Ettien Lazare Kpré
email: ettien.kpre@etu.unilim.fr

Cyril Decroze

Passively Coded Synthetic Aperture Interferometric Radiometer (CSAIR): Theory and Measurement Results

Keywords: Synthetic aperture, radiometry, Microwave imaging, microwave device

Based on the synthetic interferometric imaging technique (SAIR), a new microwave radiometer architecture is proposed in this paper to detect thermal noise sources with a passive coded measurement approach so called CSAIR. In this new system, a passive microwave device is used to intrinsically code and multiplex the antenna signals. This allows the reduction of the number of RF chains while keeping the same antenna array configuration needed in a conventional interferometric radiometers. The system principle is described and the signal processing required for the target brightness temperature rendering is also discussed. Simulation and measurement results show the effectiveness and the potential of the proposed system.

I. INTRODUCTION

Microwave radiometer are powerful sensors for highresolution imaging. Nevertheless, due to the trade off between the spatial resolution and the antennas aperture, real aperture radiometer cannot achieve the desired high spatial resolution. Synthetic aperture radiometer has been suggested as an alternate to real aperture radiometer [START_REF] Ruf | Interferometric synthetic aperture microwave radiometry for the remote sensing of the Earth[END_REF]. Indeed, it can synthesize a large aperture from a few sparsely arranging number of antennas. In this way, high spatial resolution can be achieved without any mechanical or electronically scanning [START_REF] Dong | Antenna Array Design in Aperture Synthesis Radiometers[END_REF]. The basic idea is to measure all the possible cross-correlation between each pair of electromagnetic waveforms received by the antennas. These correlation samples are usually called visibility function and their relative positions in the spatial frequency domain are called baselines. In the ideal case, the visibility function is related to the target brightness temperature through a spatial Fourier transform [START_REF] Zernike | The concept of degree of coherence and its application to optical problems[END_REF]. Numerous radiometers are based on this concept such as the Electronically Scanned Thin Array Radiometer (ESTAR) [START_REF] Le Vine | Synthetic aperture radiometer systems, Microwave Theory and Techniques[END_REF], Microwave Imaging Radiometer using Aperture Synthesis (MIRAS) [START_REF] Martin-Neira | MIRAS, a two-dimensional aperture synthesis radiometer[END_REF].

Besides the maturity of microwave technologies, high resolution imaging algorithms have greatly improved the conventional interferometric radiometer performances [START_REF] Chen | An accurate imaging algorithm for millimeter wave synthetic aperture imaging radiometer in near-field[END_REF]. Thereby, the SAIR technique can be applied to practical near-field applications such as security check, concealed weapons detection, since they require real time acquision and high spatial resolution. On the over hand, high resolution imaging requires a large number of antennas which is directly linked to the number of RF chains. As depicted in Fig. 1 a conventional interferometric radiometer architecture consists of an antenna array, where each antenna is connected to a single RF chain, insuring the down-conversion of the RF signal to a digital baseband IQ signal. Then, the ensuing process consists to perform the correlations in pair and image reconstruction. As consequence, the hardware complexity increases as well as the dimensional requirements of the antenna array.

The main contribution of this paper is to propose a new approach using a M × N (M N ≥ 1) ports microwave device whose transfer functions are uncorrelated to intrinsically code the receive waveforms from the M antennas. Herein, the device frequency diversity is associated to a spatial diversity through N output ports connected to N shared RF chains. A generalized visibility function recovering method is introduced and the image reconstruction ensuing. The simulation and experimental results indicate that the new architecture can be an alternate to drastically reduce the complexity of the conventional microwave radiometers while keeping the same antenna array geometry.

II. CSAIR FUNDAMENTALS

In this section a new radiometer architecture is introduced to reduce the number of RF chains needed in a conventional SAIR architecture. The aim of this concept is to measure the visibility function sample from a compressed acquisition performed in the physical layer. A generic setup of the passive coded SAIR (CSAIR) is described in Fig. 2

c n (f ) = M m=1 h nm (f )s m (f ) (1)
where h nm (f ) corresponds to the device transfer function between the antenna m and the device output n. s m (f ) is the received signal by the m th antenna. For the sake of efficient description of the principle, Eq.1 can be reformulated for each frequency sample as linear equation bellow:

c = Hs (2)
Where c and s are respectively N ×1 and M ×1 vectors and H is N ×M matrix. Since the principle of the SAIR is to measure the cross-correlation between each pair of the received signals, the challenge is to estimate the individual antenna signal from the aggregate signal c n (f ). Therefore, due to the reduction of the number of RF chains, the dimension of the measured signal c is significantly less than the received signals s. So the Eq.2 is underdetermined equation. Under this ill-posed problem, the most efficient method is the equalization to solve it. A standard approach is to estimate the received signal s r that minimizes the l 2 norm of the objective function defined by [START_REF] Bertero | Introduction to Inverse Problems in Imaging[END_REF]:

min||Hs -c|| 2 2 + µ||s|| 2 2 ( 3
)
where µ is the regularization parameter which can be tuned to obtain a trade-off between the regularization term and the penalty function. The choice for µ must be done before inversion. Tikhonov solution is an example of suitable equalization method which can explicitly be formulated as follow:

H + µ = (H † H + µI) -1 H † . (4)
(.) + and (.) † are respectively the equalized pseudo-inverse and the transposed conjugate operators. The received signal can be computed by :

s rµ = H + µ c (5)
The estimated signal can be derived Eq.2 and Eq.5 by the following relation:

s rµ = H + µ c = H + µ Hs = R µ s (6)
R µ denotes the pseudo-correlation matrix of the device transfer functions. Ideally this matrix tends to an identity matrix leading to a perfect estimation of the received signal. But this is hardly achievable in practice, limited by the frequency and spatial diversity of the component. Thereby it can be approximated by the use of an oversized metallic microwave cavity which transfer function are uncorrelated [START_REF] Fromenteze | Single-Shot Compressive Multiple-Inputs Multiple-Outputs Radar Imaging Using a Two-Port Passive Device[END_REF].

The visibility function between two antennas labeled i and j can be computed by:

V ij (u, v) =< s rµi (f ).s * rµj (f ) > (7)
Based on interferometry fundamentals and first order Taylor approximation, the near-field visibility function is related to the brightness temperature through the following equation [START_REF] Chen | An accurate imaging algorithm for millimeter wave synthetic aperture imaging radiometer in near-field[END_REF]:

V ij (u, v) = e -iΦ(u,v) × ξ 2 +η 2 ≤1 T M (ξ, η)r ij (- uξ + vη f 0 )e i2π(uξ+vη) dξdη (8)
where T M (ξ, η) is the modified brightness temperature given by:

