R Amraoui

A Aissat
email: aa_aissat@univ-blida.dz

J P Vilcot

D Decoster

Frequency Response Optimization of P-I-N Photodiode Based on InGaAsN Lattice Matched to GaAs for High-Speed Photodetection Applications

Keywords: InxGa1-xAs1-yNy quaternary, p-i-n photodiode, stable structure, high-speed photo-detection

This paper reports on pin photodiode frequency response optimization based on InxGa1-xAs1-yNy quaternary lattice matched to GaAs. Two transparent layers are placed on p-side and n-side in order to manage the photodiode frequency response limitations. The lattice matching condition is calculated in order to obtain stable structure. The physical and optical parameters calculations are performed at room temperature showing the impact of nitrogen on the absorption coefficient. A stable structure, having 2 % of nitrogen and 6% of indium, allows to achieve a cutoff frequency of about 116GHz and a capacitance of 5.21fF while the quantum efficiency is 41.59% for a depletion region thickness of about 0.55µm. However, in case of depletion region thickness of about 0.625µm, the cutoff frequency degrades to 98GHz while the capacitance diminishes to 4.58fF and the quantum efficiency increases to 51.56%. In addition, a comparative study with literature results has been carried out in order to show the advantages of the proposed photodiode. This comparison affirms that the proposed photodiode based on InGaAsN lattice matched to GaAs exhibits high-speed photo-detection. This work allowed us to obtain a p-i-n photodiode with stable structure suitable for photo-detection at 1.15µm.

Introduction

Nowadays, optoelectronic devices such as p-i-n photodiode become more controversial and sophisticated regarding the enormous progress in optical systems. Heterostructures, those based on III-V quaternaries have taken advantage of this development and become a major candidate in optoelectronic devices thanks to their attractive physical and optical properties. Among them, InGaAsN is considered as a promising compound for optoelectronic devices. Since the last few years, InGaAsN deposed on GaAs substrate has received an intense interest in several researches [START_REF] Gladysiewicz | Theoretical studies of optical gain tuning by hydrostatic pressure in GaInNAs/GaAs quantum wells[END_REF][START_REF] Duque | Exciton-related nonlinear optical response and photoluminescence in dilute nitrogen InxGa1-xNyAs1-y/GaAs cylindrically shaped quantum dots[END_REF][START_REF] Gladysiewicz | Material gain in Ga0.66In0.34NyAs1-y, GaNyAs0.69-ySb0.31, and GaNyP0.46Sb0.54-y comparative theoretical studies[END_REF][START_REF] Mal | Theoretical studies on band structure and optical gain of GnInAsN/GaAs cylindrical quantum dot[END_REF][START_REF] Mohamad | Characterization of III-V dilute nitride based multi-quantum well p-i-n photodiodes for next generation opto-electrical conversion devices[END_REF][START_REF] Pavelescu | Very high dose electron irradiation effects on photoluminescence from GaInNAs/GaAs quantum wells by molecular beam epitaxy[END_REF][START_REF] Sarcan | Quantum oscillations and interference effects in strained n-and p-type modulation doped GaInNAs/GaAs quantum wells[END_REF][START_REF] Nomura | Photoluminescence properties of localized states caused by nitrogen alloying in a GaInNAs/GaAs single quantum well[END_REF][START_REF] Ungan | Effects of indium and nitrogen mole concentrations on the optical properties in a GaInNAs/GaAs quantum well under the intense laser field[END_REF]. This is due to, firstly, the beneficial applications such as solar cells [START_REF] Geisz | III-N-V semiconductors for solar photovoltaic applications[END_REF][START_REF] Kosa | Defect distribution in InGaAsN/GaAs multilayer solar cells[END_REF][START_REF] Kurtz | InGaAsN solar cells with 1.0 eV band gap, lattice matched to GaAs[END_REF], photo-detection [START_REF] Aissat | Modeling of the absorption properties of Ga1-xInxAs1-yNy quantum well structures for photodetection applications[END_REF] and laser applications [START_REF] Kondow | GaInAsN: a novel material for long-wavelength-range laser diodes in the excellent high-temperature performance[END_REF][START_REF] Mal | Hydrostatic pressure dependent optoelectronic properties of InGaAsN/GaAs spherical quantum dots for laser diode application[END_REF]. Secondly, the fact that InGaAsN can be grown lattice matched to GaAs [START_REF] Sato | Metalorganic chemical vapor deposition of GaInNAs lattice matched to GaAs for long-wavelength laser diodes[END_REF][START_REF] Sato | Room-temperature operation of GaInAsN/GaInP double-heterostructure laser diodes grown by Metalorganic chemical vapor deposition[END_REF]. The lattice matched structure has been adopted in different works [START_REF] Bellaiche | Band gaps of lattice-matched (Ga,In)(As,N) alloys[END_REF][START_REF] Park | AFM study of lattice matched and strained InGaAsN layers on GaAs[END_REF][START_REF] Tan | Dark Current Mechanism in Bulk GaInNAs Lattice matched to GaAs[END_REF][START_REF] Royall | Experimental investigation and numerical modelling of photocurrent oscillation in lattice matched Ga1-xInxNyAs1-y/GaAs quantum well p-i-n photodiodes[END_REF][START_REF] Ng | Long wavelength bulk GaInNAs p-i-n photodiodes lattice matched to GaAs[END_REF].

On the other side, p-i-n photodiode is considered as a pertinent device in modern optical communication systems because of the remarkable characteristics such as high-speed photo-detection. The latter is well assessed in the photodiode frequency response simulation.

The photodiode frequency response calculation has been extensively tackled in quite number of researches. P-I-N photodiode frequency response has been calculated and optimized [START_REF] Besseghi | Frequency response modeling and optimization of a PIN photodiode based on GaN/InGaN adapted to photodetection at a wavelength of 633 nm[END_REF] using an absorbent region made of In0.5Ga0.5N and a single transparent layer on n-side. An analytical model is presented in [START_REF] Sabella | Analysis of InGaAs p-i-n photodiode frequency response[END_REF] in order to accurately study the performances of InGaAs/InP based pin photodiode. In ref [START_REF] Konno | High frequency response of p-i-n photodiodes analyzed by an analytical model in Fourier space[END_REF] a formulation was developed for carrier transport using vertical pin photodiode. Author studied the device high frequency response and cleared up its limiting parameters. Recently, author in ref [START_REF] Alshehri | Dynamic Characterization of III-nitride base high speed photodiodes[END_REF] developed a multiple quantum wells InGaN/GaN based pin photodiode in the aim of fabricating high-speed pin photodiode. A p-i-n photodiode was designed [START_REF] Elbar | Design and simulation of InGaN/GaN p-i-n photodiodes[END_REF] with structure consists of: In0.1Ga0.9N as an absorbent region having a thickness of 0.1μm placed between two layers p-type and n-type made of GaN, their thicknesses are 0.1 µm, and 0.3µm respectively. The achieved cutoff frequency was 400MHz, however for an intrinsic region thickness of 1.5µm, the cutoff frequency attained 4GHz. A new analytical model was developed in order to study the p-i-n photodiode highfrequency performance [START_REF] Torrese | A fully analytical model to describe the highfrequency behavior of p-i-n photodiodes[END_REF]. In Ref [START_REF] Loquai | A fully analytic model of large area silicon p-i-n photodiodes verified at short wavelengths[END_REF] a characterization of high-speed large-area p-i-n photodiodes was tackled. Author has proposed a fully analytic model in order to study the frequency response for a homostructure p-i-n photodiode. P-I-N photodiode having dual-depletion region has been proposed in order to improve the bandwidth [START_REF] Effenberger | Ultrafast, dual-depletion region, InGaAs/InP p-i-n detector[END_REF][START_REF] Torres | Frequency response optimization of dual depletion InGaAs/InP pin photodiodes[END_REF] taking into consideration the transit time and the capacitance effect. Finally, p-i-n photodiode frequency response was calculated analytically in [START_REF] Locovsky | Transit time considerations in p-i-n diodes[END_REF].

It is crucial to note that p-i-n photodiode frequency response is mainly limited by two parameters of upmost importance [START_REF] Besseghi | Frequency response modeling and optimization of a PIN photodiode based on GaN/InGaN adapted to photodetection at a wavelength of 633 nm[END_REF][START_REF] Sabella | Analysis of InGaAs p-i-n photodiode frequency response[END_REF][START_REF] Konno | High frequency response of p-i-n photodiodes analyzed by an analytical model in Fourier space[END_REF][START_REF] Alshehri | Dynamic Characterization of III-nitride base high speed photodiodes[END_REF][START_REF] Elbar | Design and simulation of InGaN/GaN p-i-n photodiodes[END_REF][START_REF] Torrese | A fully analytical model to describe the highfrequency behavior of p-i-n photodiodes[END_REF][START_REF] Loquai | A fully analytic model of large area silicon p-i-n photodiodes verified at short wavelengths[END_REF][START_REF] Effenberger | Ultrafast, dual-depletion region, InGaAs/InP p-i-n detector[END_REF][START_REF] Torres | Frequency response optimization of dual depletion InGaAs/InP pin photodiodes[END_REF]: the carriers' transit time and capacitance effect.

The main aim of this research is to obtain a high-speed p-i-n photodiode based on stable structure with low capacitance effect and short transit time. However, in case of conventional photodiode low capacitance effect needs thick absorbent region. This leads to increase the transit time and consequently tends to slow the photodiode response. While, high-speed communications (cutoff frequency >60GHz) need short transit time [START_REF] Decoster | Photodetectors for microwave applications: A revew and trends[END_REF] and low capacitance effect. The influence of both transit time and capacitance effect on the photodiode performances is well shown in the frequency response. In attempt to overcome this issue, a structure based on two transparent layers under field inserted on p-side and n-side of the absorbent region is proposed. In this paper we have assumed that carriers move at their saturation velocities, so the transit time becomes directly proportional to depletion region the thickness.

In the paper in hand, physical and optical calculations as well as the frequency response optimization for p-i-n photodiode are provided. The photodiode is based on InxGa1-xAs1-yNy quaternary as an absorbent region lattice matched to GaAs substrate using two transparent layers placed on p-side and n-side. In the first step, the lattice matching condition is calculated in order to obtain stable structure. After that, the bandgap energy as well as the absorption coefficient are calculated at room temperature showing the impact of nitrogen concentration (y) on the absorption coefficient. The next section, on one side, it deals with calculation of the frequency response which corresponds to the transit behavior with highlighting the influence of the absorption coefficient in order to select indium (x) and nitrogen concentrations required in the detection part. On the other side, it deals with the frequency response optimization and the impact of the transit time and the capacitance effect on the frequency response. At a later stage, a comparative study with literature results is provided to evaluate, in term of speed, the benefits of the proposed structure based on InGaAsN lattice matched to GaAs. The next section focuses on the capacitance effect which has a strong impact on the total frequency response. The last section is devoted to the total frequency response and its limitations.

Theoretical framework

This study is based on InxGa1-xAs1-yNy lattice matched to GaAs which allows to obtain a stable structure i.e. ε(x,y) = 0, where ε(x,y) represents the strain given by [START_REF] Aissat | Modeling of the absorption properties of Ga1-xInxAs1-yNy quantum well structures for photodetection applications[END_REF][START_REF] Aissat | Modeling the structure based on GaAsNBi/GaAs for solar cell[END_REF][START_REF] Chenini | Optimization of inter-subband absorption of InGaAsSb/GaAs quantum wells structure[END_REF]:

, = , , (1)
where:

as: is the lattice parameter of the substrate. ae(x,y): is the lattice parameter of InxGa1-xAs1-yNy.

The variation of the strain ε(x,y) as a function of indium and nitrogen concentrations is illustrated in figure.1.

After calculation, the condition that allows to obtain a stable structure is:

y x = . . . . . (2)
Equation.2 presents the variation of nitrogen concentration as a function of indium concentration. For each pair (x, y) we have a stable structure.

Indeed, the introduction of small quantity of nitrogen into InGaAs semiconductor results a drastic reduction in the bandgap energy. This is due to strong interaction between the conduction band and spatially localized state of nitrogen (EN) [START_REF] Pavelescu | Effects of 7-MeV electron irradiation on photoluminescence from 1-eV GaInNAs-on-GaAs epilayers[END_REF]. Consequently, the conduction band splits into two subbands E+ and E-.

