D which represents the degree of osteoarthritis are validated for the two states, 0 D = (normal cartilage) and 0.71 In recent studies, significant correlations were observed between MRI and mechanical parameters (especially Young's modulus). Therefore, MRI could be a powerful technique to access to the internal variable of damage [START_REF] Lammentausta | T2 Relaxation time and delayed gadolinium-enhancedMRI of cartilage (dGEMRIC) of human patellar cartilage at 1.5T and 9.4 T: relationships with tissue mechanical properties[END_REF][START_REF] Juras | In vitro determination of biomechanical properties of human articular cartilage in osteoarthritis using multi-parametric MRI[END_REF]).

These viscoelastic model including degradation could be easily introduced in a predictive calculation of the hip joint to simulate the effects of osteoarthritis and loading rate on the contact areas, pressures, stresses, or others values like relative sliding velocity during various activities of daily life.

List of Figures

Fig. 1 -Experimental instrumented indentation test.

EH      [ , , ] H     [ , , ] EH    [ , , ] E     [ , , ] EH     J E 330 223

1

  Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation F. Richard a,b,* , M. Villars a,b , S. Thibaud a,b a Université de Franche-Comté, 25000 Besançon, France b FEMTO-ST Institute, Department of Applied Mechanics, UMR CNRS 6174, 24 rue de l'Epitaphe, 25000 Besançon, France.

  are associated to the viscoelastic model with the parameters obtained by the identification procedure.

  Fig. 7 shows the sensitivity of the response in relation with the different parameters from indentation test. Each sensitivities j δ are equivalent in term of magnitude. Sensitivity to other

E

  and H are about 3.5 times lower OA cartilage than normal, the critical damage between the proposed model and the experiments are presented in Fig. 8. A good correlation between model and experimental data is observed. The proposed model must be validated for intermediate degrees of osteoarthritis.

  with advanced coxarthrosis) but it must be validated for intermediate degrees of osteoarthritis. Magnetic resonance imaging (MRI) offers excellent soft tissue contrast and it has been developed for noninvasive assessment of the structure of articular cartilage.

Fig. 2 -Fig. 3 -Fig. 4 -

 234 Fig. 2 -Spherical indenter.

Fig. 5 -Fig. 7 -Fig. 8 -

 578 Fig. 5 -Inverse method procedure.

Table 5

 5

	shows the means and standard deviations

Figure1 Click here to download high resolution image Figure2 Click here to download high resolution image Figure3 Click here to download high resolution image Figure4 Click here to download high resolution image Figure5 Click here to download high resolution image Figure6 Click here to download high resolution image Figure7 Click here to download high resolution image Figure8 Click here to download high resolution image 1Table 1 -Some relevant characteristics of the normal subjects and patients with OA.

 1

	Sujet	Hip	Sex	Height	Mass	Age
		replacement		(mm)	(kg)	(years)
	Normal subjects
	1	P	F	165	63	81
	2	P	F	150	50	83
	3	P	F	168	65	73
	4	P	F	160	50	85
	5	P	F	160	60	94
	6	P	F	174	83	83
	Patients with OA
	7	T	M	172	85	77
	8	T	F	160	52	74
	9	T	M	172	83	79
	10	T	F	160	89	57
		M = Male; F = Female; P = Partial ; T =Total

Table 2 -Identifiability measurement (<100 good; >1000 poor).

 2

	k	4	3	3	3	3
	parameter set [ , , , ]

Table 3 -Viscoelastic parameters for normal cartilage. *(MPa), **MPa.s

 3 .49 1.38 29.37 3.48 0.49 1.42 30.98 4.12 0.40 1.52 34.63 5.17 0.49 1.67 136.91 Mesure 2 2.91 0.45 1.19 31.79 14.12 0.48 0.34 304.38 3.75 0.49 1.49 36.45 4.73 0.48 1.62 98.42 Mesure 3 3.10 0.45 1.15 28.82 5.39 0.49 0.09 199.82 3.51 0.46 1.21 29.75 4.78 0.43 0.30 173.35 Mesure 1 15.54 0.40 4.09 234.79 6.57 0.48 2.00 65.90 10.33 0.49 3.34 208.68 9.72 0.45 3.03 136.88 Mesure 2 17.67 0.46 5.46 238.03 5.67 0.49 1.47 51.22 10.59 0.47 0.21 396.46 9.72 0.46 3.43 140.86 Mesure 3 18.45 0.48 6.01 312.33 7.30 0.49 1.95 62.43 10.64 0.48 0.72 407.46 10.75 0.47 0.37 165.38

	Table3
	Area		Superior			Inferior			Posterior			Anterior
	Parameters	E *	ν H*	η **	E *	ν H *	η **	E *	ν H *	η **	E *	ν H *	η **
	Femur 1
	Mesure 1 11.57 0.40 3.24 12.23 13.31 0.49 0.15 367.29 9.67 0.49 0.35 531.12 10.22 0.48 0.86 353.11
	Mesure 2 10.48 0.49 2.34 277.93 13.00 0.48 0.26 356.78 9.36 0.49 0.58 536.22 10.37 0.49 0.46 344.75
	Mesure 3	9.56 0.49 2.28 270.73 13.31 0.48 0.15 367.29 9.96 0.48 0.45 323.58 9.92 0.49 0.14 430.38
	Femur 2
	Mesure 1 19.49 0.37 4.20 220.09 18.21 0.48 4.46 191.64 18.96 0.48 4.61 122.29 17.47 0.48 3.74 154.80
	Mesure 2 16.68 0.48 3.50 214.77 16.48 0.48 4.16 184.55 19.50 0.49 4.00 214.89 19.18 0.48 4.01 220.96
	Mesure 3 17.57 0.37 3.90 262.40 20.25 0.48 4.45 319.05 18.48 0.48 4.34 217.93 19.87 0.48 4.12 198.79
	Femur 3
	Mesure 1	8.35 0.41 3.05 110.04 11.34 0.48 2.79 113.02 8.66 0.48 2.64 131.80 12.72 0.40 2.31 83.30
	Mesure 2	8.28 0.43 3.11 128.57 12.53 0.48 2.91 128.69 8.41 0.48 2.47 128.15 10.46 0.48 2.54 94.10
	Mesure 3				11.84 0.48 2.83 127.24 12.87 0.48 2.80 139.09 12.41 0.49 2.85 108.37
	Femur 4
	Mesure 1 3.33 0Femur 5
	Mesure 1 10.30 0.48 0.08 499.21 3.44 0.48 0.85 26.29 12.12 0.48 2.39 344.33 14.12 0.48 5.78 366.09
	Mesure 2	7.79 0.47 0.08 456.44 2.03 0.48 0.49 11.81 10.94 0.48 3.01 284.87 16.48 0.46 0.14 810.59
	Mesure 3	7.74 0.48 0.24 462.65 2.97 0.48 0.72 18.47 10.22 0.48 2.55 206.74 14.05 0.46 1.74 595.20
	Femur 6