T M (ξ, η) = T (ξ,η)F i (ξ,η)F * j (ξ,η) √ 1-ξ 2 -η 2 (9)
(u, v) is the relative antennas spacing normalized by the wavelength namely baselines, (ξ, η) are the spatial coordinates of the radiating source, T (ξ, η) is the brightness temperature, F i (ξ, η) and F j (ξ, η) are the normalized antennas radiation pattern, 1/ 1 -ξ 2 -η 2 is the oblique factor. r(τ ) = r(τ )e -j2πf0τ is the so-called fringe washfunction which account for spatial decorrelation effect and depends on the component transfer function through Eq.11 [START_REF] Fischman | How digital correlation affects the fringe washing function in L-band aperture synthesis radiometry[END_REF]:

r ij (τ ) = ∞ 0 R µi (f )R * µj (f )e j2πf τ df (10)
For an ideal radiometer, this parameter can be neglected whereas for a real CSAIR system, the fringe-wash function can rapidly be significant, thus it is important to take it in account for image reconstruction. Φ(u, v) = π(d 2 i -d 2 j )/λR is the quadratic modified term which should be considered for a near-field imaging [START_REF] Chen | An accurate imaging algorithm for millimeter wave synthetic aperture imaging radiometer in near-field[END_REF]. R and d i are respectively the imaging distance and the antenna euclidean distance from the center of the array. Formulating visibility function and brightness temperature images as vectors of respectively length P and Q, Eq.8 can be rewritten in a matrix form [START_REF] Zhang | A Robust Reweighted L1-Minimization Imaging Algorithm for Passive Millimeter Wave SAIR in Near Field[END_REF]:

V = G.T ( 11
)
Where G is the P × Q-length system matrix, P is the number of visibility samples including redundant elements, Q is the number of reconstructed pixels and G(p, q) is the p th row and q th column component of G .

G(p, q) = F i (ξ q , η q )F * j (ξ q , η q )r ij (-

u p ξ q + v p η q f 0 )
× e i(2π(upξq+vpηq)-φ(up,vp)) (12)

Finally, a conventional G matrix inversion can be applied to reconstruct the target image [START_REF] Anterrieu | Regularization of an inverse problem in remote sensing imaging by aperture synthesis[END_REF]. Counter to a simple spatial Fourier transform, the G matrix inversion helps to correct errors due to the fringe-wash function, oblique factor, and antenna patterns. This method is the basis of the image reconstruction in the following sections.

The proposed method is validated by the means of numerical simulations and experiments measurement in the following sections.

III. COMPUTATIONAL RESULTS AND ANALYSIS

In this section, two numerical scenarios are held to compare the proposed CSAIR concept and the conventional SAIR. The scenarios consist of 16 isotropic antennas arranged along a "T" shaped array as depicted in Fig. 3. The antennas are separated by an inter-elements spacing of d x = 0.7 × λ, where λ is the central wavelength in the microwave S-Band (2-4GHz). The noise sources are assumed to be able to radiate centimeter-wave with a power proportional to their brightness temperature T (ξ, η). The integration time used for these scenarios is τ = 1µs with an average factor of 128. In the first scenario, each antenna is directly connected to a single receiver channel as shown in Fig. 1. The visibility function is directly calculated by performing the correlation in pair between the received waveforms, while in the second scenario, the antennas are connected to 16 × N (i.e N = 2) microwave device as depicted in Fig. 2. A simple analytic model of the component transfer functions is numerically implemented to model an oversized microwave cavity general behavior [START_REF] David | Hill Electromagnetic Theory of Reverberation Chambers[END_REF]:

h nm (f ) = F [g nm (t)e -t/τc ] ( 13
)
with g nm (t) a random Gaussian distribution which is a convenient approximation of the oversized microwave cavity. τ c represents the decay time parameter which is linked to the component quality factor. The decay time is set to 1µs in this example. The antenna signals are estimated by the equalization method described in the previous section, the visibility function is thus calculated from Eq.7. Finally, the same imaging algorithm based on Tikhonov regularization [START_REF] Anterrieu | Regularization of an inverse problem in remote sensing imaging by aperture synthesis[END_REF] is applied to both scenarios for the brightness temperature reconstruction (Eq.11). Original point-like sources image is reconstructed considering the two scenarios, especially the conventional SAIR, and the CSAIR as shown in Fig. 4. The imaging resolution is about (18cm, 28cm) for both techniques and the image dynamic is about 14dB for the conventional SAIR and 10dB for the CSAIR. Fig. 5 bellow shows the cut-plane along X-axis of the point-like sources image. The simulation model was then extended to a 2-D 'T' shaped objects (i.e Fig. 4). Even if the dynamic of the CSAIR imaging result is degraded, the image rendering is sufficient to clearly recognize the shape of the object. This demonstrate the effectiveness of the proposed system. Nevertheless, one way to improve the image quality could be to increase the average factor or the spatial diversity (Number of the device output ports).

IV. MEASUREMENT RESULTS

A measurement platform has been setup to validate the CSAIR concept and the deconvolution strategy. The setup is the same as that introduced in simulation part with 16 Vivaldi antennas designed to match 2 -4GHz band. The antennas are arranged in 'T' shape (9 along X-axis and 7 along Y-axis) with an inter-elements spacing of dx = dy = 7cm. Three noise sources are emulated using three 12Gs/s Arbitrary Waveform Generators (Agilent M8190A) connected to three horn antennas. A 16 × 2 ports air-filled metallic cavity with outer dimensions of 0.8 × 0.8 × 1m 3 has been utilized as coding device as depicted in Fig. 6. The channel transfer functions is pre-characterized and then used as the coding matrix as defined in Eq.1. The noise time integration is τ = 170µs. The experience is validated with the 16 Vivaldi antennas connected to the cavity and only two signals are measured by a DSO90404A 20 GSa/s Agilent oscilloscope. Fig. 7 shows the imaging result of three horn antennas. As it can be noticed, the proposed system is able to detect and discriminate the three sources with only two measured waveforms. The resolution is the same as the theoretical one (18cm, 28cm). The imaging result show that the CSAIR can be an efficient alternative of a conventional SAIR since it is cost-efficient technique. Note that in this experimentation, noise signal acquisition has been performed in one shot without any average. This implies that the image dynamic could be significantly improved by implementing averaging on this experimental bench.

V. CONCLUSION

In this paper, a new synthetic aperture architecture has been proposed based on a passive coding technique performed in the physical layer. Actually a patent is pending on this new approach. In this concept, a microwave coding device is used to intrinsically code and multiplex the M antenna signals into N measured waveforms (M N ≥ 1). Equalization based on Tikhonov method in frequency domain has been applied to estimate the antenna signal needed to reconstruct the target image. The simulation and measurement results show the potential of the proposed system. Since it can drastically reduce the number of RF chains while keeping the same number of antennas, this new approach can obviously be an alternate of the conventional SAIR systems. Future works will focused on an enhanced reconstruction methods and accurate imaging algorithms to reach the conventional SAIR performances. In addition, an important metric to evaluate the performances of the proposed system is the sensitivity ∆T which represents the minimum change in temperature of the source that can be detected by the radiometer. Thereby, extended analysis to quantitatively evaluate the sensitivity of the proposed system will be implemented.

Figure 1 .

 1 Figure 1. Architecture of a conventional microwave radiometer.

  . The receiving antenna array is connected to a M ×N (Output/Input) ports microwave device which intrinsically codes and multiplexes the received M antenna signals into N M output signals, leading to the architecture simplification. Then the measured signal c n (f ) (n ∈ [1, 2, .., N ]) without noise at the device single output in frequency domain can be expressed by:

Figure 2 .

 2 Figure 2. Passive coded SAIR architecture. M received signals are coded into N measured signals.

Figure 3 .