Increasing the amount of nitrogen results a repulsion of E+ and E-above and below respectively, which has a significant effect on optical properties. The bandgap reduction due to nitrogen incorporation is well clarified by the Band Anti-Crossing model (BAC) [START_REF] Aissat | Modeling of the absorption properties of Ga1-xInxAs1-yNy quantum well structures for photodetection applications[END_REF][START_REF] Skierbiszewski | Band structure and optical properties of InyGa1-yAs1-xNx alloys[END_REF][START_REF] Gladysiewicz | Band structure and the optical gain of GaInNAs/GaAs quantum wells modeled within 10-band and 8-band k.p model[END_REF][START_REF] Zhao | Effects of nitrogen incorporation on the properties of GaInNAs∕GaAs quantum well Structures[END_REF][START_REF] Shan | Band Anti-Crossing in GaInNAs alloys[END_REF][START_REF] Koksal | The effect of dilute nitrogen on nonlinear optical properties of the InGaAsN/GaAs single quantum wells[END_REF]:

∓ = + ∓ - + 4! " (3)
= 1.52 -3.9

= -1.55 )! + * + , * (4)
! = . (5)
where:

bMN=2.3.

Eo: the conduction band minimum of InGaAs.

m*e : is electron effective mass in the ternary InGaAs.

EM: is the energy of the InGaAs matrix conduction band edge.

EN: is the nitrogen energy level relative to the top of the valence band.

VMN: is the matrix element describes the interaction between EM and EN.

In the equation.3 only E-is taken into account, because E-is responsible for the reduction in bandgap energy [START_REF] Aissat | Effect of on band alignment of compressively strained Ga1-xInxNyAs1-y-zSbz/GaAs quantum well structures[END_REF][START_REF] Ng | Comparison of electronic band structure and optical transparency conditions of In xGa1-xAs1-yNy/GaAs quantum wells calculated by 10-band, 8-band, and 6band k•p models[END_REF].

Absorbing photons energy to generate electrical signal is the basic role of the photodiode. If an incident optical beam having an energy hʋ smaller than the semiconductor bandgap, the device appears as transparent to the photons energies, therefore photons are not absorbed. By contrast, if hʋ is greater than or equal to the bandgap energy, electrons in the valence band absorb the photons energies and move towards the conduction band. The absorption coefficient α(E) in case of absorbed photons is given as follow [START_REF] Aissat | Modeling the structure based on GaAsNBi/GaAs for solar cell[END_REF][START_REF] Aissat | Efficiency Improvement of the Structure InGaN/GaN for Solar cells Applications[END_REF][START_REF] Bestam | High efficiency quadruple junction solar cells[END_REF]:

α E = α • 23 3 4 ,5 3 (7)
where:

E: is the photon energy.

αo: constant depends on the semiconductor.

Eg(x,y): is the bandgap energy of InxGa1-xAs1-yNy.

To perform the frequency response calculation, we have assumed that carriers move with their saturation velocities given by [START_REF] Palankovsky | Simulation of heterojunction bipolar transistor, for obtaining the academic degree: a doctor of technical sciences[END_REF][START_REF] Quay | Analysis and simulation of high electron mobility transistors, for obtaining the academic degree: a doctor of technical sciences[END_REF][START_REF] Adachi | Properties of group-IV, III-V and II-VI semiconductors[END_REF][START_REF] Piprek | Semiconductor Optoelectronic Devices: Introduction to Physics and Simulation[END_REF][START_REF] Quay | Gallium nitride electronics[END_REF][START_REF] Piprek | Nitride semiconductor devices: principles and simulation[END_REF] at 300K for binary compounds (InAs, GaAs, InN, GaN) and the saturation velocity of InGaAsN can be determined by using Vegard's law.

The total current circulates in the device is given as follow [START_REF] Besseghi | Frequency response modeling and optimization of a PIN photodiode based on GaN/InGaN adapted to photodetection at a wavelength of 633 nm[END_REF].

6 7 = 89: ; • < 7 + : = • > 7 ? (8)
where

8 = @ A B
, q is the electron charge, dt is the depletion region thickness.

vp, vn are holes and electrons saturation velocities respectively. P(t) and N(t) represent the number of holes and electrons in the depletion region.

Light-generated carriers transit from the absorbent region towards p-type and n-type regions. Holes response is completed after abandoning the depletion region in duration of dp/vp+ dabs/vp and electrons in duration of dn/vn+ dabs/vn. The carriers' contributions are:

• for holes:

6 ; 7 = @C D ; E F G•AH • I J K•L MN J K•L MN O J K•L MN J K•L MN 0 < 7 < R ; (9) 6
; 7 = @C D ; E F G.A B • I J K•L MN J KST D •BUL D V J KST D BUL D WL MN V O R ; < 7 < R ; + R ;, XY (10)
• for electrons:

6 = 7 = @C Z = E F G•A B • I J K•L MN J K•L MN O 0 < 7 < R = (11)
6 = 7 = @C Z = E F G•A B • I J KST Z •BUL Z WL MN V J KST Z •BUL Z WL MN V O R = < 7 < R = + R =, XY (12)
where:

τn,abs=dabs/vn, τn=dn/vn and τp, abs=dabs/vp, τp=dp/vp.

po and no represent the carriers maximum densities.

dabs is the absorbent region thickness.

dp(dn) is p-side (n-side) transparent layer thickness.

τp,abs(τn,abs) is the transit time required for holes (electrons) to cross the absorbent region.

τp(τn) are the transit time that holes(electrons) take to cross the p-side (n-side) transparent layer.

A is the junction area.

The impulse response is the sum of the electrons and holes contributions, it is given as:

[ 7 = 8′ • 96 ; 7 + 6 = 7 ? (13)
where

8′ = @• ]DB
Nopt is the photons number in the optical beam.

The frequency response allows to verify the photodiode performance. From the photodiode impulse response, we are able to obtain the frequency response by using the Fourier transform:

G _ w = a b c•A B •d • e1 + j • g h g + • i j• k + Uj•k l m i Uj• k + Wj•k l i k h i Uk h n ( 14
)
G o w = a p c•A B •d • e1 + j • g h g q • i j• k q Uj•k r m i Uj• k q Wj•k r i k h i Uk h n ( 15
)
where w is the angular frequency. a1=

Nitrogen impact on optical behavior

The first step is to calculate the bandgap energy and the absorption coefficient. The pairs (x, y) are extracted from equation.2. Indium concentration is limited at 14% and therefore nitrogen concentration is limited at 4.96%. The evolution of the absorption coefficient versus nitrogen concentration is presented in figure.2 in which some nitrogen concentrations match the stable structure are provided.

The influence of nitrogen on the absorption coefficient is well clarified in figure.2. A slight concentration of nitrogen results a notable reduction in the bandgap energy, consequently a considerable amelioration in the absorption coefficient (α). For example, in case of y=2% the bandgap energy is about 1.08eV and the absorption coefficient is of about α=4.72x10 4 cm -1 . However, for y=2.5% the bandgap energy decreases to 1.03eV, while the absorption coefficient rises to α = 4.44x10 5 cm -1 .

The physical and optical properties of InxGa1-xAs1-yNy lattice matched to GaAs for 6% ≤ x ≤ 7% and 2% ≤ y ≤ 2.5% are as follow: bandgap energy from 1.08eV down to 1.033eV, the photo-detection wavelength from 1.15µm until 1.20µm and the absorption coefficient ranging from 4.72x10 4 cm -1 to 4.44x10 5 cm -1 .

PIN photodiode frequency response optimization

The proposed structure is provided in figure .3. It consists of: n-type GaAs layer followed by the first under field transparent layer made of GaAs having a thickness of dn(µm). The absorbent region is an intrinsic InxGa1-xAs1-yNy with thickness of dabs(μm) and the second under field transparent layer is made of GaAs having a thickness of dp(µm). Finally, the p-type layer is made of GaAs. The cathode and anode contacts are made of Ti/Al/Ni/Au and Ni/Au respectively to guarantee the connection with the electrical circuit.

According to the previous results, two configurations satisfying the stable structure have been adopted

taking indium cost and nitrogen influence into consideration. The first configuration is the quaternary In0.07Ga0.93As0.975N0.025, while the second is In0.06Ga0.94As0.98N0.02. The physical and optical parameters of these two configurations are given in section 3.

Focusing on the link between the cutoff frequency and the absorption coefficient, different absorbent regions thicknesses have been examined using the first configuration (In0.07Ga0.93As0.975N0.025): dabs=0.5, 0.8 and 1µm, while both dp and dn are kept at 0.5µm. The results are as follow: for dabs=0.5µm the cutoff frequency at -3 dB is 67.46GHz, this value is unchanged either for dabs=0.8µm or 1µm. The cutoff frequency seems stuck at 67.46GHz despite using different absorbent region thicknesses. This is due to the high absorption coefficient caused by the important nitrogen concentration which has a strong impact on the bandgap energy.

The same study is applied on the second configuration: In0.06Ga0.94As0.98N0.02 which has a reduced absorption coefficient compared to the first configuration. The results are established in figure .4. According to this figure, for dabs=0.2µm the structure reveals a -3dB cutoff frequency of 59.00GHz, then the cutoff frequency degrades to 49.03GHz for dabs=0.5µm. Whereas, in case of the absorbent region is sufficiently large, dabs=0.8 and 1µm, the cutoff frequency reduces to 44.33GHz and 42.86GHz respectively. On the other hand, figure.4 affirms that the cutoff frequency depends highly on the absorbent region thickness when using low absorption coefficient (order of 10 4 ). For this reason, the second configuration (In0.06Ga0.94As0.98N0.02) is selected.

The next step is dedicated to search the optimal depletion region thickness that improves the photodiode frequency response taking into account carriers' transit time and capacitance effect. According to figure.5, in case dn=0.2µm (dashed curves) the maximum cutoff frequency is 143GHz, this is given when dp=0.15µm and dabs=0.1µm. After that, the cutoff frequency decreases to 116.20GHz for dp=0.15µm and dabs= 0.2µm, then it degrades to 98.4GHz for dp=0.125µm and dabs=0.3µm. The last depletion regions have approximately the same transparent layers thicknesses but different absorbent regions which enhance the quantum efficiency as 25.62% for dabs=0.1µm to 41.59% in case dabs=0.2µm until 51.56% for dabs=0.3µm.

Still in figure.5, in case dn=0.5µm (solid curves) the structure exhibits low cutoff frequencies. The maximum cutoff frequency is about 66.23GHz which is obtained for dp=0.375µm and dabs=0.1µm. This degradation in the cutoff frequency values is due to increasing the depletion region thickness. This fact leads to reducing the capacitance, but at the same time a long transit time which slows the photodiode response.

Referring to figure.6.a, when dp=0.375µm (dashed curves) the highest cut off frequency is 97.9GHz, this is given for dn=0.2µm and dabs=0.1µm. Enlarging the depletion region produces low cutoff frequencies as illustrated in case of dp=0.5µm (solid curves). The revealed cutoff frequency is 75.97GHz which is obtained when dn=0.275µm and dabs=0.1µm. However, figure.6.b affirms that high cutoff frequencies needs low thicknesses [START_REF] Kato | Ultrawide-band/high-frequency photodetectors[END_REF]. In this figure a very thin depletion region shows high cutoff frequencies (>160GHz).

To summarize this part, three situations are distinguished. For very thin depletion region, the capacitance effect attains its maximum since it is inversely proportional to the depletion region thickness while the transit time is very short. Enlarging the depletion region reduces gradually the capacitance effect and rises progressively the transit time. At this stage, the situation arrives to an equilibrium point between the capacitance effect and the transit time which enhances the cutoff frequency. Then, when the depletion region becomes sufficiently large, the capacitance effect diminishes while the transit time increases rapidly since it is directly proportional to the depletion region thickness. Consequently, the cutoff frequency leaves its maximum and decreases rapidly. These results show the dominance of the capacitance for thin depletion regions and the transit time for thick depletion regions, so there is a tradeoff between the transit time and the capacitance effect.