Table 4 -Viscoelastic parameters for OA cartilage. *(MPa), **MPa.s

 4

	Table4
	Area	Superior	Inferior	Posterior	Anterior
	Parameters E *	ν	H* η ** E *	ν	H * η ** E *	ν	H * η ** E *	ν	H * η **
	Femur 7
	Mesure 1		2.23 0.49 0.81 16.52 3.95 0.40 0.90 20.33
	Mesure 2		5.95 0.49 0.85 13.26 3.84 0.49 0.94 23.50
	Mesure 3				3.54 0.49 0.93 22.35
	Femur 8
	Mesure 1				5.23 0.45 1.02 97.78
	Mesure 2				3.89 0.45 0.21 82.49
	Mesure 3				4.87 0.45 0.00 149.17
	Femur 9
	Mesure 1		1.98 0.49 0.29 13.26
	Mesure 2		2.17 0.49 0.32 14.19
	Mesure 3		2.43 0.49 0.29 13.62
	Femur 10
	Mesure 1						2.61 0.45 0.49 18.85
	Mesure 2						2.15 0.45 0.79 20.17
	Mesure 3

Table 5 -

 5 Distribution of parameters E, , H and  for

normal and OA cartilage (n: number of samples, NC: Non Calculable, SD: Standard Deviation).

	Table5
	Parameter	Position	Normal			OA
			n	Mean	SD *	n	Mean	SD *
	E (MPa)	Global	24	10.9	5.0	5	3.4	1.1
		Superior	6	11.1	5.5	0	NC	NC
		Inferior	6	10.1	5.6	2	3.1	1.3
		Posterior	6	10.7	4.9	2	4.2	0.6
		Anterior	6	11.8	4.7	1	2.4	NC
		Global	24	0.47	0.02	5	0.47	0.02
		Superior	6	0.45	0.03	0	NC	NC
		Inferior	6	0.48	0.00	2	0.49	0.00
		Posterior	6	0.48	0.01	2	0.46	0.01
		Anterior	6	0.47	0.01	1	0.45	NC
	H (MPa)	Global	24	2.2	1.4	5	0.6	0.3
		Superior	6	2.7	1.7	0	NC	NC
		Inferior	6	1.8	1.7	2	0.6	0.4
		Posterior	6	2.2	1.5	2	0.7	0.4
		Anterior	6	2.2	1.2	1	0.6	NC
	 (MPa.s)	Global	24	218.7	150.6	5	36.0	41.4
		Superior	6	217.9	150.4	0	NC	NC
		Inferior	6	162.6	125.4	2	14.3	0.8
		Posterior	6	238.6	153.9	2	65.9	62.0
		Anterior	6	256.2	191.5	1	19.5	NC
								1

Table 6 -

 6 Thermodynamic variables.

	Table6
	State variables			Associated variables
	Observable	Internal
				σ
		ε	e	σ
		ε	v	σ
		α	X
		D	Y

Acknowledgments

We thank the Pr P. Garbuio for his help and all the Department of Orthopaedic Surgery of the University Hospital of Besançon for collecting femoral heads. The samples were cut at the Besançon anatomy laboratory. We also thank S. Mith, L. Carpentier and V. Gunes for their assistance on the indentation tests, P. Clément for the development of the experimental protocol and E. Fournier in the statistical analysis results.

List of Tables

 Table 1 -Some relevant characteristics of the normal subjects and patients with OA.


Alexia Bontempi

Damien Teyssieux

Jean Friedt

Laurent Thiery

Damien Hermelin

Pascal Vairac

Photo-thermal Quartz Tuning Fork excitation for Dynamic Mode Atomic Force Microscope


Frédéric Holweck
email: frederic.holweck@utbm.fr

Jean-Gabriel Luque
email: jean-gabriel.luque@univ-rouen.fr

Michel Planat
email: michel.planat@femto-st.fr

Singularity of type D 4 arising from four qubit systems

Keywords: Quantum Information Theory, Entangled states, Simple singularities of hypersurfaces, Dynkin diagrams, Hyperdeterminant PACS :02.10.Xm, 02.40.-k, 03.65.Aa, 03.65.Fd, 03.65.Ud Msc: 32S25, 32S30, 15A69, 14M17, 15A72

An intriguing correspondence between four-qubit systems and simple singularity of type D 4 is established. We first consider the algebraic variety X of separable states within the projective Hilbert space P(H) = P 15 . Then, cutting X with a specific hyperplane H, we prove that the X-hypersurface, defined from the section X ∩ H ⊂ X, has an isolated singularity of type D 4 ; it is also shown that this is the "worstpossible" isolated singularity one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of X, which is nothing but the Cayley hyperdeterminant of type 2 × 2 × 2 × 2, can be expressed in terms of the SLOCC invariant polynomials as the discriminant of the miniversal deformation of the D 4 -singularity.

I. INTRODUCTION

Several branches of geometry and algebra tend to play an increasing role in quantum information theory. We have in mind algebraic geometry for describing entanglement classes of multiple qubits [START_REF] Brody | Geometric quantum mechanics[END_REF][START_REF] Heydari | Geometrical Structure of Entangled States and the Secant Variety[END_REF][START_REF] Holweck | Geometric descriptions of entangled states by auxiliary varieties[END_REF][START_REF] Miyake | Multipartite entanglement in 2 × 2 × n quatum systems[END_REF] , representation theory and Jordan algebras for entanglement and the black-hole/qubit correspondence 3-5 , and geometries over finite fields/rings for deriving point-line configurations of observables relevant to quantum contextuality [START_REF] Levay | Three-qubit operators, the split Cayley hexagon of order two, and black holes[END_REF][START_REF] Planat | Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon[END_REF][START_REF] Saniga | Mermin's pentagram as an ovoid of PG(3, 2)[END_REF] . The topology of hypersurface singularitites, and the related Coxeter-Dynkin diagrams, represent another field worthwhile to be investigated in quantum information, as shown in this paper.