 3 Figure 3. T-shaped antenna array and the corresponding (u, v) coverage.

Figure 4 .

 4 Figure 4. Simulation results of point-like sources and a 'T' shaped 2-D source imaging with conventional SAIR and CSAIR imaging system.

Figure 5 .

 5 Figure 5. Cut-plane along X-axis of the point-like sources image.

Figure 6 .

 6 Figure 6. 'T' shaped antenna array connected to the metallic microwave cavity.

Figure 7 .

 7 Figure 7. Imaging Results. (a) Three horn antennas used in the noise source emulator. (b) Imaging result of the noise sources.

ACKNOWLEDGMENT

This work is supported by the French National Research Agency (ANR) within the framework of the funded project "PIXEL". The authors would like to thank all partners of this project.


WAGE-LED REGIME, PROFIT-LED REGIME AND CYCLES: A MODEL

Nicolas CANRY CES Matisse, Université Paris 1

We propose a dynamic model which deals with the impact of income distribution variations on growth. In that goal, we use two models : the classical Goodwin model (1967) and the Bhaduri-Marglin model (1990), which also focuses on the links between income distribution and growth, but in a Keynesian frame. We introduce Keynesian demand constraints within the Goodwin model and modify its investment function, which becomes non-linear. With these new hypotheses, we show that Goodwin cycles may either be maintained or disappear. If most trajectories oscillate around a classical equilibrium, the economy may also fall during a cycle into a Keynesian unemployment state. In that case, cycle dynamic is broken because wages are squeezed whereas the economy is in a wage-led regime. This model allows to capture some speci…c characteristics of the French economic situation that took place in the 1980s-1990s.

The impact of income distribution on growth has been studied by two di¤erent strands of economic literature, belonging respectively to the classical and post-Keynesian traditions. On the classical side, [START_REF] Goodwin R | ‹ ‹ A growth cycle››[END_REF] proposes a formalization of endogenous cycles where investment is directly related to saving. Because the propensity to save of entrepreneurs is higher than that of workers, an increase (respectively a decrease) in the pro…t share always boosts (respectively hinders) accumulation and growth. The latter assumption has however been questioned by some Keynesian economists. According to [START_REF] Bhaduri | ‹ ‹ Unemployment and the real wage : the economic bases for contesting political ideologies››[END_REF], the e¤ect on growth of a rise in pro…ts remains uncertain: it may either be positive (pro…tled regime) or negative (wage-led regime). Nevertheless, the Bhaduri-Marglin model remains incomplete as it does not tackle dynamic issues.

CES Matisse, Université Paris 1. MSE 106-112, boulevard de l'Hôpital 75013 Paris. France; e-mail address : nicolas.canry@univ-paris1.fr. I am very grateful to Robert BOYER and Dominique LEVY for their helpful comments and advice. Any errors are my own.

1

A few contributions have already introduced Keynesian hypotheses in a classical framework. [START_REF] Skott | ‹ ‹ E¤ective Demand, Class Struggle and Cyclical Growth››[END_REF] integrates a class struggle dynamics in a Kaldorian trade cycle model. Duménil and Levy (1999) build a model combining a short term Keynesian dynamics with a long term classical one, with adjusting prices. In this article, we try to incorporate both con…gurations of Bhaduri-Marglin's model within the Goodwin framework. We so underline that the e¤ects of income distribution evolutions on growth fundamentally depend on the growth regime existing in the economy. Besides, we assume that the two regimes emphasized by [START_REF] Bhaduri | ‹ ‹ Unemployment and the real wage : the economic bases for contesting political ideologies››[END_REF] are endogenous and are themselves related to the level of the wage share. Thus we propose a dynamic model where growth a¤ects income distribution, which itself may modify in return the nature of the growth regime and lead to a new income distribution. One main conclusion is that "Goodwin cycles" can either be maintained or disappear. In this last case, the dynamics jumps from one equilibrium to another that includes long lasting unemployment. This conclusion allows to capture some speci…c characteristics of the French economic situation during the 1980s-90s when both pro…t share and unemployment rate durably increased together. Such a fact cannot entirely be explained by the Goodwin model and needs therefore to take into account Bhaduri-Marglin's assumptions.

The …rst section is brie ‡y devoted to Goodwin's and Bhaduri-Marglin's models, and the way they can be articulated. This combination is then formalized in the second section. The third section emphasizes our main conclusions that are illustrated by simulations. In the fourth section our results are confronted to the French economic dynamics of the 1980s-90s. Lastly a short conclusion is given.

I. -THE GOODWIN (1967) AND BHADURI-MARGLIN (1990) MODELS

The Goodwin model is a well-known mathematical representation of endogenous cycles rooted in the Marxian tradition. According to Goodwin, cycles are the consequence of the class struggle upon the sharing of the value added between workers and capitalists (wages against pro…ts). To give a better approach of the mechanisms of this model, let us introduce a situation characterized by a high unemployment. Real wages increase less than labour productivity according to the iron law. Consequently pro…t share increases, investment being proportionally raised. Accumulation and growth then push the demand for labour that leads to a decline of unemployment. This phenomenon restores the wage bargaining power of trade-unions, which progressively brings about a pro…t-squeeze. This weaker pro…tability (and thus the weaker level of saving) leads to a drop of investment that provokes accumulation slowdown as well as an increase in unemployment. Thus, a new cycle begins.

The hypotheses of the model are:

-The production factors are labour L and capital K -Labour productivity a and labour force n grow at a constant rate and :

a = a 0 e t > 0 n = n 0 e t > 0
-Wages are entirely consumed, pro…ts entirely saved and invested, then

: K = Y wL = 1 w a Y = (1 !)Y (1)
where w is the wage rate and ! the wage share in the value added.

-The capital-output ratio is constant. Thus

: K K = : Y Y = (1 !) (2)
-The real wage growth rate

: w w is related to the employment level v : w w = + v ; > 0 (3)
By solving this model, we get a two di¤erential equations system

: v = 1 ( + ) ! v : ! = [ v ( + )] ! (4)
The equilibrium is given by ! = 1 ( + ) and v = + . At the equilibrium, the Jacobian has a trace equalling zero and a positive determinant (= ! v ). The eigenvalues are complex and conjugate, their real part equalling zero. The dynamics of the system is a cycle oscillating around the center (vortex E on …gure 1) that corresponds to the system equilibrium.

Fig. I. The Goodwin cycle.

Say's law (according to which growth is never demand constrained) is implicit in the Goodwin model and implies that the level of investment always equalsand is determined by-pro…ts (pro…ts generate investment and not the contrary like in the Keynesian analysis of Kalecki). This hypothesis has been criticized by Keynesian economists, highlighting that the amount of investment does not depend directly on saving level. Following this argument, a speci…cation of investment is needed for building a new model. However this should not prevent from taking into account pro…tability as a central determinant of both investment decision and saving behaviour, as [START_REF]‹ ‹ Model of accumulation››[END_REF] emphasized it. In that sense, the Bhaduri-Marglin model analyzes the e¤ects of income distribution on growth through the impact of pro…t-share evolutions upon investment on the one hand and saving on the other hand. The authors show that output growth can be supported either by wages increase (wage-led regime) or pro…ts increase (pro…t-led regime).