Comparative study

In order to reinforce the present work, this section focuses on the benefit of using p-i-n photodiode based InGaAsN lattice matched to GaAs having double transparent layers instead of single transparent layer or totally absorbent region.

The cutoff frequency (Fc) of p-i-n photodiode has been tackled in several works such as Ref [START_REF] Aissat | Transport optimization in pin photodiodes using mixed depletion region[END_REF] where author adopted three structures based on different depletion regions thicknesses. The results of comparison with this study are shown in figure.7.

The first structure is a conventional photodiode in which the depletion region is totally absorbent with a thickness of 0.8µm. The frequency response can be obtained by setting dp and dn to zero in equations 14 and 15.

In our work Fc is equal to 74GHz, whereas in Ref [START_REF] Aissat | Transport optimization in pin photodiodes using mixed depletion region[END_REF] Fc=37GHz.

The second structure consists of single transparent layer placed on n-side of the absorbent region. Their thicknesses are 0.2 and 0.6µm respectively. Using the proposed structure, we have found Fc=69.60GHz, whereas in Ref [START_REF] Aissat | Transport optimization in pin photodiodes using mixed depletion region[END_REF] Fc=45GHz.

The last structure is organized as follow: double transparent layers inserted on n-side and p-side of a very thin absorbent region. Their thicknesses are respectively: dn=0.45µm, dp=0.35µm and dabs=4nm. The exhibited cutoff frequency in our study is Fc=80.63GHz, whereas in Ref [START_REF] Aissat | Transport optimization in pin photodiodes using mixed depletion region[END_REF] Fc=62GHz.

Another comparison with literature results has been performed. Ref [START_REF] Elbar | Design and simulation of InGaN/GaN p-i-n photodiodes[END_REF] adopted a conventional photodiode structure in which the absorbent region thickness is dabs=1.5µm and the cutoff frequency was found 4GHz.

To accomplish this comparison, we have kept the absorbent region thickness at 1.5µm (dabs=1.5µm), the results of this comparison are illustrated in figure.8 for dp and dn ranging from 0 to 0.4µm. From the obtained results we have selected some transparent layers thicknesses (found in section.4) that improve the photodiode response to show the influence of the transparent layers on the photodiode response.

In case dp=0.375µm and dn=0.2µm, the obtained cutoff frequency is about 51.5GHz, whereas, when dp=0.125µm and dn=0.2µm Fc reaches 56.2GHz. However, for dp=0.15µm and dn=0.2µm Fc is about 56.23GHz.

This section asserts that the p-i-n photodiode based InGaAsN lattice matched to GaAs using double transparent layers exhibits high-speed compared to both the conventional one and that whose depletion region consists of single transparent layer.

Capacitance effect

The capacitance is directly proportional to the area, but inversely to the depletion region thickness. It is given as [START_REF] Besseghi | Frequency response modeling and optimization of a PIN photodiode based on GaN/InGaN adapted to photodetection at a wavelength of 633 nm[END_REF][START_REF] Alshehri | Dynamic Characterization of III-nitride base high speed photodiodes[END_REF][START_REF] Yang | High-Power Photodiodes and their Application in Analog Photonic Links[END_REF][START_REF] Xu | Design considerations for 1.3µmGaNAsSb-GaAs high speed and high quantum efficiency waveguide photodetectors[END_REF][START_REF] Decoster | Optoelectronics sensors[END_REF]:

C = ε | • ε } • ÃB ( 16
)
where εo is the permittivity of free space, εr is the relative permittivity of InxGa1-xAs1-yNy. In case of x=0.06 and y=0.02 the relative permittivity is 12.95.

A is the junction area and dt is the depletion region thickness.

Figure.9 illustrates the variation of the capacitance as a function of the depletion region thickness and the area. As shown in this figure, important capacitance is obtained for thin depletion regions and big areas. The variation of the capacitance with the depletion region thickness and the area is resumed in table.1.

According to table.1, structures whose depletion regions have thicknesses of dti; i=1…6 exhibit high capacitance by incrementing 'i'. Besides, this table shows the impact of the area on the capacitance. Areas of 100µm 2 show important capacitance compared to those given for areas of 25µm 2 .

The total frequency response

This section is devoted to the total frequency response calculation and simulation with highlighting its limitations. The total cutoff frequency is given as [START_REF] Besseghi | Frequency response modeling and optimization of a PIN photodiode based on GaN/InGaN adapted to photodetection at a wavelength of 633 nm[END_REF][START_REF] Yang | High-Power Photodiodes and their Application in Analog Photonic Links[END_REF][START_REF] Xu | Design considerations for 1.3µmGaNAsSb-GaAs high speed and high quantum efficiency waveguide photodetectors[END_REF]:

f €t• = 2' ƒ" U+ m ' … U+ (17)
where ft : is the transit cutoff frequency which is related much more to the transit phenomenon.

fRC: is the capacitive cutoff frequency, it is directly proportional to the depletion region thickness and inversely to the area. It is given as [START_REF] Besseghi | Frequency response modeling and optimization of a PIN photodiode based on GaN/InGaN adapted to photodetection at a wavelength of 633 nm[END_REF][START_REF] Alshehri | Dynamic Characterization of III-nitride base high speed photodiodes[END_REF][START_REF] Yang | High-Power Photodiodes and their Application in Analog Photonic Links[END_REF][START_REF] Xu | Design considerations for 1.3µmGaNAsSb-GaAs high speed and high quantum efficiency waveguide photodetectors[END_REF][START_REF] Decoster | Optoelectronics sensors[END_REF]:

f † ‡ = A B •ˆ• †•‰ E •‰ Š •~ (18)
Where, R is the photodiode total resistance and the load resistance.

The variation of the total cutoff frequency as function of dn is presented in figure. to 50µm 2 , this is due to the capacitance effect increases, which is directly proportional to the area. For areas ranging from 50 to 100µm 2 the capacitance effect increase considerably, thereby a severe degradation is observed in the total cutoff frequency.

The degradation is less important in figures 10.a and 10.b, however in figure 10.c the degradation is severe. In figure 10.a, area of 100µm 2 allows a degradation of 4% in the total cutoff frequency relative to area of 25µm 2 . In figure 10.b, area of 100µm 2 allows a degradation of 10% in the total cutoff frequency relative to area of 25µm 2 , while in figure 10.c it is 35% relative to area of 25µm 2 . The same is in figure 11; the degradation in the total cutoff frequency in both figures 11.a and 11.b is slight compared to figure 11.c. In figure 11.a, area of 100µm 2 allows a degradation of 11% in the total cutoff frequency relative to area of 25 µm 2 . In figure 11.b, for A=100µm 2 the degradation is 18% in the total cutoff frequency relative to area of 25µm 2 , while in figure 10.c it is 28% relative to area of 25µm 2 . These results confirm the impact of increasing the area on the total frequency response.

Indeed, the total cutoff frequency depends not only on the area but also on the depletion region thickness, the latter influences more on the carriers' transit time than the capacitance. In In summary of these results, in figures 10 and 11, areas less than or equal 25µm 2 (blue curves) exhibit low capacitance effect. This fact increases significantly the capacitive cutoff frequency versus the transit one, thereby the total cut of frequency becomes strongly dominated by the transit cutoff frequency which related to the transit phenomenon, this dominance is as follow f † ‡ 1 + f † ‡ /f OE ⁄ . By contrast, areas greater than 25µm 2 increase the capacitance effect, thereby, decrease the capacitive cutoff frequency. With increasing the area till 100µm 2 , the capacitance effect becomes important which decreases significantly the capacitive cutoff frequency versus the transit component. Thus, the total cutoff frequency becomes dominated by the capacitive cutoff frequency as in f OE 1 + f OE /f † ‡ ⁄ .

Conclusion

In this work, the lattice matched InxGa1-xAs1-yNy/GaAs based pin photodiode whose depletion region consists of double transparent layers is proposed in order to improve the device performances. Besides, the link between the absorption coefficient and the device response has been tackled in order to select the suitable nitrogen and indium concentrations. For a concentration of nitrogen at 2.5% allows to get an absorption coefficient of 4.44x10 5 cm -1 and an unchangeable cutoff frequency of about 67.46GHz regardless the absorbent region thickness. However, 2% of nitrogen leads to an absorption coefficient of about 4.72.10 4 cm -1 and an improvement in the cutoff frequency by varying the absorbent region thickness as follow: 42.86GHz for 1µm to 49.03GHz for 0.5µm and 59.00GHz for 0.2µm with keeping the transparent layers thicknesses dp and dn at 0.5µm. In addition, this study affirms that thin depletion regions make it possible to achieve cutoff frequencies (>100GHz). It has been also found that the total cutoff frequency completely depends on the transit cutoff frequency for areas less than 25µm 2 . Besides, the proposed structure is able to be adopted for ultra-high-speed applications. Table .1. Impact of the area and the depletion region thickness on the capacitance.

Figure captions

Figure. 5

 5 depicts the cutoff frequency values variation as function of dp where dabs ranging from 0.1 to 0.4µm, and dn=0.2, 0.5µm. Figure.6 provides the cutoff frequency values variation as function of dn where dabs ranging from 0.1 to 0.4µm, while dp=0.5, 0.375 µm in figure.6.a and dp=0.2µm in figure.6.b.

  10 (a,b,c) for different areas: 25, 50 and 100µm 2 using different depletion regions thicknesses: in (a) dp=0.50µm and dabs=0.1µm, in (b) dp=0.375µm and dabs=0.1µm and in (c) dp=0.20µm and dabs=0.1µm. The same is in figure.11 (a,b,c), it illustrates the variation of the total cutoff frequency depending on dp for different areas: 25, 50 and 100µm 2 using different depletion regions thicknesses, for (a) dn=0.20µm and dabs=0.30µm, for (b) dn=0.20µm and dabs=0.20µm, and for (c) dn=0.20µm and dabs=0.10µm. The total cutoff frequency shows degradation when increasing the area from 25

  figure 11.b respectively. These two thicknesses can be written as function of d't,c as follow: d't,a = 1.388d't,c and d't,b = 1.166d't,c. Thus, using thin depletion regions and at the same time high areas increases the severity of degradation in the total cutoff frequency. Whereas, a relatively thick depletion regions (dt,a, dt,b in figure10 and d't,a, d't,bin figure11) make it possible to compensate the increase of the capacitance effect caused by increasing the area, thereby, reducing the severity of degradation in the total cutoff frequency.On the other hand, very high speed responses need low thicknesses[START_REF] Decoster | Photodetectors for microwave applications: A revew and trends[END_REF]. This can be observed if we compare for example blue curve (A=25 µm 2 ) of figure10.a (or figure 11.a) with blue curve (A=25 µm 2 ) of figure 10.c (or figure 11.c). In figure 10.c (figure 11.c) the depletion region has low thickness and it exhibits high cutoff frequencies compared to figure 10.a (or figure 11.a) in which we have relatively thick depletion region.

Figure. 1 .

 1 Figure.1. Variation of the strain versus the indium and nitrogen concentrations, we are interested in ε(x,y)=0.

Figure. 2 .

 2 Figure.2. Variation of the absorption coefficient versus nitrogen concentration using different indiumconcentrations providing some nitrogen concentrations for the stable structure and E=1.08eV.

Figure. 3 .

 3 Figure.3. Structure of the proposed InGaAsN/GaAs p-i-n photodiode.

Figure. 4 .

 4 Figure.4. Frequency response simulationforIn0.06Ga0.94As0.98N0.02/GaAs p-i-n photodiode depending on different absorbent region thicknesses, dp and dn are kept at 0.50µm.

Figure. 5 .

 5 Figure.5. Variation of the cutoff frequency as a function of the p-side transparent layer thickness using different absorbent region thicknesses ;dn=0.20µm for dashed curves and 0.50µm for solid curves.

Figure. 6 .

 6 Figure.6. (a, b). Variation of the cutoff frequency as a function of the n-side transparent layer thickness using different absorbent region thicknesses, a)dp=0.375µm for dashed curves and 0.5µmfor solid curves. b) dp=0.20µm.

Figure. 7 .