Dynkin diagrams are well known for classifying simple Lie algebras, Weyl groups, subgroups of SU(2) and simple singularities, i.e. isolated singularities of complex hypersurfaces that are stable under small perturbations. More precisely, if we consider simple-laced Dynkin diagrams, i.e. diagrams of type A -D -E, we find objects of different nature classified by the same diagrams: Type Lie algebra Subgroup of SU(2) Hypersurface with simple singularity

A n sl n+1 (C) cyclic group x n+1 1 + x 2 2 + • • • + x 2 k = 0 D n so 2n (C) binary dihedral group x n-1 1 + x 1 x 2 2 + x 2 3 + • • • + x 2 k = 0 E 6 e 6 binary tetrahedral x 4 1 + x 3 2 + x 2 3 + • • • + x 2 k = 0 E 7
e 7 binary octahedral

x 3 1 x 2 + x 3 2 + x 2 3 + • • • + x 2 k = 0 E 8 e 8 binary icosahedral x 5 1 + x 3 2 + x 2 3 + • • • + x 2 k = 0
A challenging question in mathematics is to understand these ADE-correspondences by establishing a direct construction from one class of objects to the other. For instance, the construction of surfaces with simple singularities from the corresponding subgroup of SU (2) is called the McKay correspondence. A construction due to Grothendieck allows us to recover the simple singularities of a given type from the nullcone (the set of nilpotent elements) of the corresponding simple Lie algebra. For an overview of such ADE correspondences, see

Ref [START_REF] Slodowy | Simple singularities and simple algebraic groups[END_REF][START_REF] Slodowy | Platonic solids, Kleinian singularities, and Lie groups[END_REF] and references therein.

Another construction connecting simple Lie algebras and simple singularities is due to Knop [START_REF] Knop | Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten[END_REF] . In his construction, Knop considers a unique smooth orbit, X, for the adjoint action of Lie group G on the projectivization of its Lie algebra P(g) and cuts this variety by a specific hyperplane. The resulting X-hypersurface has a unique singular point of the same type as g.

Looking at ADE-correspondences in the context of QIT is a way to understand the role played by those diagrams in this field. In different classification schemes of four-qubit systems, the Dynkin diagram D 4 has already appeared thanks to the role played by the Lie algebra so (8) (that is the type D 4 ). In the present paper, we will establish a correspondence between four-qubit systems and D 4 -singularities by using a construction inspired by Knop's paper. In other words, we will establish an ADE-type correspondence between SO(4, 4) and singularities of type D 4 using the Hilbert space of four qubits.

Let H = C 2 ⊗ C 2 ⊗ C 2 ⊗ C 2 be
(C) × SL 2 (C) × SL 2 (C) × SL 2 (C). It is well
known that G acts transitively on the set of separable states. The projectivization of the corresponding orbit -also called the highest weight orbit -is the unique smooth orbit X

for the action of G on P(H), that is

X = P(G.|0000 ) = {The set of separable states} ⊂ P 15 .
A parametrization of X is given by the Segre embedding of four projective lines [START_REF] Heydari | Geometrical Structure of Entangled States and the Secant Variety[END_REF][START_REF] Holweck | Geometric descriptions of entangled states by auxiliary varieties[END_REF] φ :

   P 1 × P 1 × P 1 × P 1 → P 15 ([w 0 : w 1 ], [x 0 : x 1 ], [y 0 : y 1 ], [z 0 : z 1 ]) → [w 0 x 0 y 0 z 0 : • • • : W J : • • • : w 1 x 1 y 1 z 1 ]
where W J = w i x j y k z l for J = {i, j, k, l} ∈ {0, 1} 4 and the monomial order is such that

W J 1 ≺ W J 2 if 8i 1 + 4j 1 + 2k 1 + l 1 ≤ 8i 2 + 4j 2 + 2k 2 + l 2 .
A hyperplane equivalently, X ∩ Ψ|, will be the hypersurface of X given by

Ψ|φ(P 1 × P 1 × P 1 × P 1 ) = 0≤i,j,k≤1 h ijkl w i x j y k z l = 0. (1)
To state our main Theorem, let us recall that the ring of polynomials invariant under G is generated by 4 invariants [START_REF] Luque | The polynomial invariants of four qubits[END_REF] . Let us denote by Ĩ1 , Ĩ2 , Ĩ3 , Ĩ4 a choice of four generators of the ring of invariants (that choice will be explained in Section III B), i.e.

C[H] G = C[ Ĩ1 , Ĩ2 , Ĩ3 , Ĩ4 ].
The quotient map Φ : H → C 4 is defined by Φ(x) = ( Ĩ1 (x), Ĩ2 (x), Ĩ3 (x), Ĩ4 (x)). The main result of this article is the following theorem: Let us denote by (f, 0) the germ of a holomorphic function, f : (C k , 0) → (C, 0) at 0, and by O k the set of all those germs. We consider the group D k of biholomorphic maps

Theorem 1. Let H = Ψ|
g : (C k , 0) → (C k , 0) acting of O k such that g.f = f • g -1 .
A singularity is an equivalence class of a germ (f, 0) such that ∂f ∂x i (0) = 0 for i = 1, . . . , k. In other words, a singularity is an orbit in O k and we will write [(f, 0)] for the orbit of the representative (f, 0). We denote by S k ⊂ O k the set of all singular germs. Let f be a representative of a singularity and let us denote by

A = ∂ 2 f ∂x i ∂x j (0) i,j
the corresponding Hessian matrix. The corank of the germ (f, 0) is the dimension of the kernel of A. From the definition of the action of D k it follows that equivalent germs will have the same corank, which means that the corank is an invariant of a singularity.

Definition II.1. A singularity is said to be non-degenerate, or quadratic, or of the Morse type, if, and only if, its corank is zero.

The Morse Lemma [START_REF] Milnor | Singular Points of Complex Hypersurfaces[END_REF] ensures that if (f, 0) is a non-degenerate singular germ, then f ∼

x 2 1 + • • • + x 2 k . The non-degenerate singularity is a dense orbit in S k . Assume that [(f, 0)] is a singularity of corank l, a generalization of Morse's Lemma 1 tells us that f ∼ h(x 1 , . . . , x l ) + x 2 l+1 + • • • + x 2
k and leads to an equivalence relation between germs of distinct number of variables.

Definition II.2. Two function germs f : (C k , 0) → (C, 0) and g : (C m , 0) → (C, 0), with k < m, are said to be stably equivalent if, and only if,

f (x 1 , . . . , x k ) + x 2 k+1 + • • • + x 2 m ∼ g(x 1 , . . . , x m ).
Remark II.1. In terms of the last definition we can compare singularities of functions which do not have the same number of variables. Adding quadratic terms of full rank in new variables do not affect the classification of the singular type.