In their model, consumption is given by:

C = (1 s w )W + (1 s r )R (5)
where W is the payroll, R are the total pro…ts (Y = R + W ), s w and s r are saving propensities of workers and capitalists respectively, with s r > s w . It follows that:

S K = 1 [(s r s w ) z + s w z] (6)
with K, the level of physical capital of the economy,

S the saving, = R Y , z = Y Y
(where Y is the output level when the capital stock is fully used), and

= K Y (constant) y .
The investment function has two explicative variables: the pro…t rate R K and the rate of capacity utilization z, which captures the demand e¤ect:

I K = i 0 + i r R K + i z Y Y = i 0 + z(i r + i z ) i r ; i z > 0 (7)
As I K = S K , we get a relation between z and :

z = i 0 (s r s w i r ) + (s w i z ) (8)
An economy is pro…t-led if @z @ > 0 and wage-led otherwise. The nature of the economic regime rests only on the sensitivity of both saving and investment to variations of : the economy is wage-led if i r > s r s w (we suppose i 0 > 0). Therefore the market equilibrium condition implies either an increasing or a decreasing relation between wage share and output. The e¤ect of pro…t on growth is then mitigated.

In the Goodwin model, recession periods generate their own recovery thanks to investment takeo¤ (boosted by higher pro…ts). Here, this recovery might not happen because:

-either investment grows less than savings when pro…t share increases, so that global e¤ect on growth is negative. This arises if the economy is wage-led.

-or recession has a direct negative impact on investment due to aggregate demand decrease. Investment evolutions depend on two antagonistic e¤ects: the positive e¤ect of pro…t recovery (Goodwin), and the negative impact of economic activity slowdown (negative Keynesian accelerator).

In his review of the Bhaduri-Marglin model, [START_REF] Taylor | Income distribution, in ‡ation and growth. Lectures on structuralist macroeconomic theory[END_REF] suggests that the nature of the economic regime could directly depend on the "initial"level of wages: the greater it is, the more likely is the economy to be pro…t-led. This rather intuitive interpretation can be illustrated by an inverted U-shaped curve, which can be used in a fruitful manner in a dynamic model z .

II. -THE PROPOSED MODEL

On Figure 2, we combine the "Taylor curve" with a wage-setting (WS ) curve (i.e. a wage equation where the level of real wage depends positively on output, see [START_REF] Jackman | Unemployment. Macroeconomic Performance and the Labour Market[END_REF]). In this framework, output is determined by aggregate demand which itself depends on the wage share. There is a wage share level ! which separates wage-led and pro…t-led areas from each other. As long as wage share is below ! , the positive e¤ect of wage share increase on consumption outweighs the (possible) negative e¤ect on investment (note that investment itself may even be supported by an accelerator e¤ect in this case). On the contrary, beyond ! , pro…ts are threatened to be squeezed so that every wage increase will depress investment more than it boosts consumption. Only a decrease of the wage share could recover investment and growth in this area. It is important to note that the dynamical dimension of our model comes from the endogeneity of the wage share. We see that this dynamics generates cycles around one …rst equilibrium but may bifurcate to another one, where unemployment and pro…t share are both very high. Indeed, a too large wage share reduction during the recession period of the cycle may prevent investment to recover (in spite of the increase of pro…t share), that actually breaks the cycle. This new equilibrium is characterized by unemployment and weakness of global demand. This is the reason why we call it a "Keynesian equilibrium" (see …gure 2 below). In the rest of this article, we propose a macroeconomic model which captures these dynamics. For that purpose, we keep the framework of the Goodwin model, so the wage equation is a Phillips curve (not a WS curve). Nevertheless, as we shall see, we modify the investment function, which allows us to establish an inverted U-shaped relation between wage share and output growth rate.

Our model is built in discrete time. All coe¢ cients are supposed positive. The …rms production function is given by:

Y t = a t L t (9)
The rates of growth of labour productivity a t and labour force n t are and respectively. We call w the real wage rate, ! the wage share and the pro…t share.

II.1. Goods market

We suppose that production depends on aggregate demand and not on the current stock of capital as in Goodwin model. At each period t, consumption and investment determine production, which is then shared between wages and pro…ts. Consumption and investment of the following period are thus directly related to the income distribution of the current period.

We suppose that saving propensities of workers and capitalists are respectively equal to 0 and 1. Consumption in (t + 1) equals the payroll of period t:

C t+1 = w t L t (10)
Investment level in (t + 1) depends on pro…tability and on a variable capturing the demand faced by …rms. Pro…tability is measured by the pro…t share of the preceding period t. Consumption of the current period replaces the capacity utilization rate to capture the demand e¤ect. Nevertheless, demand a¤ects investment as soon as pro…tability is high enough to allow …rms to invest (for example, imagine that banks demand a minimum level of pro…tability to lend). That's why we propose a multiplicative form between pro…tability and demand variables in the function:

I t+1 = C t+1 t = C t+1 (1 ! t ) > 0 (11)
In the following, we will suppose > 1. From ( 10) and ( 11), the goods market equilibrium Y = C + I is given by:

Y t+1 = Y t [1 (1 ! t )(1 ! t )] (12)
This is an inverted U-shaped curve between the output growth rate and the wage share. Both pro…t-led and wage-led con…gurations don't depend only on model parameters anymore (like in the Bhaduri-Marglin model) but on an endogenous variable, i.e. the wage-share. Taking (9) into account, we easily obtain from (12) a relation between v t+1 and v t (v is the employment rate).

II.2. Labour market

We keep the Goodwin wage equation (in discrete time):

w t+1 w t w t = v t+1 ; > 0 (13)
From ( 13), it follows:

! t+1 = 1 + v t+1 + 1 1 + ! t (14)
Note immediately that 1 1+ < 1. In the following, we suppose that 1 < 1 1+ < 1. We …nally obtain a two equations system: 8 > < > :

v t+1 = [1 (1 !t)(1 !t)] (1+ )(1+ ) v t = f (v t ; ! t ) ! t+1 = 1 1+ h [1 (1 !t)(1 !t)] (1+ )(1+ ) v t + (1 ) i ! t = g(v t ; ! t ) (15)
In the following, we note = (1 + )(1 + ) > 1.

II.3. Equilibria determination

Solving the system for …xed points, i.e., v t+1 = v t and ! t+1 = ! t (demonstrations of this section are in Appendix 1), we then can draw the corresponding phase diagram (…gure 3). On …gure 3, we have replaced employment rate v by unemployment rate u = 1 v.

Fig. III. Phase Diagram

Note that if u = 1 (v = 0), then : u = 0 ; if ! = 0, then : ! = 0. Therefore (u = 1; ! = 0) is one equilibrium of the system (E 1 ). The other equilibria are a saddle-point (E 2 ) and a center (locally), E 3 . Thus the system has a center -in accordance with the Goodwin model -but exhibits two other equilibria, especially a locally stable one, E 1 . This latter result contradicts the Goodwin model.

III. -SIMULATIONS

In this section we have calibrated our model so as to exhibit various dynamics that we can obtain from our model. All these simulations are presented in the plan (wage share, unemployment rate).