 7 Figure.7. Frequency response of theIn0.06Ga0.94As0.98N0.02/GaAs p-i-n photodiode for different depletion region thicknesses compared to that in Ref [53]..

Figure. 8 .

 8 Figure.8. Frequency response variation of the In0.06Ga0.94As0.98N0.02/GaAs p-i-n photodiode for dabs=1.50µm.

Figure. 9 .

 9 Figure.9. Variation of the capacitance as a function of the n-side and p-side transparent layers thicknesses. The area A is 25µm 2 and 100µm 2 and the absorbent region thickness dabs is 0.1µm and 0.3µm.

Figure. 10 .

 10 Figure.10. (a,b, c)Variation of the total cutoff frequency as a function of n-side transparent layer thickness using different areas, a) dp=0.50µm and dabs=0.10µm, b) dp=0.375µm and dabs=0.10µm, c) dp=0.20µm and dabs=0.10µm.

Figure. 11 .

 11 Figure.11. (a,b,c).Variation of the total cutoff frequency as a function of p-side transparent layer thickness using different areas, dn=0.20µm and dabs=0.30µm, b) dn=0.20µm and dabs=0.20µm, c) dn=0.20µm and dabs=0.10µm.

Figure 1

 1 Figure 1

Figure 5 Figure 6

 56 Figure 2

Figure 7 Figure 8

 78 Figure 7

Figure 9

 9 Figure 9

Figure 10

 10 Figure 10.a


Henri Azaïs

Lise Lecointre

Geoffroy Canlorbe
email: geoffroy.canlorbe@aphp.fr

What's new in the management of endometrial cancer? Update on the new European recommendations for 2021

Quelles nouveautés pour la prise en charge du cancer de l'endomètre ? Le point sur les recommandations européennes de 2021

INTRODUCTION

Les nouvelles recommandations européennes concernant la prise en charge des patientes présentant un cancer de l'endomètre ont été publiées début 2021 (1). Ces recommandations sont conjointes aux trois sociétés savantes européennes d'oncologie gynécologique (ESGO), de radiothérapie et d'oncologie (ESTRO) et d'anatomo-pathologie (ESP). Au nom de la Société Française d'Oncologie Gynécologique (SFOG), nous souhaitons apporter à la connaissance du lectorat francophone les principales mesures pour la prise en charge des patientes qui représentent une avancée par rapport aux recommandations de 2016 (2).

DIAGNOSTIC HISTOLOGIQUE ET MARQUEURS MOLECULAIRES

Le diagnostic de cancer de l'endomètre reste basé sur l'analyse anatomopathologique du matériel obtenu lors d'une biopsie d'endomètre. D'après les nouvelles recommandations, on note la modification du grade qui est désormais qualifié de bas grade (ancien grade 1-2) ou de haut grade (ancien grade 3). Aussi, les types histologiques sont maintenant regroupés en endométrioïde ou non-endométrioïde (séreux, cellules claires, carcinosarcomes), plutôt qu'en type 1 et type 2.

SYNDROME DE LYNCH

Pour toute patiente présentant un cancer de l'endomètre, quel que soit le sous-type histologique, il est recommandé de rechercher un instabilité des microsatellites (MSI) ou de réaliser une analyse immunohistochimique des protéines du système MMR (MLH1, PMS2, MSH6, MSH2 avec le statut de méthylation du promoteur de MLH1 en de perte d'expression en immunohistochimie de MLH1/PMS2). Une consultation d'oncogénétique sera proposée selon les résultats.

BILAN PREOPERATOIRE COMPLEMENTAIRE

Le bilan général doit renseigner l'histoire familiale, le bilan des comorbidités avec évaluation gériatrique si indiquée, et l'examen physique comprenant un examen pelvien.

Le bilan d'imagerie doit comprendre une échographie pelvienne endovaginale ou endorectale ou une IRM pelvienne afin de déterminer la profondeur d'invasion myométriale et l'atteinte ou non du stroma cervical. En fonction des constatations cliniques et de l'analyse anatomopathologique, d'autres examens d'imagerie (tomodensitométrie thoraco-abdomino-pelvienne, IRM, TEP scanner, échographie) peuvent être demandés pour évaluer une atteinte ovarienne, ganglionnaire, ou une autre localisation métastatique. A noter qu'il n'est pas recommandé de demander un examen extemporané de la profondeur d'invasion myométriale sur la pièce d'hystérectomie car cette analyse et peu reproductible et interfère avec la qualité de l'analyse anatomo-pathologique.

ETABLISSEMENT DU PRONOSTIC SELON 4 GROUPES DE RISQUE

Les nouvelles recommandations ont établi une classification en 4 groupes de risque (faible, intermédiaire, haut-intermédiaire et élevé) basés sur les données de la classification FIGO 2009, l'analyse anatomopathologique (type, grade binaire, emboles), et également les données de l'analyse moléculaire (Figure 1). Il est important de préciser que les emboles décrits comme absents ou focaux sont considérés comme négatif, et ceux décrits comme substantiels sont considérés comme positifs.

Cette classification servira à adapter le traitement chirurgical initial et également le traitement adjuvant.

On remarque que cette nouvelle classification se différencie de celle des recommandations de 2016 qui n'intégraient pas les données moléculaires et qui étaient constituée de 3 groupes en pré opératoire, et de 4 groupes en post opératoire.

PRISE EN CHARGE CHIRURGICALE INITIALE

La nouvelle classification pré opératoire, d'emblée en 4 groupes à risque (Faible, intermédiaire, Haut Intermédiaire et élevé) va guider la prise en charge chirurgicale initiale. On note que ces recommandations, par rapport à celles de 2016, ont considérablement élargi le champ des indications du ganglion sentinelle par rapport aux curages complets, pour les types histologiques endométrioïdes et non endométrioïdes.

Concernant les patientes présentant apparemment un stade FIGO I ou II.

La chirurgie mini-invasive est recommandée, y compris pour les patientes du groupe haut risque.

Toute procédure risquant de favoriser une dissémination intra abdominale de la maladie doit être évitée, notamment le morcellement (même protégé dans un sac). Il est à noter que l'extension de la maladie en dehors de l'utérus et du col (sauf l'envahissement ganglionnaire) est une contreindication relative à la voie mini-invasive. -Curages ganglionnaires : ils doivent comporter des curages pelviens bilatéraux et un curage lombo-aortique étendu jusqu'à la veine rénale.

-A noter que si un envahissement ganglionnaire est mis en évidence en per opératoire, il n'est pas utile de poursuivre les curages pelviens de façon systématique. On procède, dans ce contexte, à l'exérèse des adénopathies suspectes volumineuses et à un curage lomboaortique. Aussi, la présence de macro métastase ou de micro métastase doit être considérée, pour la suite de la prise en charge, comme un envahissement ganglionnaire métastatique. -Évaluation de la fertilité avant et après le traitement -Résection hystéroscopique de la lésion suspecte -Traitement recommandé par medroxyprogesterone acetate (400-600 mg/jour) ou megestrol acetate (160-320 mg/jour). On peut également discuter un traitement avec un dispositif intra utérin ou levonorgestrel en association au traitement par progestérone orale, avec ou sans agonistes de la GnRH. Un suivi à 3-4 mois, puis à 6 mois est recommandé avec des biopsies endométriales guidées par hystéroscopie + imagerie.

-Une hystérectomie est recommandée en l'absence de réponse au traitement à 6 mois.

-Dans les suites, il est recommandé de poursuivre le traitement hormonal si la patiente ne souhaite pas de grossesse dans l'immédiat. Le suivi consiste alors à une surveillance tous les 6 mois comportant un examen physique et une échographie pelvienne endo-vaginale. Une hystéroscopie avec biopsies ne sont recommandées qu'en cas de saignements anormaux.

Concernant les patientes présentant apparemment un stade FIGO III ou IV

Une chirurgie de résection complète, incluant l'exérèse des adénopathies suspectes, doit être proposée si la tolérance de la patiente le permet. Les curages ne sont pas recommandés à titre systématique, mais seule une exérèse des adénopathies suspectes doit être proposée.

En cas de tumeur non résécable d'emblée, il convient de discuter en RCP les modalités du traitement par radiothérapie-curiethérapie suivie de chimiothérapie (ou bien une chimiothérapie première si on envisage une chirurgie d'exérèse au décours en cas de bonne réponse).

Concernant le cas particulier de maladie résiduelle après chirurgie (marges de résection positives, atteinte de la paroi pelvienne, envahissement vaginal).

Une prise en charge personnalisée validée en RCP doit être proposée, comprenant de la radiothérapie, ou de la chimiothérapie ou une association des deux.

TRAITEMENT ADJUVANT

Les indications de traitement adjuvant sont maintenant basées selon la nouvelle définition des groupes à risque de récidive incluant la classification moléculaire si les analyses sont disponibles (tableau 1).

Prises en charge spécifiques selon la classification moléculaire

Cinq classifications moléculaires ont été définies : 1) les tumeurs ultramutées / avec mutations POLE pathogéniques (POLEmut) ; 2) les tumeurs hypermutées avec micro-satellites instables (MSI)/MMRd ; 3) avec un nombre élevé de copies / p53abn ; 4) avec un nombre faible de copies / NSMP ; et 5) avec classifications multiples.

Tumeurs POLEmut

Le pronostic est excellent pour ce type de tumeurs.

Si le risque faible a été déterminé sur la base de la mutation du domaine exonucléase de POLE, l'abstention thérapeutique est recommandée pour les stades I et II en raison d'un très faible risque de récidive.

Pour les tumeurs POLEmut de stade III-IVA, les données de la littérature semblent insuffisantes pour privilégier l'abstention thérapeutique. Un traitement adjuvant par radio chimiothérapie est recommandé en l'absence de données suffisantes. De plus, l'inclusion dans des essais thérapeutiques est encouragée.

Tumeurs p53abn

Le pronostic pour ce type de tumeurs apparaît plus sombre.

Les tumeurs p53abn sont considérées comme à haut risque de récidive en cas d'invasion myométriale. Il existe donc un bénéfice à un traitement par radio chimiothérapie adjuvante pour les stades I à III (Essai PORTEC-3).

Néanmoins, en cas de tumeur p53abn limitée à un polype endométrial ou sans invasion myométriale

  Il est maintenant recommandé de réaliser une analyse complémentaire associant les marqueurs immunohistochimiques (p53, MLH1, PMS2, MSH6, et MSH2) et un test moléculaire (analyse de mutation du domaine d'exonucléase du POLE). Cette classification moléculaire est particulièrement encouragée pour les tumeurs de haut grade. L'analyse des mutations POLE peut être omise dans les cas de tumeurs à faible risque et risque intermédiaire avec un bas grade histologique.

L

  'hystérectomie totale avec annexectomie bilatérale est le geste de référence. La préservation ovarienne peut être proposée chez les patientes non ménopausées de moins de 45 ans avec cancer de bas grade et un stade FIGO IA. A noter que la préservation ovarienne n'est pas recommandée en cas d'histoire familiale augmentant le risque de cancer de l'ovaire (ex : mutation BRCA, Lynch notamment). Une hystérectomie élargie n'est réalisée que si cela est nécessaire pour assurer le caractère in sano de la résection. Une omentectomie infra-colique doit être proposée pour les types histologiques suivants : séreux, carcinosarcome et indifférencié. Une reprise chirurgicale pour stadification peut être proposée si celle-ci n'a pas été complète lors de la chirurgie initiale pour les patientes présentant un risque « haut-intermédiaire » ou « élevé » et si cela peut avoir une implication pour les modalités du traitement adjuvant. La chirurgie de stadification ganglionnaire peut comporter, selon les indications, les gestes suivants : -Procédure du ganglion sentinelle : o Proposée pour les groupes à risque « bas » ou « intermédiaire », quel que soit le type histologique. La procédure peut cependant être évitée en cas d'absence d'envahissement myométrial. o Recommandée pour les groupes à risque « haut-intermédiaire » et « élevé », quel que soit le type histologique. o Il est recommandé d'en faire la détection avec le vert d'indocyanine. Des réinjections per opératoires sont possibles en cas d'absence de détection. Un curage pelvien homolatéral doit être réalisé en l'absence de détection pour les groupes à risque « haut-intermédiaire » et « élevé ». Une analyse par ultrastadifiction est recommandée sur les ganglions sentinelles prélevés.