Another important invariant of singular germs is the famous Milnor number [START_REF] Milnor | Singular Points of Complex Hypersurfaces[END_REF] . Let (f, 0) be a singular germ and consider

I ∇f = O k < ∂f ∂x 1 (0), . . . , ∂f ∂x k (0) > the gradient ideal.
Definition II.3. The Milnor number µ of a singular germ (f, 0) is equal to the dimension of the local algebra of (f, 0), i.e. the quotient of the algebra O k by I ∇f ,

µ = dim C (O k I ∇f ) .
The critical point 0 of the function f will be isolated if, and only if, its Milnor number is finite.

Let us now state what, in the sense of Vladimir Arnol'd, a simple singularity is .

Definition II.4. The orbit [(f, 0)] is a simple singularity if, and only if, a sufficiently small neighborhood of (f, 0) intersects S k with a finite number of non-equivalent orbits.

Remark II.2. If we consider a representative of a non-degenerate singularity

f ∼ x 2 1 + • • • + x 2
k , a small perturbation of f in S k , i.e. f + εh with h ∈ S k , will still have a Hessian of full rank for ǫ small. Thus f ∼ f + εh, which means that non-degenerate singularity is the most stable type of singularity. We can rephrase Definition II.4 by saying that [(f, 0)] is a simple singularity if, and only if, a small perturbation of a representative f will only lead to a finite number of non-equivalent singularities.

In his classification of simple singularities 1 , Arnol'd proved that being simple is equivalent to the following conditions:

• µ < +∞, • corank ∂ 2 f ∂x i ∂x j (0) ≤ 2,
• if corank ∂ 2 f ∂x i ∂x j (0) = 2 the cubic term in the degenerate direction of the Hessian is non-zero,

• if corank = 2 and the cubic term is a cube then µ < 9.

With these conditions Arnol'd obtained the classification of simple singularities into five different types (Table I).

Type A n D n E 6 E 7 E 8
Normal forms x n+1 x n-1 + xy 2 x 3 + y 4 x 3 + xy 3 x 3 + y 5

Milnor number n n 6 7 8

Table I. Simple singularities.

Remark II.3. The functions given in Table I are stably equivalent to the hypersurfaces given in the introduction. They are also clearly equivalent to the rational double points of algebraic surfaces.

The classification given by Arnol'd furnishes an algorithm to test if a singularity is simple or not.

Algorithm II.1. Let (f, 0) be a singularity.

• Compute µ; if µ = ∞ the singularity is not isolated (and not simple),

• If not, compute r = corank(Hess(f, 0)).

if r ≥ 3, the singularity is not simple,

-if r = 1, the singularity is of type A µ ,
if r = 2, then * if the cubic term in the degenerate directions is non-zero and is not a cube, then the singularity is of type D µ , * if the cubic term in the degenerate directions is a cube and µ < 9, then the singularity is of type E µ , * if not, the singularity is not simple.

In the next section we will follow this algorithm to compute the singular type of a given hyperplane section.

B. Computing singularities of hyperplane sections

Before we prove the first proposition, let us consider two examples in order to explain how we calculate the singular type of a hyperplane section.

Example II.1. Let H ∈ P(H * ) be a hyperplane, or a linear form, given by H = Ψ 1 | = 0011| + 1100|. The corresponding hyperplane section X ∩ H is tangent to |1111 . Indeed, a tangent vector to X at |1111 will be of the form |v = α|0111 + β|1011 + γ|1101 + δ|1110 and it is clear that Ψ 1 |v = 0. The homogeneous form of the linear section X ∩ H corresponds to its restriction to (the cone over) X, that is to

f (w 0 , w 1 , x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) = w 0 x 0 y 1 z 1 + w 1 x 1 y 0 z 0 .
In a non-homogeneous form f can be written in the chart corresponding to w 1 , x 1 , y 1 , z 1 = 1 as f (w 0 , x 0 , y 1 , z 1 ) = w 0 x 0 + y 0 z 0 . In this chart the point |1111 has coordinates (0, 0, 0, 0)

and (we can forget about the subscripts) the hyperplane section is a hypersurface of X defined (locally) by the equation

f (w, x, y, z) = wx + yz = 0.
This hypersurface has a unique singularity ∂f ∂w (a), ∂f ∂x (a), ∂f ∂y (a), ∂f ∂z (a) = (0, 0, 0, 0) ⇔ a = (0, 0, 0, 0), which corresponds to |1111 , and the Hessian matrix

       0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0       
is of the full rank. One concludes that (X ∩ H, |1111 ) is an isolated singularity of type A 1 and we denote it by (X ∩ H, |1111 ) ∼ A 1 , or, equivalently, by

(X ∩ Ψ 1 |, |1111 ) ∼ A 1 .
Example II.2. Let us consider the hyperplane section defined by H = Ψ 2 | = 0000| + 1011| + 1101| + 1110| ∈ H * . This section X ∩ H is tangent to |0111 . It is clear that a tangent vector to X at |0111 will be of the form |v = α|1111 + β|0011 + γ|0101 + |0110 and H|v = 0. The homogeneous linear form corresponding to X ∩ H is f (w 0 , w 1 , x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) = w 0 x 0 y 0 z 0 + w 1 x 0 y 1 z 1 + w 1 x 1 y 0 z 1 + w 1 x 1 y 1 z 0 . In the chart one can check that µ x=(0,0,0,0) (f ) = 4 and the rank of the Hessian

w 0 = x 1 = y 1 = z 1 =
       0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0        is 2. Thus, we conclude that (X ∩ H, |0111 ) ∼ D 4 , or, equivalently, (X ∩ Ψ 2 |, |0111 ) ∼ D 4
(i.e. the unique isolated singularity where the corank equals 2 and µ = 4).

We can now prove our first proposition.

Proposition II.1. Let X ∩ H be a singular hyperplane section of the variety of separable states for four-qubit systems, i.e. X = P 1 × P 1 × P 1 × P 1 , with an isolated singularity

x ∈ X ∩ H. Then the singularity (X ∩ H, x) will be of type A 1 , A 2 , A 3 or D 4 and each type can be obtained by such a linear section of X.

Proof. To prove Proposition II.1, we compute the singular type of all possible hyperplane sections of X. As the variety X is G-homogeneous, the singular type of X ∩ H will be identical for any representative of the G orbit of H. By the duality of the Hilbert space, a hyperplane H corresponds to a point h ∈ P(H). But the G orbits of P(H) have been classified

by Verstraete et al. [START_REF] Verstraete | Four qubits can be entangled in nine different ways[END_REF] (with a corrected version provided by Chterental and Djokovic 7 ).