As noted by [START_REF] Harvie | ‹ ‹ Testing Goodwin : growth cycles in ten OECD countries››[END_REF], the econometric estimates for the center in the Goodwin model are far from actual values, what is true for our own simulations too. Concerning and in the wage equation, we have taken values between 0,5 and 1, like in [START_REF] Harvie | ‹ ‹ Testing Goodwin : growth cycles in ten OECD countries››[END_REF] estimates (for OECD countries where this relation is significant). Labour force and labour productivity growths are respectively of 2 % and 3 % ( = 0; 02 and = 0; 03) so that ' 1; 05. The coe¢ cient in investment function has been chosen such as ! 1 and ! 2 remain real ( > 1; 56). has been …xed at 1; 6. We suppose that, at equilibrium E 3 , v is equal to 0; 95. To keep v = + constant, we must modify when we modify (i.e. the sensibility of real wage growth to employment). In our di¤erent simulations we take as initial conditions v 0 = 0; 98 (unemployment rate of 2 %) and ! 0 = 0; 83 (! 0 is between ! 1 and ! 2 ). The unique aspect distinguishing our di¤erent simulations is the varying value for (and consequently the value of too, to keep v = 0; 95), which increases from 0,5 to 1.

The …rst simulation ( = 0; 5, see …gure 4) shows that our model allows us to reproduce Goodwin cycles. In the third simulation (…gure 6), the dynamics "bifurcates" and is attracted by the equilibrium E 1 (0; 0) and there is no cycle anymore. Here cycle dynamics is interrupted because of a too important wage squeeze during a recession phase. As a result, consumption brings down investment in its fall in spite of pro…t share recovery. As soon as investment increases when consumptions drops, cycles are maintained although the model is demand constrained. However, if consumption becomes too weak, that may o¤set the positive e¤ect of pro…t recovery on investment. Consumption and investment may thus fall together what induces a cumulative slump. Cycles disappear because neither consumption nor investment can restore growth. Slowdown exacerbates wage-share fall, whereas the economy is in the wage-led area (i.e. wages should increase to boost growth). Moreover, the labour force growth exceeds the growth of labour demand as soon as wage-share is lower than ! 1 . That is the reason why slump is cumulative and the economy collapses to E 1 (0; 0) (which does not appear on the …gure because of the retained scaling. See appendix 1). The dynamics leading to this equilibrium is thus characterized by a form of Harrodian instability (which reminds us Keynes's banana economy parable in A Treatise on Money (1930)): higher unemployment induces weaker wages, which strengthens recession and unemployment (in wage-led regime), and so on. Such an instability could be softened by supposing that high and persistent unemployment has no e¤ect on wages anymore. Moreover, economic policy is a very e¢ cient tool against Harrodian instability [START_REF] Harrod R | Towards a dynamic economics[END_REF]), even if our model does not emphasize it.

Thanks to this model, we propose an interpretation of growing unemployment which characterized the French economy in the 1980s and 1990s.

IV. -THE CASE OF FRENCH UNEMPLOYMENT IN THE 1980s AND 1990s Figure 7 sketches the French trajectory between 1960 and 2000 as it can be seen with the French Goodwin curve that we present. In the 1960s, unemployment remained below 3 %. This situation strengthened the bargaining power of trade unions in a context of labour productivity growth slowdown. That partly explained the takeo¤ of wage-share in the 1970s which squeezed pro…ts and reduced investment [START_REF] Bruno | The economics of worldwilde stag ‡ation[END_REF]. At the beginning of the 1980s, the competitive disin ‡ation has been run with the purpose to diminish unemployment by promoting exportations through in ‡ation …ght and wage austerity [START_REF] Blanchard | ‹ ‹ Competitiveness through disin ‡ation : an assessement of the french macroeconomic strategy››[END_REF]. Unfortunately, the pro…t-share recovery in the 1980s came along with a very short decrease of unemployment. Therefore, since the middle of 1980s, we can observe a sharp breakdown in the French trajectory with the theoretical dynamics of Goodwin model. Our model proposes an explanation of the lasting and joint increase of unemployment and pro…t-share (which has reached historical high levels in the 1990s without sensitive e¤ect on investment) from 1982 to 1997. Note however that our model can explain neither the pro…t-share increase in the 1960s nor the coming back of sustained growth from 1997 (not shown on the …gure).

V. -CONCLUSION

In this article, Keynesian demand constraints have been introduced in the Goodwin model. We built a non-linear investment function including pro…tability and accelerator components. From the goods market equilibrium we thus get a mathematical relation between output growth rate and wage share. This relation is …rst increasing then decreasing, what corresponds to two distinct growth regimes studied by Bhaduri-Marglin: the wage and pro…t-led regimes. We keep the wage equation of Goodwin model to make our model dynamic. Therefore, the economy can be alternatively wage and pro…t-led. We then show that Goodwin cycles can be maintained in this new framework. However cycles are not the only possible trajectories of the system anymore: the economy may also be attracted by a cumulative slump path. This results from the drop of the too classical and mechanistic hypothesis of ex ante identity between pro…t and investment postulated by the Goodwin model. In our model a large wage decrease can reduce consumption so much that investment will not take o¤ despite pro…t restoration. In that case investment and consumption decline together as in a typical Keynesian unemployment case. Nothing can then restore growth and the economy collapses toward a zero output level. We have not succeeded in stabilizing this "Keynesian"equilibrium before zero yet, which remains a weakness of our model. Thanks to it, we manage to sketch the French economic dynamics between 1970 and 1997 and the long lasting growth of both unemployment and pro…t share from the end of 1980s. Nevertheless, it cannot explain the economic recovery observed after 1997 in that country.

APPENDIX A MATHEMATICAL ANALYSIS OF THE DYNAMIC (SECTION II).

We suppose that:

= 1 + and = 1 1 + : From (15) it results that v t+1 = v t if v = 0 or if (1 ! )(1 ! ) = 1
, where roots are called ! 1 and ! 2 (! 1 ! 2 ) and belong to [0; 1].

The nature (real or complex) and values of ! 1 and ! 2 depend only on and : We will now consider that both ! 1 and ! 2 are real, positive and inferior to 1, which implies restrictive conditions on . For example, if = 1; 05, it is easy to show that must be greater than 1; 56 (in the unlikely case where = 1; 1, we must have > 1; 86) x . In these conditions ! 1 ! 2 = , therefore ! 1 1=2 and ! 2 1=2

.

! t+1 = ! t if ! = 0 or if v = (1 ) [1 (1 ! )(1 ! )]
, which is …rstly decreasing then increasing in !.

We …nally obtain three equilibria :

E 1 (v = 0; ! = 0) E 2 (v = (1 ) ; ! = ! 1 ) and E 3 (v = (1 ) ; ! = ! 2 )
The Jacobian is equal to:

J = 1 [1 (1 ! )(1 ! )] 1 v ( + 1 2 ! )
(! ) 2 (( + 1) ! ) v ! (2( + 1) 3 ! ) +

!

In the neighborhood of E 1 , both eigenvalues are real and lower than 1. E 1 is locally a stable equilibrium.

In the neighborhood of E 2 , the trace of J E 2 is strictly greater than 2 since ! 1 1=2

. Consequently, E 2 is a saddle-point.