  Concernant le cas particulier des patientes présentant des co-morbidités majeures contre-indiquant la chirurgie classique, on peut leur proposer : -Une éventuelle voie d'abord vaginale pour hystérectomie et annexectomie bilatérale, -Si contre-indication à chirurgie, une radiothérapie définitive, comportant : o Pour les bas grade : curiethérapie exclusive o Pour les haut grade et/ou tumeur infiltrant profondément le myomètre : association de radiothérapie externe et curiethérapie -Si contre-indication à tout autre traitement : un traitement anti hormonal Une préservation de la fertilité peut être proposée, après discussion en centre spécialisé, aux patientes présentant une tumeur de grade 1 ou une hyperplasie atypique (prouvé préférentiellement par une biopsie réalisée sous contrôle hystéroscopique, avec relecture par un pathologiste expert), sans infiltration du myomètre (sur IRM pelvienne ou échographie par radiologue expert) et sans facteurs de risque génétique, avec information de l'indication d'hystérectomie après grossesse ou en cas d'échec de grossesse. La préservation de fertilité comprend la prise en charge suivante :

(

  risque intermédiaire), une abstention thérapeutique peut être discutée en raison de l'absence d'inclusion de ces patientes dans des essais thérapeutiques randomisés et d'un bénéfice incertain de la chimiothérapie ainsi que de la curiethérapie vaginale (Barney 2013). Tumeurs MMRd et NSMP Les tumeurs MMRd et NSMP sont en grande partie à risque intermédiaire et bénéficient par conséquent d'une curiethérapie adjuvante. Cependant, ces tumeurs sont considérées à haut risque si le stade est III-IVA en résection complète. Il existe un bénéfice de la radiothérapie pour les tumeurs MMRd et NSMP à risque élevé, par conséquent elle est recommandée. La chimiothérapie concomitante semble avoir un bénéfice pour les tumeurs NSMP notamment pour le stade III. Néanmoins, la place de la chimiothérapie pour les tumeurs MMRd n'est pas établie à ce jour (de Boer 2018, Léon-Castillo 2020 PORTEC-3) . 2. Prises en charge spécifiques sans la classification moléculaire : a) Stades I et stades II Le groupe à risque faible Il n'y a pas de modification de prise en charge adjuvante par rapport aux recommandations de 2016. L'abstention thérapeutique reste recommandée dans le groupe à risque faible. En effet, aucun bénéfice à la radiothérapie externe ou à la curiethérapie vaginale n'a été démontré dans ce sousgroupe de patientes. Le groupe à risque intermédiaire grades et les carcinomes endométrioïdes de stade II grade 1, la curiethérapie reste recommandée afin de réduire les récidives locales. La place de la chimiothérapie a également évolué. En 2016, la chimiothérapie adjuvante n'était pas recommandée mais la participation à des essais cliniques était encouragée, alors qu'en 2021, la chimiothérapie trouve une indication pour les tumeurs de haut grade et/ou en cas d'invasion lymphovasculaire substantielle. Le groupe à risque élevé Les critères d'éligibilité des patientes à haut risque à un traitement adjuvant ont nettement évolué dans ces nouvelles recommandations de 2021. De façon beaucoup plus consensuelle et unanime, une radiochimiothérapie concomitante ou séquentielle est désormais recommandée afin d'améliorer le contrôle loco-régional ainsi que la survie globale et la survie sans progression. La chimiothérapie seule reste néanmoins une alternative possible sur la base des résultats de l'étude GOG-258 sur les stades III-IV. De plus, les carcinosarcomes doivent être considérées comme des cancers de l'endomètre à haut risque et non pas comme des sarcomes. b) Stades avancés III et IV Pour les patientes présentant des cancers de l'endomètre de stades avancés, il n'y a eu de modifications majeures dans les nouvelles recommandations par rapport à 2016. En effet, l'association d'une chimiothérapie et d'une radio-curiethérapie reste recommandée après discussion pluridisciplinaire. SURVEILLANCE Sur le plan de la surveillance, la nouveauté de 2021 est l'émission de recommandations concernant les femmes porteuses d'un variant délétère germinal d'un gène associé au syndrome de Lynch. En effet, il est actuellement recommandé de proposer une consultation oncogénétique aux patientes ayant un cancer de l'endomètre avec risque de syndrome de Lynch, notamment le type MMRd. La surveillance endométriale des patientes mutées doit débuter à partir l'âge de 35 ans, toutefois cet âge limite pourra être modulé selon les facteurs personnels et le contexte familial. Cette surveillance associera une échographie endométriale par voie transvaginale annuelle à une biopsie de l'endomètre annuelle ou tous les 2 ans. Une chirurgie prophylactique (de réduction de risque du cancer endométrial et ovarien), comprenant une hystérectomie totale et une annexectomie bilatérale, pourrait être réalisée une fois le projet parental accompli et de préférence avant l'âge de

Comme en 2016, la curiethérapie vaginale reste recommandée pour le groupe à risque intermédiaire afin de diminuer le risque de récidive vaginale avec une moindre morbidité en comparaison à la radiothérapie externe. Une abstention thérapeutique peut être discutée pour les patientes âgées de moins de 60 ans.Le groupe à risque haut-intermédiaireEn 2016, la seule présence d'emboles lymphovasculaires lors de l'analyse histologique de la pièce opératoire indiquait la réalisation d'un traitement adjuvant. En 2021, il convient de distinguer la caractère focal ou substentiel de ces emboles pour guider le choix thérapeutique. En l'absence d'atteinte ganglionnaire après stadification chirurgicale (pN0), la curiethérapie reste recommandée pour diminuer le risque de récidive vaginale. Cependant, dans les nouvelles recommandations de 2021, il est également possible de réaliser une radiothérapie externe en cas d'invasion lymphovasculaire substantielle ainsi que pour les stades II, afin de réduire les risques de récidives ganglionnaires pelviennes et para-aortiques. De même, une chimiothérapie peut être discutée en cas de grade élevé et/ou d'invasion lymphovasculaire substantielle. Enfin, de manière similaire à 2016, l'abstention thérapeutique reste une option. En cas de statut ganglionnaire inconnu (cN0/pNx) quand la stadification ganglionnaire chirurgicale n'a pas été effectuée, de manière superposable à 2016, une radiothérapie externe est recommandée en cas d'invasion lympho-vasculaire substantielle, afin de réduire les récidives pelviennes ainsi qu'en cas de stade II. En l'absence d'emboles lymphovasculaires pour les hauts

chirurgie chez les femmes non ménopausées.

CONCLUSION

Les nouvelles recommandations européennes concernant la prise en charge des patientes présentant un cancer de l'endomètre insistent sur la généralisation de la recherche de MSI ou de l'analyse immunohistochimique des protéines du système MMR pour toutes les patientes. Aussi, la classification en 4 groupes à risque pronostique intègre les données de l'analyse moléculaire (p53, MLH1, PMS2, MSH6, MSH2, POLE) pour guider la prise en charge chirurgicale initiale ainsi que les modalités du traitement adjuvant. La procédure du ganglion sentinelle au vert d'indocyanine est par ailleurs devenue la technique de référence pour la stadification ganglionnaire des stades FIGO I et II quel que soit le type histologique.

On rappelle que la prise en charge doit être assurée dans un établissement spécialisé par une équipe spécialisée dans la prise en charge des cancers gynécologiques, en particulier pour les patientes à haut risque et/ou présentant un cancer avancé. BIBLIOGRAPHIE 1) Concin N, Matias-Guiu X, Vergote I, Cibula D, Mirza MR, Marnitz S, Ledermann J, Bosse

FIGURES ET TABLEAUX


Amel Kime
email: amel.kime@gmail.com

Julie Dremaux

Jean-Marc Régimbeau

Mathurin Fumery

Jean-Fortuné Ikoli

Denis Chatelain

Une sténose iléale inhabituelle An unusual ileal stenosis

Keywords: Maladie de Crohn, maladie inflammatoire chronique de l'intestin, adénocarcinome Crohn's disease, inflammatory bowel disease, low grade tubuloglandular adenocarcinoma, IDH1 mutation, small bowel adenocarcinoma

Nous rapportons le cas d'un adénocarcinome tubulo-glandulaire de bas grade iléal chez un homme de 53 ans suivi pour maladie de Crohn. L'adénocarcinome tubulo-glandulaire de bas grade est rare et souvent associé à une maladie inflammatoire chronique de l'intestin. Il est caractérisé par des glandes très bien différenciées et peu d'atypies cytologiques. Récemment, des mutations de IDH1 ont été observées dans des carcinomes coliques et pourraient être spécifiques de ce sous-type et constituer ainsi une potentielle cible thérapeutique. Les pathologistes doivent connaître cette entité rare qui peut être de découverte fortuite.

Observation

Un homme de 53 ans suivi pour une maladie de Crohn était hospitalisé en urgence pour un syndrome occlusif. Il présentait une altération de l'état général depuis un mois avec une perte de 13 kg, associée à des douleurs abdominales et un syndrome inflammatoire biologique. Le scanner abdominal montrait une sténose inflammatoire de la dernière anse iléale et des trajets fistuleux communiquant avec la paroi sigmoïdienne. Une résection iléo-caecale était réalisée. L'examen macroscopique retrouvait une paroi iléale épaissie, indurée et blanchâtre, avec une lumière iléale sténosée (Figure 1). Microscopiquement, la paroi iléale était infiltrée par des glandes tubulées, très bien dif férenciées, de petite taille, avec souvent une disposition organoïde, organisées en lobules et travées (Figure 2A). Elles étaient bordées de cellules cubocylindriques au cytoplasme éosinophile, au noyau élargi, avec des atypies modérées, sans mitose (figure 2B). Il existait au contact immédiat des glandes une stroma réaction inflammatoire, peu desmoplasique. Elles s'insinuaient à travers toute la paroi iléale (Figure 2A).

A l'étude immunohistochimique, la tumeur exprimait de façon intense et diffuse CK, CDX2, S100P, mésothéline, p53 (Figure 3) et plus focalement CK20. Elle n'exprimait pas CK7, SATB2, BRAF et IDH1. L'index de prolifération Ki67 était évalué à 10%. La tumeur présentait un phénotype microsatellite stable (expression nucléaire conservée de MSH2, MSH6, MLH1 et PMS2).

L'étude en biologie moléculaire révélait des mutations de IDH1 (c.394C>T, p.Arg132Cys) et de TP53 (c.818G>A, p.Arg273His) sans mutation de KRAS, BRAF et des gènes du système MMR.

Quel est votre diagnostic ?

Diagnostic proposé

Adénocarcinome tubulo-glandulaire de bas grade iléal compliquant une maladie de Crohn.