According to Verstraete et al.'s classification, the G-orbits of the four-qubit Hilbert space consist of 9 families (3 families are parameter free and 6 of them depend on parameters) and normal forms for each family are known [START_REF] Chterental | Normal forms and tensor ranks of pure states of four qubits[END_REF][START_REF] Verstraete | Four qubits can be entangled in nine different ways[END_REF]

L 0 5⊕3 0000| + 0101| + 1000| + 1110| non-isolated L 0 3⊕1 0 3⊕1 0000| + 0111| non-isolated
Table II. Hyperplanes and the corresponding sections which do not depend on parameters.

Remark II.4. Tables III, IV, V show that the classification of entangled states into 9

families can be refined according to the singular type of the corresponding section. The singular type of the linear section X ∩ Ψ| is an invariant of the G-orbit of |Ψ and may be used to distinguish two non-equivalent classes of entanglement. Thus, the values of the parameters which distinguish the sections indicate how we can decompose further the classification. However, to fully distinguish non-equivalent sections from their singular type, it would be necessary to investigate more precisely the non-isolated singular sections.

Remark II.

+ a-b 2 ( 0110| + 1001|) a = b = 0 non-isolated + i √ 2 ( 0001| + 0010| -0111| -1011|) L a 2 b 2 a(|0000 + |1111 ) + b(|0101 + |1010 ) a, b generic smooth section +|0110 + |0011 a = 0 or b = 0 non-isolated a = b = 0 non-isolated L abc 2 a+b 2 ( 0000| + 1111|) + a-b 2 ( 0011| + 1100|) a, b, c generic A 1 (a unique singularity) c( 1010| + 0101|) + 0110| a = ±b A 1 c = 0 A 1 a = ±b = ±c non-isolated a = c = 0 or b = c = 0 non-isolated a = b = c = 0 non-isolated G abcd a+d 2 (|0000 + |1111 ) + a-d 2 (|0011 + |1100 ) a, b, c, d generic smooth section + b+c 2 (|0101 + |1010 ) + b-c 2 (|0110 + |1001 ) see Table IV A 1 see Table V non-isolated
Table III. Hyperplanes and the corresponding sections which do depend on parameters.

III. THE CAYLEY 2 × 2 × 2 × 2 HYPERDETERMINANT AND THE D 4 -DISCRIMINANT
Another fundamental concept associated with a simple singularity is its discriminant, i.e.

the locus that parametrizes the deformation of the singular germs. In this section, we will show that the discriminant of the D 4 -singularity is linked to the dual variety, in the sense of the projective duality, of the set of separable four-qubit states.

A. Discriminant of the miniversal deformation of the singularity

Consider a holomorphic germ f : (C k , 0) → (C, 0) with a simple isolated singularity of Milnor number µ(f, 0) = n. A miniversal deformation 1 of the germ f is given by

f + λ i g i , where (g 1 , . . . , g n ) is a basis of O k I ∇f . Definition III.1. The discriminant Σ ⊂ C n is the subset of values (λ 1 , . . . , λ n ) ∈ C n such that the miniversal deformation f + λ i g i is singular, i.e. Σ = {(λ 1 , . . . , λ n ) ∈ C n , ∆(f + n i=1 λ i g i ) = 0},
where ∆ is the usual notion of discriminant.

Remark III.1. The discriminant parametrizes all singular deformations of (f, 0). It is known [START_REF] Wirthmüller | Singularities determined by their discriminants[END_REF] that for hypersurfaces endowed with a simple singularity, the discriminant of the singularity characterizes its type.

Example III.1. Let (f, 0) be a singularity of type A n , i.e. f ∼ x n+1 . Then O 1 I ∇x n+1 =< 1, x, . . . , x n-1 >. Thus, a miniversal deformation of f is

F (x, λ) = x n+1 + λ 1 x n-1 + λ 2 x n-2 + • • • + λ n .
The corresponding discriminant is the hypersurface Σ An ⊂ C n defined by

∆(x n+1 + λ 1 x n-1 + λ 2 x n-2 + • • • + λ n ) = 0.
In the case where n = 2, i.e. when f ∼ x 3 is a singularity of type A 2 , then its discriminant is given by ∆(x 3 +λ 1 x+λ 2 ) = 0, i.e. the discriminant is a cubic curve defined by -4λ 3 1 -27λ 2 2 = 0.

The following example will be useful to prove the main result of the next section.

Example III.2. Consider now a singular germ (f, 0) of type D n ; then f ∼ x n-1 + xy 2 .

A basis of the local algebra O 2 I ∇(x n-1 +xy 2 ) is (1, x, . . . , x n-2 , y) and, hence, a miniversal deformation is

F (x, y, λ) = x n-1 + xy 2 + λ 1 x n-2 + . . . λ n-2 x + λ n-1 + λ n y.
Its discriminant is given by

∆(x n-1 + xy 2 + λ 1 x n-2 + . . . λ n-2 x + λ n-1 + λ n y) = 0.
(2)

The following lemma proposes an alternative expression of the discriminant of the D n singularities.

Lemma 1. The discriminant of the miniversal deformation of f ∼ x n-1 + xy 2 is the hypersurface Σ Dn ⊂ C n defined by

∆(λ 1 , . . . , λ n ) = ∆(t n + λ 1 t n-1 + • • • + λ n-1 -( 1 2 λ n ) 2 ) = 0. (3)
Proof. Let us denote by Σ ⊂ C n the locus defined by eq. (3). To prove that equations ( 2)

and (3) are equivalent, we will show that Σ = Σ Dn .

To this end, let us characterize the hypersurfaces Σ and Σ Dn . Given the definition of the discriminant, the expression ∆(F (t, λ)) = 0 means there exists t 0 such that F (t 0 ) = 0 and ∂F ∂t (t 0 ) = 0. In other words, (λ 1 , . . . , λ n ) ∈ Σ if, and only if, there exists t 0 such that   

t n 0 + λ 1 t n-1 0 + • • • + λ n-1 t 0 -( 1 2 λ n ) 2 = 0, nt n-1 0 + (n -1)λ 1 t n-2 0 + • • • + λ n-1 = 0.    (4)
Similarly, (λ 1 , . . . , λ n ) ∈ Σ Dn if, and only if, there exists (x 0 , y 0 ) such that F (x 0 , y 0 , λ) = ∂F ∂x (x 0 , y 0 , λ) = ∂F ∂y (x 0 , y 0 , λ) = 0, i.e.

         x n-1 0 + x 0 y 2 0 + λ 1 x n-2 0 + • • • + λ n-2 x 0 + λ n-1 + λ n y 0 = 0, (n -1)x n-2 0 + y 2 0 + (n -2)λ 1 x n-3 0 + . . . λ n-2 = 0, 2x 0 y 0 + λ n = 0.          (5)
Let us assume that λ n = 0, then if (λ 1 , . . . , λ n ) ∈ Σ there exists t 0 such that the system (4) is satisfied. It is obvious that λ n = 0 implies t 0 = 0 and thus one can check that the system (5) is also satisfied for (x 0 , y 0 ) = (t 0 , -λ n 2t 0