In the neighborhood of E 3 , the trace of J E 3 is lower than 2 since ! 2 > 1=2 . Roots are thus complex and conjugate. Their modulus is 1 so that E 3 is locally a center.

x This restriction on is coherent with equation ( 11). In this model, the entire wage bill is spent on consumption (of the next period). Then new investment can only be …nanced by past pro…ts and new bank credits. We then may expect that It+1 Yt > t . To …t this consistency condition, our investment behaviour equation (11) must verify > 1 ! (note that is a constant parameter and that is credit which is the adjusting variable here). Empirically is very plausibly much bigger than 1 ! .

FIGURES

Fig

  Fig. II. Combining the "Taylor curve" with a wage-setting (WS) curve.

Fig

  Fig. IV. First simulation: Goodwin cycles.

Figure 5 (

 5 Figure5(where = 0; 7) exhibits the attractive e¤ect of E 2 on these cycles.

Fig. V .

 V Fig. V. Second simulation: Goodwin cycles and attracting e¤ect of E 2 .

Fig

  Fig. VI. Third simulation: The Goodwin cycle disappears.

Fig

  Fig. VII. French "Goodwin curve", 1960-2000.

Fig. I .

 I Fig. I. The Goodwin cycle.

Fig. II .

 II Fig. II. Combining the "Taylor curve" with a wage-setting (WS) curve.

Fig. III .

 III Fig. III. Phase Diagram.

Fig. IV .

 IV Fig. IV. First simulation: Goodwin cycles. = 0:50 ' 0:45 = 1:60 = 1:05 ! 0 = 0:83 v 0 = 0:98 v ' 0:95 ! 1 ' 0:75 ! 2 ' 0:88

y Note that in Goodwin model, Y always equals Y .

z Note that this representation does not correspond at all to the analytical model of Bhaduri-Marglin, since the sign of z from (8) does not depend on . In the next section, we will try to give a mathematical formalization of this "Taylor curve".

SECOND SIMULATION (SECTION III).

Here we reproduce the same …gure as in the text (…gure 6) but on a larger scale, so as to represent the complete trajectory towards E 1 (0; 0):


Nicolas Candau
email: nicolas.candau@mines-paristech.fr

Jean-Luc Bouvard
email: jean-luc.bouvard@mines-paristech.fr

Edith Peuvrel-Disdier
email: edith.peuvrel-disdier@mines-paristech.fr

Rudy Valette
email: rudy.valette@mines-paristech.fr

Christophe Pradille
email: christophe.pradille@mat-xper.com

Noëlle Billon
email: noelle.billon@mines-paristech.fr

COUPLED THERMAL AND VOLUME CHANGE MEASUREMENTS DURING STRETCHING OF FILLED EPDM RUBBERS

Keywords: EPDM, strain field, volume change, VIC-3D, thermomechanical analysis

Introduction

Ethylene-propylene-diene-monomer rubber (EPDM) has been widely studied in the recent years. Thanks to its high resistance to ageing and mechanical properties, it is one of the best candidates for the automotive industry. For a better understanding of its mechanical properties, a suitable characterization of its thermomechanical behavior is mandatory. For this purpose, the combination of infrared thermography and strain field measurement is commonly used in literature [START_REF] Chrysochoos | Use of Full-Field Digital Image Correlation and Infrared Thermography Measurements for the Thermomechanical Analysis of Material Behaviour[END_REF]. Over the last decade, image correlation approach has been widely improved. It now allows precisely estimating the volume change during the deformation [START_REF] Crevoisier | Volume changes in a filled elastomer studied via digital image correlation[END_REF]. However, to the authors' knowledge, no study was already devoted to the combined analysis of both thermal and volume change. This approach should be useful to identify the respective contributions of the reversible and irreversible mechanisms associated with the deformation of elastomers.

Objectives and Methodology

The present work proposes an experimental approach for the combined measurement of both volume and thermal changes during stretching of elastomers. This is done thanks to the set-up of a specifically dedicated device which consists of a four camera stereovision system and an infrared camera (figure 1a). The stereovision system is made up of two independent camera pairs -allowing neglecting the out of plane displacements -which are positioned to measure the complete strain displacement field (front and side faces of the samples) and thus the volume change of the sample. A speckle pattern is put on the surface sample and Digital Image Correlation is used to analyze the displacement field. The IR camera is switched 2h before tensile tests to stabilize its internal temperature. To reduce the impact of external heat sources, a black box surrounding the sample is used during each test. The studied rubbers are carbon black filled EPDM obtained by sulphur vulcanization of the gum. They contain 80phr of carbon black (80 g of fillers per 100 g of rubber). EPDM samples are stretched uniaxially with an INSTRON tensile test machine at room temperature with the strain rate 1.s -1 up to four different maximum strains (figures 1b, 1c and 1d). Mullins type tests are also performed. They consist of four series of three cycles with an increasing maximum strain.

Results and analysis

Figures 1b-d present the evolution of the volume change and self-heating during stretching of EPDM rubbers at various maximum strains. Volume and temperature increase during loading and decrease during unloading. At a given strain, volume and temperature are found higher during the unloading phase respectively due to the remaining open cavities and the accumulation of heating on the surface sample. Following Martinez et al. [START_REF] Martinez | Mechanisms of deformation in crystallizable natural rubber. Part 2: quantitative calorimetric analysis[END_REF], temperature rise in filled elastomers is mainly due to entropic elasticity and viscosity but damage mechanisms are assumed to be non-negligible. With an appropriate treatment of thermomechanical equations (i.e. taking into account the non-adiabatic conditions), the impact of damage on the mechanical and the thermal response could be quantified. This point is of matter of importance concerning the Mullins effect, whose physical meaning is not totally understood [START_REF] Diani | A review on the Mullins effect[END_REF]. During such a test, the accumulation of cycles at a fixed strain is associated with a progressive decrease of the stress (so-called Mullins effect). Our results show that the main volume change is observed at the end of the first cycle, but still progressively increases with the number of cycles, suggesting that irreversible damages still occur. We think that the present work would be interesting to identify the contributions of reversible and irreversible mechanisms associated with the deformation of elastomers. This approach will be improved by testing different experimental conditions such as various strain rates or loading conditions. as well as EPDM obtained for different , i.e. different sulfur and carbon black contents.

Fig. 1 .

 1 Fig.1. Experimental set up (a). Stress-strain curves for cyclic tests at room temperature and the strain rate 1.s -1 (b). Volume change (c). Self-heating (d).


Georges Gras

OF AN ALGEBRAIC NUMBER by

Keywords: 2000 Mathematics Subject Classification. -Primary 11R04, Secondary 11R16, order modulo p, Frobenius automorphisms, probabilistic number theory

published or not. The documents may come

Résumé. -Soit K/Q Galoisienne, et soit η ∈ K × de conjugués multiplicativement indépendants. Pour un premier p, non ramifié, étranger à η, soient np le degré résiduel de p et gp le nombre de p|p, puis op(η) et op(η) les ordres de η modulo p et p respectivement. En utilisant les automorphismes de Frobenius, nous montrons que pour tout p ≫ 0, certains diviseurs explicites de p np -1 ne peuvent réaliser ni op(η) ni op(η), et nous donnons une borne inférieure de op(η). Ensuite nous obtenons que Prob(op(η) < p) ≤ 1 p gp (np -1)-ε , où ε = O 1 log 2 (p) , pour tout p ≫ 0 tel que np > 1 ; sous l'heuristique de Borel-Cantelli, ceci conduit à op(η) > p pour tout p ≫ 0 tel que gp(np -1) ≥ 2, ce qui couvre les cas "limites" des corps cubiques avec np = 3 et des corps quartiques avec np = gp = 2, mais non celui des corps quadratiques avec np = 2. Dans le cas quadratique, la conjecture naturelle est, au contraire, que op(η) < p pour une infinité de p inertes. Des calculs sont donnés via des programmes PARI.