Discussion

L'adénocarcinome tubulo-glandulaire de bas grade (ATGBG) est une entité rare, représentant moins de 1% des carcinomes colorectaux, mais environ 10% des carcinomes intestinaux associés aux maladies inflammatoires chroniques de l'intestin (MICI). Il survient habituellement lorsque la maladie évolue depuis plus de 20 ans (1) Vingt-sept cas ont été publiés dans la littérature, chez 13 hommes et 10 femmes, âgés de 28 à 58 ans (44,2 ans en moyenne d'âge). Dans 39% des cas les patients présentaient une MICI à type de maladie de Crohn, 56,5% à type de rectocolite ulcéro-hémorragique et 0.5% à type de colite indéterminée. Les tumeurs siégeaient au niveau de l'iléon (22%) et du colon (78%) avec une distribution égale entre colons droit et gauche et mesuraient de 0,4 à 15 cm. L'ATBGB prend le plus souvent la forme d'une tumeur sténosante, parfois exophytique mais il peut aussi être plan et difficile à visualiser à l'examen endoscopique [START_REF] Levi | Intestinal Low-grade Tubuloglandular Adenocarcinoma in Inflammatory Bowel Disease[END_REF][START_REF] Grolleau | Small bowel adenocarcinoma complicating Crohn's disease: a single-centre experience emphasizing the importance of screening for dysplasia[END_REF]. Le diagnostic microscopique d'invasion se fait au faible grossissement. Les glandes tumorales sont très bien différenciées, uniformes, arrondies ou tubulées avec des atypies cytonucléaires de bas grade. Le stroma présente peu ou pas de réaction desmoplasique (3). Sur 24 ATGBG décrits dans la littérature, 58 % étaient en continuité avec un adénocarcinome de type conventionnel, évoquant un continuum lésionnel. On dénombrait 3 tumeurs de stade T1, 7 tumeurs de stade T2, 13 tumeurs de stade T3 et 1 tumeur de stade T4. Parmi elles, 18% présentaient au moins une métastase ganglionnaire et seul un cas des métastases hépatiques. Une co-expression de CK7 et CK20 est fréquente en immunohistochimie (69% des cas). Une perte d'expression de hMLH1 en immunohistochimie est présente dans 55% des ATGBG [START_REF] Levi | Intestinal Low-grade Tubuloglandular Adenocarcinoma in Inflammatory Bowel Disease[END_REF]. Cette forte prévalence est en faveur d'une instabilité microsatellitaire observée dans 15% des cancers colorectaux compliquant une MICI. Dans une série de huit adénocarcinomes associés aux MICI, trois avaient une mutation somatique du gène IDH1 (p.R132C, c.394C>T). L'isocitrate deshydrogénase (IDH) est une enzyme du cycle de Krebs qui catalyse l'oxydation d'isocitrate en α-cétoglutarate. En cas de mutation de son isoforme cytoplasmique (IDH1), l'isocitrate devient 2-hydroxyglutarate et ce dernier peut induire une hyperméthylation des ilots CpG et ainsi une instabilité génomique et une altération de la transcription de certains gènes. La mutation du gène IDH1 au niveau de tumeurs coliques serait spécifique des ATGBG (4). De plus, ont été rapportés deux adénocarcinomes mutés pour IDH1 avec une mutation activatrice de KRAS associée (p.G12D, c.35G>A). L'analyse moléculaire représente donc une aide au diagnostic dans les cas difficiles. Cette mutation de IDH1 est également présente dans d'autres cancers, notamment dans certains gliomes et cholangiocarcinomes intra-hépatiques. Récemment, l'ivosidenib, un inhibiteur de IDH1 a montré son efficacité dans le cholangiocarcinome intra-hépatique résistant aux chimiothérapies classiques, et pourrait constituer une alternative thérapeutique dans les ATGBG de stade avancé [START_REF] Abou-Alfa | Ivosidenib in IDH1mutant, chemotherapy-refractory cholangiocarcinoma (ClarIDHy): a multicentre, randomised, double-blind, placebo-controlled, phase 3 study[END_REF].

Bien que peu de données existent sur la survie de ces cancers, il semblerait que l'ATGBG ait un pronostic favorable sauf en cas d'association à un adénocarcinome colorectal de type conventionnel. Notre patient ne présentait aucun signe de récidive ni de métastase après 6 mois de suivi. L'ATGBG représente un challenge diagnostique pour le pathologiste, en raison de son caractère très bien différencié, et parfois de la difficulté de caractériser son aspect invasif sur des biopsies superficielles. Lorsqu'elles sont réalisées, les biopsies pré-opératoires intéressent en majorité des lésions sans caractère invasif évident (23,5% des lésions interprétées comme dysplasiques de bas grade, 12% des lésions dysplasiques de haut grade, et 23,5% des lésions de bas et de haut grade) et dans seulement 41 % des cas un adénocarcinome invasif typique Il est important pour les pathologistes de connaitre l'ATGBG, une entité rare. Il survient préférentiellement dans un contexte de MICI, et peut être de découverte fortuite aussi bien à l'examen macroscopique qu'en microscopie. Pour confirmer son diagnostic, le pathologiste pourra s'aider d'une étude immunohistochimique avec la p53, et pourra éventuellement s'appuyer d'une analyse moléculaire.

Liens d'intérêt

Les auteurs déclarent ne pas avoir de liens d'intérêts.

Légendes des figures

Immunohistochemistry with the anti-P53 antibody Intense circumferential staining with the anti-P53 antibody.


Pavao Mardešić

Dmitry Novikov
email: dmitry.novikov@weizmann.ac.il

Laura Ortiz-Bobadilla

Jessie Pontigo-Herrera

NILPOTENCE OF ORBITS UNDER MONODROMY AND THE LENGTH OF MELNIKOV FUNCTIONS

Keywords: 1991 Mathematics Subject Classification. 34C07; 34C05, 34C08, 14D05 Iterated integrals, Melnikov function, displacement function, abelian integrals, limit cycles, nilpotence class

Let F ∈ C[x, y] be a polynomial, γ(z) ∈ π 1 (F -1 (z)) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation dF + ϵω = 0 of the integrable foliation given by F .

We study different invariants: the orbit depth k, the nilpotence class n, the derivative length d associated with the couple (F, γ). These invariants bound the length ℓ of the first nonzero Melnikov function of the deformation dF + ϵω along γ. We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon-Vey sequence. We formulate a conjecture motivated by the study of this example.

Introduction, Main Results and Conjectures

This work is motivated by the 16-th Hilbert's problem or rather its infinitesimal version. As it is known, the second part of Hilbert's 16-th problem asks for an upper bound in terms of the degree for the number of real limit cycles, i.e. isolated periodic orbits of polynomial vector fields in the plane. The problem is far from been solved and the existence of such a number is open even for quadratic vector fields.

Arnold formulated the infinitesimal Hilbert's problem, which asks for a bound on the number of (real) limit cycles that can arise under polynomial deformations from an integrable polynomial differential equation in the plane. This bound must be uniform in the sense that it must depend exclusively on the degree of the integrable 1-form defining the corresponding foliation, and the degree of the polynomial that realizes the perturbation.

In one of its forms Arnold's infinitesimal Hilbert problem [START_REF] Arnold | Arnold's problems. Translated and revised edition of the 2000 Russian original[END_REF] studies the following situation.

Let F (x, y) ∈ R[x, y] be a square-free polynomial, z a regular value of F and γ(z) ⊂ F -1 (z) a continuous family of real cycles of F -1 (z). We will consider the complexification of the polynomial F which, for the sake of simplicity, we will denote again by F . The polynomial F defines a singular fibration and hence a singular foliation given by the integrable one-form dF = 0 .

(1.1)

Consider the polynomial deformations depending on the parameter ϵ dF + ϵω = 0 .

(1.2)

As usual, one defines the displacement function (holonomy map minus identity)

∆ ϵ of the deformation (1.2) along γ(z). The isolated zeros of the displacement map ∆ ϵ are in correspondence with the limit cycles of (1.2). The Taylor series expansion with respect to ϵ at 0 of ∆ ϵ is given by

∆ ϵ (z) = ∑ i=µ ϵ i M i (z). (1.3)
The functions M i are called Melnikov functions. We assume M µ ≡ 0 and call it the first non-zero Melnikov function. By the implicit function theorem, for regular values of z, it carries the main information on the number of limit cycles that can arise under the deformation. This motivates the study of this function M µ , the subject of the infinitesimal Hilbert 16th problem.

It is known that M µ is an iterated integral [START_REF] Gavrilov | Higher order Poincaré-Pontryagin functions and iterated path integrals[END_REF]. Let ℓ denote its iterated integral length, or shorter, simply length. It measures the complexity of M µ . Bounding this length, would be a key step towards the bound of the limit cycles arising under perturbation. In [START_REF] Gavrilov | The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields[END_REF], Gavrilov and Iliev gave a condition which guarantees that M µ is an abelian integral (iterated integral of length 1). We showed the necessity of this condition in [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF].

Let z be a generic value of F . Consider the fundamental group π 1 of F -1 (z), and let L j be its lower central sequence;

L j = [L j-1 , π 1 ], where L 1 = π 1 .
(

Using the fibration given by F , we define the monodromy group, which is a subgroup of the group of automorphisms of π 1 . Then denote by O = O γ the normal subgroup of π 1 generated by the orbit of γ under the action by the monodromy group (see [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF]).

In [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF], we defined the orbit depth k of the cycle γ of F by

k = sup{j ≥ 1|O ∩ L j ⊂ [O, π 1 ]} = min{j ≥ 1|O ∩ L j+1 ⊂ [O, π 1 ]}, (1.5)
and showed that the orbit depth k bounds the length of the iterated integral M µ , for the displacement map ∆ ϵ of any polynomial deformation (1.2), i.e. that ℓ ≤ k.

(1.6)

In [START_REF] Mardešić | Infinite orbit depth and length of Melnikov functions[END_REF], we gave an example of a system having unbounded depth and formulated the following conjectures: Conjecture 1.1.

(i) For any polynomial F and any non-trivial cycle γ of F , either the depth is unbounded, or it is 1, or 2. (ii) For any F and its cycle γ, either there exist deformations ω, whose first non-zero Melnikov function M µ is of arbitrary high length, or for any deformation ω, the length of M ω is 1 or 2.

This conjecture is similar in spirit to the result of Casale [START_REF] Casale | Suites de Godbillon-Vey et intégrales premières. (French) [Godbillon-Vey sequences and first integrals[END_REF] , concerning the first integral and the length of the corresponding Godbillon-Vey sequence of a system or the Tits alternative [START_REF] Tits | Free subgroups in linear groups[END_REF].

One of the central components in the study is the orbit complement abelianization group or O-abelianization

π O-ab 1 := π 1 O ∩ L 2 . (1.7)
Note that it is associated to the couple (F, γ) and does not depend on the deformation form ω.

In order to give some steps towards the proof of the above conjecture, we consider some other, more classical invariants associated to the group π O-ab 1 : its nilpotence class n and its derivative length d, which we relate to the orbit depth k and length ℓ of the first nonzero Melnikov function (see section 2 for definitions). Proposition 1.2. Let F ∈ R[x, y] be a polynomial, γ ∈ π 1 (F -1 (z)) be a real cycle and ω a polynomial 1-form as above, and let π O-ab 1 be the group defined in (1.

ℓ ≤ k ≤ n + 1, d ≤ n.
(1.8)

Remark 1.3. Although d ≤ n, there is no evident relationship between d and k.

General approach 1.4. The Poincaré (holonomy) map of a deformation (1.2) induces a homomorphism

P ω : π 1 → C[ϵ] ⊗ Diff (C, 0) , P ω (δ) = { z → z + ϵ ∫ δ ω + O(ϵ 2 ) } , (1.9)
where P ω (δ) is the Poincaré map with respect to the foliation (1.2) along the cycle δ.

Assume that the deformation (1.2) preserves the continuous families of cycles corresponding to elements of O ∩ L 2 . This means that P ω (O ∩ L 2 ) = {Id} and the map (1.9) descends to the homomorphism

P ω : π O-ab 1 → C[ϵ] ⊗ Diff(C, 0). (1.10)
Moreover, if one defines a deformation form ω in such a way that the obtained subgroup of diffeomorphisms is parabolic, then the group π O-ab 1 inherits properties of subgroups of parabolic diffeomorphisms. Using these properties, one could conclude that either π O-ab 1 is abelian, and so the orbit depth is less or equal to 2, or it is non-solvable.

However, the difficulty here lies in finding a one-form ω such that, for a cycle σ ∈ [π 1 , π 1 ] \ O ∩ L 2 , P ω (σ) = id, but it is the identity along any commutator in the orbit.

Here, we realize the above approach in the case of a Hamiltonian F given as a product of four real lines

F (x, y) = f 1 (x, y)f 2 (x, y)f 3 (x, y)f 4 (x, y), f i = a i x + b i y + c i , (a i , b i ) = (0, 0), (a i , b i ) = (a j , b j ), for i = j.
(1.11)

Recall that, as studied in [START_REF] Mardešić | Infinite orbit depth and length of Melnikov functions[END_REF], in the above case, if f i , i = 1, . . . , 4, consist of two pairs of parallel lines, the orbit depth k is infinite. Any Hamiltonian given by a product of four different lines is of one of the following three types:

(1) The four lines are in generic position (no parallel lines among f i , i = 1, . . . , 4, for different indices).