). This proves that (λ 1 , . . . , λ n ) ∈ Σ Dn . On the other hand, if (λ 1 , . . . , λ n ) ∈ Σ Dn and (x 0 , y 0 ) is a solution of (5), then necessarily y 0 = -λn x 0 . One can further show that t 0 = x 0 is a solution of (4) and, therefore, (λ 1 , . . . , λ n ) ∈ Σ. Let us now consider the case λ n = 0. Then (λ 1 , . . . , λ n ) ∈ Σ for a given t 0 implies (λ 1 , . . . , λ n ) ∈ Σ Dn for (x 0 , y 0 ) = (t 0 , 0). On the other hand, let us assume (λ 1 , . . . , λ n ) ∈ Σ Dn for a given (x 0 , y 0 ).

The equation 2x 0 y 0 + λ n = 0 forces x 0 or y 0 to be zero. But if x 0 = 0 then necessarily also a n-1 = 0 and t 0 = 0 is a solution of (4), proving (λ 1 , . . . , λ n ) ∈ Σ. If x 0 = 0, then y 0 = 0 and t 0 = x 0 is a solution of (4), proving again

(λ 1 , . . . , λ n ) ∈ Σ. ✷ B. Hyperdeterminant of format 2 × 2 × 2 × 2 and D 4 -discriminant
The hyperdeterminant of format 2×2×2×2 is a SLOCC-invariant polynomial generalizing the ideas of Cayley for defining a higher dimensional counterpart of the determinant for multimatrices. From a geometrical perspective, the hyperdeterminant and its generalization have been studied by Gelfand, Kapranov and Zelevinsky 10 in terms of the concept of dual varieties. The geometric definition is the following one: Let X ⊂ P(V ) be a (smooth) projective variety, we denote by X * the dual variety of X, defined by

X * = {H ∈ P(H * ), ∃x ∈ X, T x X ⊂ H}.
For the case X = P 1 × P 1 × P 1 × P 1 , the dual variety, denoted X * , is a SLOCC-invariant hypersurface, whose equation is called the hyperdeterminant of format 2 × 2 × 2 × 2. This invariant polynomial, denoted as ∆ 4 , is an irreducible polynomial (X * is irreducible because X is), its degree is 24, and the corresponding hypersurface is singular [START_REF] Miyake | Multipartite entanglement in 2 × 2 × n quatum systems[END_REF][START_REF] Weyman | Singularities of Hyperdeterminants[END_REF] in codimension 1.

By definition, X * parametrizes the singular hyperplane sections of X (alternatively, H / ∈ X * is equivalent to saying that X ∩ H is a smooth section).

It would be difficult to quote all the papers in QIT (as well as in theoretical physics)

referring to the concept of hyperdeterminant [START_REF] Borsten | Black holes, qubits and octonions[END_REF][START_REF] Borsten | Explicit orbit classification of reducible Jordan algebras and Freudenthal triple systems[END_REF][START_REF] Holweck | Geometric descriptions of entangled states by auxiliary varieties[END_REF][START_REF] Levay | Three-qubit operators, the split Cayley hexagon of order two, and black holes[END_REF][START_REF] Luque | The polynomial invariants of four qubits[END_REF][START_REF] Miyake | Classification of multipartite entangled states by multidimensional determinants[END_REF][START_REF] Miyake | Multipartite entanglement in 2 × 2 × n quatum systems[END_REF] , but it is clear that this invariant polynomial plays a central role in understanding the symmetries involved in the SLOCC group action.

In the case of four-qubit systems, the ring of polynomials invariant under the group SLOCC was determined by Luque and Thibon [START_REF] Luque | The polynomial invariants of four qubits[END_REF] . It is a finitely-generated ring with four generators B, L, M and D, of respective degrees 2, 4, 4 and 6 (explicit expressions, with the same notations, can be found in Ref [START_REF] Holweck | Entanglement of four qubit systems: a geometrical atlas with polynomial compass I (the finite world)[END_REF] ). In other words, any SLOCC-invariant polynomial P

over H = C 2 ⊗ C 2 ⊗ C 2 ⊗ C 2 belongs to C[B, L, M, D].
In particular, the hyperdeterminant of format 2 × 2 × 2 × 2 can be expressed as a polynomial in the generators of the ring of invariants and one gets 18 Lévay's motivation to define this new set of generators was to obtain a more geometrical and uniform description of those polynomials, as it is shown in his paper [START_REF] Lévay | On the geometry of four-qubit invariants[END_REF] . These news invariants I 1 , I 2 , I 3 , I 4 allow one to get a new expression of ∆ 4 . In particular, Lévay proved (Eq (56) [START_REF] Lévay | On the geometry of four-qubit invariants[END_REF] ) that

∆ 4 = 1 256 ∆(t 4 -(4I 1 )t 3 + (6I 2 )t 2 -(4I 3 )t + I 2 4 ) (6)
(where ∆ is the discriminant of the polynomial in the t variable). This particular finding leads to the following claim:

Proposition III. (6) appears also in the conclusion of a previous paper involving the first two authors [START_REF] Holweck | Entanglement of four qubit systems: a geometrical atlas with polynomial compass I (the finite world)[END_REF] . When we evaluate this quartic on the G abcd state, i.e. when we consider the quartic

Q(t) = t 4 -(4I 1 (G abcbd ))t 3 + (6I 2 (G abcd ))t 2 -(4I 3 (G abcd ))t + I 4 (G abcd ) 2 , one obtains Q(t) = (t -a 2 )(t -b 2 )(t -c 2 )(t -d 2 ).
The state G abcd will cancel ∆ 4 if and only if the quartic Q has (at least) a repeated root, i.e.

there is (at least) a relation (among the parameters) of type m = ±n with m ∈ {a, b, c, d} and n ∈ {a, b, c, d} \ m. Obviously this condition is satisfied by all values of the parameters {a, b, c, d} of Tables IV and V because the corresponding states belong to the dual of X (and thus vanish ∆ 4 ). However the relations between the hyperplane sections of Tables IV and V and the number of repeated roots of the quartic Q is probably worth to be further investigate.

Remark III.4. Proposition III.1 establishes a connexion between two types of discriminant.