1. Frobenius automorphisms 1.1. Generalities. -Let K/Q be Galois of degree n, of Galois group G. Denote by h a possible residue degree of an unramified prime ideal of K, that is to say a divisor of n for which there exists a cyclic subgroup H of G of order h. Indeed, one knows that, for any generator s of H, there exist infinitely many prime numbers p, unramified in K/Q, such that s is the Frobenius s p of a prime ideal p | p in K/Q ; H is then the decomposition group H p of p. Reciprocally, any unramified p has a cyclic decomposition group H p with a canonical generator s p (the Frobenius).

Of course, if s 1 and s 2 are two distinct generators of H, the sets of corresponding primes p are disjoint (e.g., take the cyclotomic field K = Q(ζ 5 ) of fifth roots of unity and H = G (cyclic of order h = 4), with ζ s 1 5 = ζ 2 5 and ζ s 2 5 = ζ 3 5 ; this characterizes the sets {p : p ≡ 2 (mod 5)} and {p : p ≡ 3 (mod 5)}, respectively); see, e.g., [START_REF] Narkiewicz | Elementary and Analytic Theory of Algebraic Numbers[END_REF]Section 3 of Chapter 7], [START_REF]Introduction to Cyclotomic Fields[END_REF]Section 3 of Appendix], or [3,Sections 1.1,1.2,4.6,of Chapter II] for Chebotarev's density theorem and properties of Frobenius automorphisms. But we consider such a fixed residue degree h | n since we shall see that the statement of our main result, on the order of η ∈ K × modulo a prime ideal p, does not depend on the conjugate of the decomposition group H p of p, nor on its Frobenius s p , but only on the residue degree n p of the corresponding prime number p under p (in other words, we shall classify the set of unramified prime ideals p of K by means of the sole criterion n p = h; so, any of the p, with n p = h, would have the common property, depending on h, given by our main theorem). Since the h | n are finite in number, everything is effective (e.g., h ∈ {1, 2, 3} for a dihedral group G = D 6 , but h ∈ {1, 2} for the Galois group of any compositum of quadratic fields).

1.2. Orders modulo p and modulo p. -Let η ∈ K × . In the sequel we shall assume that the multiplicative Z[G]-module η G generated by η is of Z-rank n (i.e., η G ⊗ Q ≃ Q[G]), but this is not needed for the following definition.

Definition 1.1. -Let p be a prime number, unramified in K/Q, prime to η, and let p be a prime ideal of K dividing p. We define the order of η modulo p (denoted o p (η)) to be the least nonzero integer k such that η k ≡ 1 (mod p). We define the order of η modulo p (denoted o p (η)) to be the least nonzero integer k such that η k ≡ 1 (mod p).

Of course, o p (η) and o p (η) = lcm (o p (η), p | p) divide p np -1, where n p is the residue degree of p in K/Q, but we intend to prove (see Theorem 2.1 for a more complete and general statement):

Let h | n be a possible residue degree in K/Q. Let η ∈ K × be such that the multi- plicative Z[G]-module generated by η is of Z-rank n.
Then for all large enough prime p (denoted p ≫ 0 in all the paper) with residue degree n p = h, the orders o p (η) for any p | p, and o p (η) do not divide any of the integers

D h,δ (p) := p h -1 Φ δ (p) , δ | h,
where Φ δ (X) is the δth cyclotomic polynomial.

Consider, for any unramified prime p, the characteristic property of the Frobenius automorphism s p of p | p in K/Q, η sp ≡ η p (mod p).

Let H p := s p be the decomposition group of p (denoted H to simplify) and let σ ∈ G/H (or a representative in G); the Frobenius s p σ of p σ is s σ p := σ •s p •σ -1 and we get η s p σ ≡ η p (mod p σ ). So, if s p and σ commute this leads to s σ p = s p σ = s p and η sp ≡ η p (mod p σ ). In other words, we have

η sp ≡ η p mod σ∈G/H σ.sp=sp.σ p σ .
In the Abelian case, we get (independently of the choice of p | p)

η sp ≡ η p (mod p). Lemma 1.1. -Let η ∈ K × be such that the multiplicative Z[G]-module η G is of Z-rank
; let f (X) ∈ Z[X] be a given polynomial such that f (s) = 0 in Z[H].
Then, for all prime numbers p ≫ 0 such that there exists a prime ideal p | p for which s p = s, whenever ζ ∈ µ(K) we have

η f (p) ≡ ζ (mod p).
Proof. -We have η f (p) ≡ η f (s) (mod p); thus, if η f (p) ≡ ζ (mod p) for some ζ, this leads to η f (s) -ζ ≡ 0 (mod p) giving, by the norm in K/Q,

N K/Q (η f (s) -ζ) ≡ 0 (mod p |H| ).
Since η G is of multiplicative Z-rank n and f (s) = 0, we have η f (s) / ∈ µ(K); then N K/Q (η f (s) -ζ) is a nonzero rational constant depending only on η, f (s), ζ, and whose numerator is in p |H| Z (a contradiction for p ≫ 0).

The statement of the lemma does not depend on the choice of s generating H, nor on the choice of the prime ideal p | p such that s p = s (in the Abelian case, any

p | p is suitable since s p σ = s p for all σ ∈ G). If s ∈ G is of order h ≥ 1, any nonzero element of Z[H]
can be writen f (s) where f (X) is of degree < h; if we take f (X) of degree 0, then we have f (s) = f ∈ Z\{0} regardless of h and s, giving the obvious result

η f ≡ ζ (mod p) for any p ≫ 0.
Naturally, an interesting application of this Lemma is when f

(X) | X h -1 in Z[X], f (X) = X h -1,
and when the degree of f (X) is maximal. This explains why the case h = n p = 1 (p totally split in K/Q) is uninteresting since f (X) | X -1, with f (X) = X -1, gives f = 1 and the same conclusion as above.

2. Consequences for the values of o p (η) and o p (η)

We have the factorization

p h -1 = δ|h Φ δ (p),
where Φ δ (X) is the δth cyclotomic polynomial (see [START_REF]Introduction to Cyclotomic Fields[END_REF]Chapter 2]). So we can consider the divisors δ∈I Φ δ (p), where I is any strict subset of the set of divisors of h. Of course, it will be sufficient to restrict ourselves to maximal subsets I, which

gives the divisors D h,δ (p) := p h -1 Φ δ (p) , δ | h.
For instance, if h = 6, we get the set

{p 5 + p 4 + p 3 + p 2 + p + 1, p 5 -p 4 + p 3 -p 2 + p -1, p 4 -p 3 + p -1, p 4 + p 3 -p -1},
giving the complete set of "polynomial divisors" of p 6 -1, Proof.