(2) One of the bounded domains bounded by these lines is a quadrilateral with exactly one pair of parallel opposite sides (we call it a trapezoid).

(3) One of the bounded domains bounded by these lines is a parallelogram (we call it a parallelogram).

Theorem 1.5. Let F be the product of four real lines (1.11), and γ ∈ π 1 (F -1 (z)) a continuous family of real cycles of F .

(1) In the case that the four lines are in generic position, then π O-ab 1 is abelian, so its nilpotence class n and derived length d are equal to 1. Hence the orbit depth k ≤ 2, and the length of the first non-zero Melnikov function is ℓ ≤ 2.

(2) In the trapezoid case π O-ab

1 is non-solvable d = n = ∞, but the orbit depth is k ≤ 2.
Hence, the length of the first non-zero Melnikov function of any deformation is bounded by 2, ℓ ≤ 2.

(3) In the parallelogram case π O-ab

1 is non-solvable d = n = ∞ and the orbit depth is infinite k = ∞.
Remark 1.6. We stress that the realization of the length is required in order to prove the conjecture in [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF] that states that the orbit depth is an optimal bound for the length of Melnikov functions. In [START_REF] Mardešić | Infinite orbit depth and length of Melnikov functions[END_REF] we provide a deformation of the parallelogram that has a first non-zero Melnikov function of length 3, which already distinguishes this case from the trapezoid, in terms of the length, where the length is less than or equal to 2.

The case of four lines in generic position has been studied in [START_REF] Uribe | Principal Poincaré-Pontryagin function associated to polynomial perturbations of a product of (d + 1) straight lines[END_REF][START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF]. We recall it here for completeness. In this work, our aim is to focus on more degenerated situations, the trapezoid and parallelogram cases.

Since the nilpotence class n of π O-ab 1 provides an upper bound for the orbit depth k, we also want to understand conditions under which π O-ab 1 is non-nilpotent (or non-solvable). From the definition of the homomorphism in (1.10) we see that the nilpotence class of π O-ab 1 is greater than the nilpotence class of its image under P ω . In this sense, the class of nilpotence of π O-ab 1 is related with the type of deformations preserving centers in O ∩ L 2 , and therefore, with the type of integrability of these deformations.

Theorem 1.7.

(1) In the parallelogram and the trapezoid cases, there exist polynomial 1-forms ω such that the deformations (1.2) preserve the pairs of parallel lines and have a first integral of Riccati type.

(2) Moreover, there are deformations, preserving the pairs of parallel lines, having Godbillon-Vey sequences of any finite length.

Remark 1.8. In [START_REF] Cerveau | Complex Codimension one singular foliations and Godbillon-Vey sequences[END_REF] the authors define the length of a foliation as the minimal length among all Godbillon-Vey sequences for the foliation. They mention that they do not know any example of finite length greater than 4. Deformations in the second part of Theorem 1.7 could provide such examples if one can prove that its Godbillon-Vey sequence has optimal length. Conjecture 1.9.

( Here by parallel curves, we mean two level curves f -1 (c 1 ) and f -1 (c 2 ), of the same function f , for c 1 = c 2 . Parallel curves interesect only at the line at infinity. (i) Given a group G, let G i be its lower central sequence:

G = G 1 ⊃ G 2 ⊃ • • • , G j+1 = [G j , G].
If there exists j ∈ N such that G j = {e}, we say that the group is nilpotent and define its nilpotence class n = n(G) as

n = min{j ≥ 1|G j+1 = {e}}, (2.1)
where e is the identity element in G.

(ii) Similarly, the upper central sequence G j is G = G 0 ⊃ G 1 ⊃ • • • , G j+1 = [G j , G j ].
If there exists j such that G j = {e}, we say that the group is solvable and define its derived length d as

d = min{j ≥ 1|G j = {e}}. (2.2)
Note that

G j+1 ⊃ G j . (2.3)
This gives d ≤ n and in particular any nilpotent group is solvable.

We apply these two notions to the orbit complement abelianization group π O-ab

1
given in (1.7).

Proof of Propostion 1.2. In order to prove that k ≤ n + 1, it suffices to prove that

(π O-ab 1 ) j = {ē} implies O ∩ L j+1 ⊂ [O, π 1 ]
, for any j. We claim first that the assumption (π O-ab 1

) j = {ē}, implies L j ⊂ O.
(2.4) Hence,

Indeed, let σ = [[• • • [γ 1 , γ 2 ], • • • , γ j-1 ], γ j ]
L j+1 = [L j , π 1 ] ⊂ [O, π 1 ], showing that O ∩ L j+1 ⊂ [O, π 1 ].
The relation d ≤ n comes from expression (2.3). On the other hand, it is known from [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF] that the length ℓ of the first non-zero Melnikov function is bounded by the orbit depth k. Therefore, ℓ ≤ k ≤ n + 1. □

Germs of diffeomorphisms

Given a germ of diffeomorphism f ∈ Diff(C, 0), we say that it is parabolic if it is of the form f (z) = z + o(z). If f is not the identity, then f = z + az p+1 + o(z p+1 ), with a = 0. We call p the level of f . Let Diff 1 (C, 0) ⊂ Diff(C, 0) denote the subgroup of parabolic germs.

The general approach given in 1.4 is based on the following well-known facts about the solvability of the group of parabolic germs Diff 1 (C, 0). Lemma 3.1. (Proposition 6.11, [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF])

Let f = z+az p+1 +• • • and g = z+bz q+1 +• • • . Then [f, g](z) = z + ab(p -q)z p+q+1 + o(z p+q+1 ).
Proposition 3.2. (Lemma 6.13 [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF]) Let G be a finitely generated subgroup of parabolic germs Diff 1 (C, 0). Then (i) or (ii) holds (i) G is abelian, i.e. of nilpotence class n(G) ≤ 1.

(ii) G is not solvable (hence not nilpotent).

Lemma 3.3. [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF] Let G be a finitely generated subgroup of parabolic germs Diff 1 (C, 0). Then G is solvable if and only if it is abelian. Moreover, one of the following statements holds:

(i) G is not abelian and there exist two diffeomorphisms f and g in G of different levels. (ii) G is abelian and all diffeomorphisms in G are of the same level.

Proof. Suppose G = {id}. We now consider two cases: either there exist two germs f = z + az p+1 + o(z p+1 ) and g = z + bz q+1 + o(z q+1 ) in G of different level q = p, or all germs are of the same level.

In the first case, by Lemma 3.

1, h = [f, g](z) = z +ab(p-q)z p+q+1 +o(z p+q+1 ) = z. Therefore, G 1 = [G, G] = {id}.
Applying inductively the same reasoning on h and g, one can show that G ℓ = {id} and G ℓ = {id}, for all ℓ. Thus, G is not nilpotent, nor solvable. Now, suppose that all the elements of G have the same level p. Then, by Lemma 3.1, given f = z + az p+1 + o(z p+1 ) and g = z + bz p+1 + o(z p+1 ), the element [f, g] is either of a level strictly greater than p or the identity. The first option is impossible by the assumption, so it follows that G 2 = G 1 = {id}. Thus G is abelian. □ 4. Proof of Theorem 1.5

Proof. In the case (1), of product of lines in generic position, it has been proved in [START_REF] Uribe | Principal Poincaré-Pontryagin function associated to polynomial perturbations of a product of (d + 1) straight lines[END_REF] that

[π 1 , π 1 ] ⊆ O. Therefore, π O-ab 1
is abelian, and k ≤ 2. Hence, the orbit depth of the real cycle, as well as the length of the first non-zero Melnikov function for any deformation, is bounded by 2, by [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF].

To prove that π O-ab 1 is non-solvable for cases (2) and (3) we follow the same strategy. In both cases, by an affine change of coordinates we can assume that the Hamiltonian is given by F = (x -1)(x + 1)f 3 f 4 , where f 3 and f 4 are linear factors (non-parallel in case (2) and parallel in case (3)). Now, consider the foliation F ω = {dF + ϵω = 0}, where ω = F 2 ( dx x-1 + F dx x+1 ), and the homomorphims defined by the holonomy with respect to F ω :

P ω : π O-ab 1 → C[ϵ] ⊗ Dif f 1 (C, 0).
We define the subgroup G := P ω (π O-ab 1 ) of the group of parabolic difeomorphims with coefficients depending on ϵ. Note that the morphism P ω is well defined. Namely, since F ω has a reflexion symmetry with respect to y-axis, then it preserves the center at the origin. Moreover, since

P ω [δ 1 , δ 2 ] = z+ϵ 2 W (z 2 , z 3 ) ∫ [δ1,δ2] dx x-1 dx x+1 + O(ϵ 3 ) = z,
where W is the Wronskian, then G is not Abelian, and therefore nonsolvable.

In [START_REF] Mardešić | Infinite orbit depth and length of Melnikov functions[END_REF] it is proved that the parallelogram has unbounded orbit depth. It remains to prove that for the trapezoid the orbit depth is k ≤ 2. By an affine change of coordinates we may assume that F is defined by

F = (x -1)(x + 1)f 3 f 4 , (4.1)
and has the configuration given in Figure 1. The zero level F -1 (0) consists of four lines, enclosing a quadrilateral and a triangle. Let γ be the real cycle in F -1 (z ′ ), z ′ > 0, close to zero and enclosed by the quadrilateral, and let γ 1 be the real cycle in F -1 (z ′′ ), z ′′ < 0, close to zero and enclosed by the triangle, both with positive orientation. Let p i be the vertices of the quadrilateral oriented positively with p 2 and p 3 the commun vertices with the triangle and let p 5 be the last vertex of the triangle. Let δ i , be the vanishing cycles at the vertices p i , i = 1, . . . , 5 taken so that the intersection numbers verify (γ, δ i ) = 1, i = 1, . . . , 4 and (γ 1 , δ 5 ) = 1.

5

.

Figure 1. The real loops γ(z) and γ 1 (z) and the complex vanishing loops δ i (z) as elements of π 1 ( F -1 (z), p 0 ) .

Using the Gauss-Manin connection, we continue analytically γ, γ 1 , δ i to a nonsingular curve

F -1 (z), 0 < |z| 1, along segments [z ′ , z] and [z ′′ , z].
Recall that, by [START_REF] A'campo | Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I[END_REF], the intersection numbers verify:

(γ, γ 1 ) = 1 (4.2)
and

(γ 1 , δ 2 ) = (γ 1 , δ 3 ) = 1. (4.3)
Remark 4.1. Theorem 1.5 is also true in the complex case if conditions (4.2) and (4.3) are fulfilled.

In [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF], we obtained the orbit of the real cycle for a first integral of triangle type, that is

O γ1 =< γ 1 , δ 2 δ 3 δ , [δ 2 , δ 3 ] > mod [O γ1 , π 1 ],
and in [START_REF] Mardešić | Infinite orbit depth and length of Melnikov functions[END_REF], for the quadrilateral type. The quadrilateral case in a neighborhood of the quadrilateral is the same as in the parallelogram case. Now, due to (4.2), in our case (4.1), the orbit of γ is generated by the union of the orbits in these two cases. This gives:

O γ [O γ , π 1 ] = =< {γ, δ 1 δ 2 δ 3 δ 4 , [δ 1 δ 2 , δ 2 δ 3 ], [δ 1 δ 2 , [δ 2 , δ 2 δ 3 ]], . . . , [δ 1 δ 2 , [δ 2 , . . . [δ 2 , δ 2 δ 3 ]], . . .} ∪ O γ1 > .
We know that [δ 2 , δ 3 ] belongs to the orbit of γ 1 and hence to O γ . Hence,

[δ 2 , δ 2 δ 3 ] = δ 2 [δ 2 , δ 3 ]δ -1 2
belongs to the orbit as well. This gives that in the above expression for the orbit

O γ the commutator [δ 1 δ 2 , [δ 2 , δ 2 δ 3 ]] belongs to [O, π 1 ].
The same is therefore true for all terms which contain it (all the terms following it in the above braces). We know [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF] that the depth of O γ1 is two. All the terms following the term

[δ 1 δ 2 , δ 2 δ 3 ] are in [O, π 1 ],
showing that the depth k verifies k ≤ 2 and hence by [START_REF] Mardešić | Bounding the length of iterated integrals of the first non-zero Melnikov function[END_REF] the length of the first non-zero function M µ of any deformation ω is ℓ ≤ 2. □

Types of integrability of the deformations

We will study the Godbillon-Vey sequence for the foliation

dF + ϵω = 0, (5.1)
with

F = f 1 f 2 f 3 f 4 and ω = p 1 (F ) df1 f1 + p 2 (F ) df2 f2 , where f 1 = x -1, f 2 = x + 1, p 1 , p 2 are
polynomials in F , and f 3 , f 4 are linear factors different from f 1 and f 2 . For the parallelogram, f 3 and f 4 define parallel lines, while for the trapezoid they do not. By explicit computation of Godbillon-Vey sequences we will show that the foliation dF + ϵ (

df1 f1 + F df2 f2 ) = 0 is Liouville integrable, while dF + ϵ ( F df1 f1 + F 2 df2 f2 ) = 0 is Riccati integrable. Moreover, for ω = p 1 (F ) df1 f1 + p 2 (F ) df2
f2 , with n = deg{deg p 1 , deg p 2 }, foliation (5.1) admits a Godbillon-Vey sequence of length ≤ n, which increases complexity of the first integral for those cases. Therefore, for n ≥ 3 the foliation (5.1) could have a first integral which is not of Riccati type, with infinite dimensional pseudo-group of holonomy.