As pointed out earlier, the dual variety of X is a discriminant in the sense that it parametrizes the singular hyperplane sections of X. The D 4 -discriminant parametrizes the singular deformation of the germ x 3 +xy. The most singular deformation of x 3 +xy 2 +λ 1 x 2 +λ 2 x+λ 3 +λ 4 y is obtained for (λ 1 , λ 2 , λ 3 , λ 4 ) = (0, 0, 0, 0) . The preimage via the quotient map of (0, 0, 0, 0) is given by the zero-locus of (all) invariant polynomials

Φ -1 (0, 0, 0, 0) = {|Ψ , Ĩ1 (|Ψ ) = Ĩ2 (|Ψ ) = Ĩ3 (|Ψ ) = Ĩ4 (|Ψ ) = 0}.
This set does not depend on our choice of Φ and, after projectivization, it corresponds to a well-known variety N ⊂ P(H), the nullcone, which was already invoked to describe the entanglement classes of a four-qubit system [START_REF] Borsten | Four-Qubit Entanglement Classification from String Theory[END_REF][START_REF] Holweck | Entanglement of four qubit systems: a geometrical atlas with polynomial compass I (the finite world)[END_REF] . As first pointed out in Ref

X ∩ H ∼ D 4 ←→ H ∈ N smooth ⊂ X *   Φ x 3 + xy 2 ←→ (0, 0, 0, 0) ∈ Σ D 4 ⊂ C 4 .

IV. CONCLUSION

We have introduced a new construction that assigns to any quantum state |Ψ a complex hypersurface defined by the hyperplane section X ∩ Ψ| of the set X of all separable states. This hypersurface may have singular points, which can be studied using the theory of singularity. Because the variety of separable states is G-homogeneous, this construction is G-invariant and two states |Ψ 1 and |Ψ 2 which do not define equivalent (singular) hyperplane sections will not be SLOCC equivalent. For four qubits, this construction allowed us to realize the singularity of type D 4 as a specific hyperplane section and we also proved that no "higher" isolated singularities can be obtained by this construction.

The D 4 singularity is obtained only when we consider the section X ∩ Ψ|, where |Ψ is a point of an orbit of maximal dimension of the nullcone [START_REF] Holweck | Entanglement of four qubit systems: a geometrical atlas with polynomial compass I (the finite world)[END_REF] (i.e. a smooth point of the nullcone).

This is emphasized when we rephrase the notion of Cayley 2 × 2 × 2 × 2 hyperdeterminant, i.e. the dual equation of the set of separable states, in terms of the discriminant of a D 4singularity. The stratification of the discriminant Σ D 4 in terms of mutiplicities induces a stratification of the dual variety X * -a variety that is of great relevance in the study of entanglement of four qubits, as pointed out by Miyake [START_REF] Miyake | Classification of multipartite entangled states by multidimensional determinants[END_REF][START_REF] Miyake | Multipartite entanglement in 2 × 2 × n quatum systems[END_REF] .

Although the correspondence between four qubits and simple Lie algebra of type

  For instance, Verstraete et al's classification 26 is based on the classification of the SO(4) × SO(4) ⊂ SO(8) orbits on M 4 (C). Chterental and Djokovic 7 use the same group action and refer to (Remark 5.3 of Ref 7) the Hilbert space of four qubits as a subspace of so(8) whose SLOCC orbits arise from the trace of the adjoint SO(8) orbits. In their study of the four-qubit classification from the string theory point of view, Borsten et al 2 employ a correspondence between nilpotent orbits of so(4, 4) (the real form of so(8) with signature (4, 4)) and nilpotent orbits of four-qubit systems. Last but not least, the relation between so(8) and four-qubit systems has been pointed out byLévay 15 in his paper on the black-hole/qubit correspondence. In this paper Lévay describes the Hilbert space of four qubits as the tangent space of SO(4, 4)/(SO(4) × SO(4)).

1

 1 the form becomes a hypersurface defined by xyz + wx + wy + wz = 0 and (0, 0, 0, 0) is the only singularity of this hypersurface. Using the software SINGULAR[START_REF] Decker | Schönemann SINGULAR 3.1.6: a computer algebra system for polynomial computations[END_REF]

2 - 6 (B 2 +

 262 4(L + M)) 2 -24(BD + 2LM) and T = 1 216 ((B 2 -4(L + M)) 3 -3(B 2 -48(L + M))(BD + 2LM) + 216D 2 ). In his attempt to give a geometric meaning of the invariants of Luque and Thibon, Lévay 16 introduced some alternatives generators which are related to the previous ones as I 1 = 1 2 B, I 2 = 1 2L -4M), I 3 = D + 1 2 BL and I 4 = L.

  2 , the nullcone admits a stratification into 9 distinguished classes of orbits which relate to the 9 families of Verstraete et al.'s classification. To emphasize the connexion with the D 4 singular type, let us point out that H = Ψ 2 | = 0000| + 1011| + 1101| + 1110| (the hyperplane of Example II.2) is a smooth point of N and this characterizes the hyperplanes of X with a D 4 -singular point. This correspondence can diagrammatically be sketched as:

D 4

 4 is now clear from the action of the SLOCC group, the correspondence established in this paper between four qubits and a simple singularity of type D 4 is rather surprising and points out to a novel relationship between simple Lie algebra and simple singularity of type D 4 .{a = 0, b = 0, c = 0, d = d}, {a = 0, b = 0, c = c, d = 0}, {a = 0, b = b, c = 0, d = 0}, {a = a, b = 0, c = 0, d = 0}, {a = a, b = d, c = d, d = d}, {a = a, b = -c, c = c, d = -c}, {a = a, b = -d, c = d, d = d}, {a = a, b = -d, c = -d, d = d}, {a = b, b = b, c = 0, d = b}, {a = b, b = b, c = 0, d = -b}, {a = c, b = 0, c = c, d = c}, {a = c, b = 0, c = c, d = -c}, {a = c, b = c, c = c, d = d}, {a = c, b = -c, c = c, d = d}, {a = d, b = b, c = d, d = d}, {a = d, b = d, c = c, d = d}, {a = d, b = d, c = d, d = d}, {a = d, b = -d, c = d, d = d}, {a = -b, b = b, c = 0, d = b}, {a = -b, b = b, c = 0, d = -b}, {a = -b, b = b, c = c, d = -b}, {a = -c, b = 0, c = c, d = c}, {a = -c, b = 0, c = c, d = -c}, {a = -c, b = b, c = c, d = -c}, {a = -c, b = c, c = c, d = d}, {a = -c, b = c, c = c, d = -c}, {a = -c, b = -c, c = c, d = d}, {a = -c, b = -c, c = c, d = -c},{a = -d, b = b, c = d, d = d}, {a = -d, b = b, c = -d, d = d}, {a = -d, b = d, c = c, d = d}, {a = -d, b = d, c = d, d = d}, {a = -d, b = d, c = -d, d = d}, {a = -d, b = -d, c = c, d = d}, {a = -d, b = -d, c = d, d = d}, {a = -d, b = -d, c = -d, d = d} Table V. Hyperplane sections of type G abcd with non-isolated singularities.