We recall that a Gobdillon-Vey sequence for a meromophic 1-form η 0 is a sequence {η 0 , η 1 , ...}, such that

dη 0 = η 0 ∧ η 1 dη 1 = η 0 ∧ η 2 dη 2 = η 0 ∧ η 3 + η 1 ∧ η 2 dη 3 = η 0 ∧ η 4 + 2η 1 ∧ η 3 . . . dη n = η 0 ∧ η n+1 + n ∑ k=1 ( n k ) η k ∧ η n-k+1 (5.2)
It is of length ℓ if η j = 0 for all j ≥ ℓ.

It is known that η 0 has a first integral of Liouville type (respectively of Riccati type), if and only if, it admits a Godbillon-Vey sequence of length 2 (respectively of length 3).

5.1. Degree on F equals 1. Let us start by analyzing the case ω = F df1 f1 . We denote η 0 := dF + ϵF df1 f1 . Then

dη 0 = ϵdF ∧ df 1 f 1 .
Thus, we can take

η 1 := ϵ df1 f1 = df ϵ 1 f ϵ 1 .
Then, dη 1 = 0, which means η 2 = 0. Hence, η 0 admits a Godbillon-Vey sequence of length 2, and therefore (5.1) is Liouville integrable. Moreover, Casale's article [START_REF] Casale | Suites de Godbillon-Vey et intégrales premières. (French) [Godbillon-Vey sequences and first integrals[END_REF] provides a way to compute from this sequence the first integral. This is given by dH = Gη 0 , where η 1 = dG G . In this case, we have G = f ϵ 1 . Therefore the first integral H is given by ) , which will give us a way to proceed for higher degree. By substituting f 1 = x-1, f 2 = x+1, we obtain

f ϵ 1 η 0 = f ϵ 1 (dF + F df ϵ 1 f ϵ 1 ) = f ϵ 1 dF + F df ϵ 1 = d(f ϵ 1 F ); thus, H = f ϵ 1 F ,
η 0 = dF + ϵ ( 1 x-1 + F x+1 ) dx. Denote φ(x, F ) = 1 x-1 + F x+1 .
Then η 0 = dF + ϵφdx, and dη 0 = ϵdφ ∧ dx.

(5.3)

Note that dφ = φ F dF + φ x dx, where φ F = ∂φ ∂F (x, F ) = 1 x+1 , and

φ x = ∂φ ∂x (x, F ) = -1 (x-1) 2 -F (x+1) 2 .
Then, (5.3) is equal to dη 0 = ϵφ F dF ∧ dx.

(5.4)

Define η 1 := ϵφ F dx. It satisfies the equation dη 0 = η 0 ∧η 1 = (dF +ϵφdx)∧ϵφ F dx. On the other hand, dη 1 = ϵdφ F ∧ dx, where dφ F = φ F F dF + φ F x dx. That is, dη 1 = ϵφ F F dF ∧ dx, which is zero because φ F F = 0. Then, we can take η 2 = 0. Thus, η 0 admits a Godbillon-Vey sequence of length 2, and therefore is Liouville integrable. To compute the first integral H, we have to solve the equations dH = Gη 0 , and η 1 = dG G . For this purpose, we note that

η 1 = ϵ dx x+1 = df ϵ 2 f ϵ 2 . Then, dH = f ϵ 2 η 0 = f ϵ 2 ( dF + ϵ ( df1 f1 + F df2 f2
)) , which corresponds to

f ϵ 2 ( dF + df ϵ 1 f ϵ 1 + F df ϵ 2 f ϵ 2 ) = f ϵ 2 dF + f ϵ 2 df ϵ 1 f ϵ 1 + F df ϵ 2 = d ( f ϵ 2 F + ∫ f ϵ 2 df ϵ 1 f ϵ 1
) .

Therefore,

H = f ϵ 2 F + ∫ f ϵ 2 df ϵ 1 f ϵ 1 . By substituting the expressions of f 1 , f 2 , we get ∫ f ϵ 2 df ϵ 1 f ϵ 1 = ∫ ϵ(x + 1) ϵ dx
x-1 . We stress that the construction above is analogous for f 1 and f 2 defining two arbitrary parallel lines.

Proof of Theorem 1.7

Proof. We consider the deformation dF + ϵ(F df1 f1 + F 2 df2 f2 ), where f 1 = x -1, f 2 = x + 1, and define

η 0 := dF + ϵ ( F x -1 + F 2 x + 1 ) dx. (6.1) Denote φ(x, F ) = F x-1 + F 2 x+1
. Then η 0 = dF + ϵφdx, and since dφ = φ F dF + φ x dx, where φ F is the partial derivative of φ with respect to F : φ F = 1

x-1 + 2F x+1 , we have

dη 0 = ϵφ F dF ∧ dx. (6.2)
Thus, we define η 1 := ϵφ F dx. It satisfies the equation dη 0 = η 0 ∧ η 1 . Now we consider dη 1 = ϵdφ F ∧ dx. Then, again, writing dφ F = φ F F dF + φ F x dx, we have dη 1 = ϵφ F F dF ∧ dx. So, we can take η 2 := ϵφ F F dx. It satisfies the equation dη 1 = η 0 ∧ η 2 . Continuing with the Godbillon-Vey sequence we consider dη 2 = ϵdφ F F ∧ dx. But, since the degree of φ in F is 2, φ F F depends only on x, therefore dφ F F ∧ dx = 0. Now, for the equation dη 2 = η 0 ∧ η 3 + η 1 ∧ η 2 , since η 1 ∧ η 2 = 0, we can take η 3 = 0. Hence, η 0 admits a Godbillon-Vey sequence of length 3, and therefore it has a first integral of Riccati type. According to Casale's method [START_REF] Casale | Suites de Godbillon-Vey et intégrales premières. (French) [Godbillon-Vey sequences and first integrals[END_REF] we can compute the first integral by solving the following equations:

dH = G 1 η 0 dG 1 = G 1 ( η 1 + 2 G 2 η 0 ) dG 2 = - G 2 2 2 η 2 -G 2 η 1 -η 0 , (6.3)
with

η 0 = dF + ϵ ( F x-1 + F 2 x+1 ) dx, η 1 = ϵ ( 1 x-1 + 2F x+1 ) dx, and η 2 = ϵ ( 2 x+1
) dx.

One can check that H = -1

F f ϵ 1 + ∫ df ϵ 2 f ϵ 1 f ϵ 2 , G 1 = 1 F 2 f ϵ 1
and G 2 = -F satisfy equations (6. 3). Now, we analyze what happens when we increase the degree in F for the foliation (5.1). Let

η 0 := dF + ϵ ( F 2 df 1 f 1 + F 3 df 2 f 2 ) . (6.4) Substituting f 1 = x -1, f 2 = x + 1, we obtain η 0 = dF + ϵ ( F 2 x-1 + F 3 x+1
) dx. Following the same process as before, we write η 0 = dF + ϵφdx, where φ(x, F ) = F 2

x-1 + F 3 x+1 . Since dφ = φ F dF + φ x dx, we have dη 0 = ϵφ F dF ∧ dx, (

where φ F = 2F

x-1 + 3F 2 x+1 . Thus, we define η 1 := ϵφ F dx, note that deg F φ F = 2. It satisfies the equation dη 0 = η 0 ∧ η 1 . Now we consider dη 1 = ϵdφ F ∧ dx. Writing dη 1 = φ F F dF + φ F x dx, we have dη 1 = ϵφ F F dF ∧ dx. So, we can take again η 2 := ϵφ F F dx, where now deg F φ F F = 1. It satisfies the equation dη 1 = η 0 ∧ η 2 . Then dη 2 = ϵdφ F F ∧ dx = ϵφ F F F dF ∧ dx. We want η 3 to be such that dη 2 = η 0 ∧ η 3 + η 1 ∧ η 2 . Since η 1 ∧ η 2 = 0, we can take η 3 = ϵφ F F F dx. Consider dη 3 = ϵdφ F F F ∧ dx. This is zero, since φ F F F depends only on x. We want now η 4 verifying the equation dη 3 = η 0 ∧ η 4 + 2η 2 ∧ η 3 . But, since η 2 ∧ η 3 = 0, we can take η 4 = 0. Moreover, from this construction η k ∧ η ℓ = 0 for all k, ℓ ≥ 1, hence this Godbillon-Vey sequence has length 4. Therefore, η 0 could have a first integral which is not of Riccati type. With the same procedure we can obtain a deformation with a Godbillon-Vey sequence of any finite length n, by taking the degree in F of the deformation equal to n -1.

We stress that the above computations do not depend on the factors f 3 and f 4 , therefore they are valid for the parallelogram, as well as for the trapezoid. □ Remark 6.1. From these cases one can observe that the length of the Godbillon-Vey sequence for the foliation (5.1) increases with the degree in F of the function φ(x, F ). So, increasing the degree in F for the deformation, we can have a first integral of more complicated type.

In particular, for n = deg F φ > 4 one should obtain a Godbillon-Vey sequence of finite length n higher than 4. In the paper [START_REF] Cerveau | Complex Codimension one singular foliations and Godbillon-Vey sequences[END_REF] (p. 25) the authors say they do not know any example of finite length greater than 4. However, the length is defined as the minimal length among all the Godbillon-Vey sequences that the foliation can admit. So, in order to prove that these examples for n > 4 have Godbillon-Vey length higher than 4, one should prove that the constructed Godbillon-Vey is optimal. That is that there does not exist some other Godbillon-Vey sequence of smaller length.

1 , 1 ,

 11 7). Let n be the nilpotence class of π O-ab d the derivative length of π O-ab k the orbit depth and ℓ the length of the first nonzero Melnikov function of the deformation (1.2) along γ. Then the following inequalities hold

)

  The non-solvability of the group π O-ab 1 is characterized by the presence of a pair of parallel curves in the Hamiltonian foliation. (2) The non-bounded orbit depth is characterized by the presence of two pairs of parallel curves in the Hamiltonian foliation. (3) The type of singularity given by a pair of parallel curves at the line at infinity characterizes the non-solvability of π O-ab 1 .

2 .

 2 Nilpotence class and derivative length Definition 2.1.

1 ). Then σ ∈ ( π O-ab 1 )

 11 be a generator of L j and let σ ∈ π O-ab 1 be the class of this σ i in (π O-ab j = {e}, i.e. σ ∈ O ∩ L 2 . We thus have(2.4).

  which lies in a Liouvillian extension.Now we analyze the following case η0 := dF + ϵ ( df1 f1 + F df2 f2

This work was supported by Israel Science Foundation grant 1167/17, Papiit (Dgapa UNAM) IN110520, Croatian Science Foundation (HRZZ) grants no. 2285, UIP 2017-05-1020, PZS-2019-02-3055 from Research Cooperability funded by the European Social Fund.

1