  X is the hypersurface of X defined by the restriction of L H to X. Due to the duality of Hilbert spaces, for any H ⊂ P(H) there exists a state |Ψ ∈ P(H) such that H is defined by the linear form Ψ|. In what follows, we will often identify the hyperplane H and the linear form defining it, and write H = Ψ| = 0≤i,j,k,l≤1 h ijkl ijkl| with h ijkl ∈ C. The hyperplane section X ∩ H, or,

H ⊂ P(H) is the set of states |Φ ∈ P(H) on which a linear form L H ∈ H * vanishes. Given H ⊂ P(H), the hyperplane section X ∩ H ⊂

  be a hyperplane of P(H) tangent to X and such that X ∩ H has only isolated singular points. Then the singularities are either of types A 1 , A 2 , A 3 , A 4 , or of type D 4 , and there exist hyperplanes realizing each type of singularity. Moreover, if we denote by X * ⊂ H the cone over the dual variety of X, i.e. the zero locus of the Cayleyhyperdeterminant of format 2 × 2 × 2 × 2, then the quotient map Φ : H → C 4 is such that Φ( X * ) = Σ D 4 ,where Σ D 4 is the discriminant of the miniversal deformation of the D 4 -singularity.

	III, we will establish a dual version of Proposition II.1. We will first define the notion of
	discriminant of a singularity (see Section III A) and then show how it allows us to give a
	new expression for the Cayley hyperdeterminant ∆ 4 (Section III B) and prove Proposition
	III.1 about the relation between ∆ 4 and Σ D 4 . Propositions II.1 and III.1 lead to the proof
	of Theorem 1.
	II. SIMPLE SINGULARITIES AND HYPERPLANE SECTIONS OF
	SEPARABLE STATES
	A. Simple singularities following Arnol'd classification
	Simple singularities have been studied from an algebraic geometrical viewpoint as ratio-
	nal double points of algebraic surfaces, Du Val singularities, and from a complex analytic
	perspective as critical points of holomorphic functions in several variables. These approaches
	lead to many equivalent characterizations of what a simple singularity is 9 . Here, we select the
	complex analytic approach introduced by Vladimir Arnol'd. We first recall the ingredients
	The paper is organized as follows. In Section II, we will give the definition of a simple
	singularity and the invariants that follow from the Arnol'd classification 1 (Section II A).
	Then we will compute the singularity type of any hyperplane section of the set of separable
	states featuring only isolated singularities (see Section II B Proposition II.1). In Section

of Arnol'd classification of simple singularities 1 .

  Then we look at isolated singular points of each hyperplane section and we calculate the corresponding singular type with a formal algebra system following the procedure described in examples II.1, II.2 and Algorithm II.1. For the normal forms depending on parameters, the singular type of the hyperplane sections will depend on values of the parameters. The results of our calculations are given in Tables II and III and provide a proof of the proposition.✷

	Verstraete et al.'s notation	Hyperplane	Singular type of the hyperplane section
	L 0 7⊕1	0000| + 1011| + 1101| + 1110|	D 4 (a unique singularity )

. From each of Verstraete et al.'s normal forms |Ψ we compute the corresponding hyperplane section X ∩ Ψ|.

  5. It is worthwile to point out that the different isolated singular types we obtain by this construction (A 1 , A 2 , A 3 and D 4 ) are exactly the possible degenerations of the D 4 -singularity. In particular, any small neighborhood of the singularity of type D 4 will

	Verstraete's	Hyperplane	parameters	Singular type
	notation
	L a 2 0 3⊕1 a( 0000| + 1111|) + 0011| + 0101| + 0110|	a generic	A 1
			a = 0	non-isolated
	L a 4	a( 0000| + 0101| + 1010| + 1111|)	a generic	A 3 (a unique singularity)
		+i 0001| + 0110| -i 1011|	a = 0	non-isolated
	L ab 3	a( 0000| + 1111|) + a+b 2 ( 0101| + 1010|)	a, b generic	A 2 (a unique singularity)

meet, in S k , the orbits corresponding to the singular types A 1 , A 2 and A 3 as shown in the adjacency diagrams of Arnold's classification (Corollary 8.7 in Ref

1

). The fact that D 4 is the "worst-possible" isolated singularity we get from the hyperplane sections of the set of separable states will be lighted with Proposition III.1.

  1. Let us consider the quotient map Φ : H → C 4 defined by ) = Σ D 4 . Proof. According to Lévay's equation for the hyperdeterminant ∆ 4 , it is clear that our choice of Φ implies that the equation of Φ( X * ) ⊂ C 4 is

	Φ(|Ψ ) = ( Ĩ1 (|Ψ ), Ĩ2 (|Ψ ), Ĩ3 (|Ψ ), Ĩ4 (|Ψ ),
	where Ĩ1 = -4I 1 , Ĩ2 = 6I 2 , Ĩ3 = -4I 3 and Ĩ4 = I 4 . Then, Φ( X 1 i 2 256 ∆(t 4 + λ 1 t 3 + λ 2 t 2 + λ 3 t -( 1 2 λ 4 ) 2 ) = 0,

*

where (λ 1 , λ 2 , λ 3 , λ 4 ) are coordoninates on C 4 . But Lemma 1 implies that this zero locus is the discriminant of the D 4 simple singularity, i.e. the hypersurface Σ D 4 . ✷ Remark III.2. Propositions II.1 and III.1 prove Theorem 1. Remark III.3. The quartic t 4 -(4I 1 )t 3 + (6I 2 )t 2 -(4I 3 )t + I 2 4 of Eq

Borsten L., Dahanayake D., Duff M. J., Ebrahim H. and Rubens W., "Freudenthal triple classification of three-qubit entanglement", Physical Review A 80, no 3 (2009):032326.

Acknowledgment

This work is partially supported by the CNRS grant PEPS-ICQ 2013, project CoGIT (Combinatoire et Géométrie pour l'InTrication).

The authors thank Metod Saniga and Jean-Yves Thibon for their questions and fruitful comments on earlier version of this work.

Appendix A: Hyperplane sections of type G abcd

In this appendix, we will give the different values of the parameters a, b, c, d of the hyperplanes of type G abcd which lead either to hyperplane sections with only A 1 singular points (Table IV) or hyperplane sections with non-isolated singularities (Table V).