{1, p -1, p + 1, p 2 -1, p 2 -p + 1, p 3 -2 p 2 + 2 p -1, p 3 + 1, p 4 -p 3 + p -1, p 2 + p + 1, p 3 -1, p 3 + 2 p 2 + 2 p + 1, p 4 + p 3 -p -1, p 4 + p 2 + 1, p 5 -p 4 + p 3 -p 2 + p -1, p 5 + p 4 + p 3 + p 2 + p + 1}.
-Let k ′ = gcd (k, p h -1). Then we have k ′ = λ k + µ (p h -1), λ, µ ∈ Z, and
η k ′ ≡ η λ k ≡ ζ λ =: ζ ′ (mod p); but k ′ | k, so k = k ′ | p h -1. Suppose that k divides some D h,δ (p) = p h -1 Φ δ (p) = δ ′ |h, δ ′ =δ Φ δ ′ (p).
Let s be the Frobenius of p and H = s its decomposition group (of order h). Thus

η k ≡ ζ (mod p) yields η D h,δ (p) ≡ ζ ′ (mod p), ζ ′ ∈ µ(K), giving η D h,δ (p) ≡ η D h,δ (s) ≡ ζ ′ (mod p). From Z[H] ≃ Z[X]/(X h -1) Z[X], we get in Z[H] D h,δ (s) = δ ′ |h, δ ′ =δ Φ δ ′ (s) = 0 since D h,δ (X) / ∈ (X h -1) Z[X]; the polynomial D h,δ (X) ∈ Z[X] being independent of p, Lemma 1.1 applied to f (X) = D h,δ (X)
gives a contradiction for all p ≫ 0 with residue degree n p = h.

If η G is not of Z-rank n, a statement does exist which depends on the Grepresentation η G ; for instance, let K = Q( √ m) and η ∈ K × \ µ(K):

-If N K/Q (η) = ±1, then o p (η) ∤ D 2,2 (p) = p -1 for all prime p ≫ 0, inert in K/Q. -If η 1-s = ±1, then o p (η) ∤ D 2,1 (p) = p + 1 (e.g., η = √ m, m = -1).
The expression "for all p ≫ 0 of residue degree n p = h" in the theorem is effective and depends, numerically, only on h and the conjugates of η.

The theorem gives the generalization of the particular case h = 2 in [START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF].

In the above case h = 6 and p ≫ 0 (with n p = 6), the orders o p (η) are divisors of p 6 -1 which are not divisors of any of the integers in the set:

{ p 5 + p 4 + p 3 + p 2 + p + 1, p 5 -p 4 + p 3 -p 2 + p -1, p 4 -p 3 + p -1, p 4 + p 3 -p -1}.
For p = 7 and h = 6, we have 60 divisors of p 6 -1 = 2 4 • 3 2 • 19 • 43, and the distinct divisors of these 4 polynomials are the 52 integers:

1 , 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 24, 36, 38, 43, 48, 57, 72, 76, 86, 114, 129, 144, 152, 171, 172, 228, 258, 304, 342, 344, 387, 456, 516, 684, 688, 774, 817, 912, 1032, 1368, 1634, 2064, 2451, 2736, 3268, 4902, 6536, 7353, 9804,14706,19608. So the remaining (possible) divisors of p 6 -1 are , 3096, 6192, 13072, 29412, 39216, 58824, 117648. Of course, in our example, the prime p = 7 is too small regading η, but the interesting fact (which is similar for larger p and any integer h) is the great number of impossible divisors of p h -1 for small numbers η.

1548

For p = 1093 (resp. 504202701918008951235073), only 76 (resp. 242424) divisors are possible among the 384 (resp. 518144) divisors of p 6 -1.

The case h = ℓ (a prime) implies that o p (η) is not a divisor of p -1 nor a divisor of p ℓ-1 +• • •+p+1 for p ≫ 0 with residue degree n p = ℓ; this means that

o p (η) = d 1 d 2 with d 1 | p -1, d 1 = 1, d 2 | p ℓ-1 + • • • + p + 1, d 2 =
1 (taking care of the fact that when p ≡ 1 (mod ℓ), we have the relation

gcd (p -1, p ℓ-1 + • • • + p + 1) = ℓ). Remark 2.1. -It is clear that if r ∈ N \ {0}
is small, Theorem 2.1 implies that for all prime p ≫ 0 with residue degree n p = h and for any p | p, the least integer k ≥ 1 for which there exists

ζ ∈ µ(K) such that η k ≡ ζ (mod p) cannot divide any of the integers r • D h,δ (p), δ | h (indeed, η r is still small in an Archimedean point of view). This makes sense only when r = r δ is choosen, for each D h,δ (p), as a small divisor of Φ δ (p). So the probability of o p (η) | r δ • D h,δ (p) increases (from 0 to 1) when the factor r δ | Φ δ (p) increases (from r δ = 1 to r δ = Φ δ (p)). In the example h = ℓ, where o p (η) = d 1 d 2 , d 1 | p -1, d 2 | p ℓ-1 + • • • + p + 1, we have d 1 & d 2 → ∞ for p → ∞.

A numerical example

Let K = Q(x) be the cyclic cubic field of conductor 7 defined by x = ζ 7 + ζ -1 7 from a primitive seventh root of unity ζ 7 ; its irreducible polynomial is X 3 +X 2 -2X -1. Let η = 8 x + 5 of norm -203; then for p < 200, inert in K (i.e., p 2 ≡ 1 (mod 7)), we obtain the exceptional example o 17 (η) = 307 = p 2 + p + 1 and no other when p increases; we get some illustrations with a small r | p -1, r > 1 (e.g., p = 101, r = 2, with o p (η) = r • (p 2 + p + 1)), according to the following numerical results; note that when p ≡ 1 (mod 3), we have o p (η) = [START_REF] Gras | Étude probabiliste des quotients de Fermat[END_REF]3,[START_REF] Narkiewicz | Elementary and Analytic Theory of Algebraic Numbers[END_REF]11,17,19,23,31,37 giving a solution up to 10 7 .

A lower bound for o p (η)

When η is fixed in K × , very small orders are impossible as p → ∞ because of the following theorem giving Archimedean constraints; in this result none hypothesis is done on the rank of the multiplicative Z[G]-module generated by η (except that this Z-rank is assumed to be = 0) nor on the field K itself. We denote by Z K the ring of integers of K.

Theorem 4.1. -Let µ(K) be the group of roots of unity contained in K. Let η ∈ K × \ µ(K) and ν ∈ N \ {0} be such that ν η ∈ Z K . Then, for any p prime to η and ν, the congruence

η k ≡ ζ (mod p), ζ ∈ µ(K), k ≥ 1, implies the inequality k ≥ log(p) -log(2) max log(ν • c 0 (η)), log(ν)
,

where c 0 (η) = max σ∈G (|η σ | . If η ∈ Z K (i.e., ν = 1), then we get k ≥ log(p -1) log(c 0 (η)) . In other words, if Z K,(p) is the ring of p-integers of K, the order of the image of η in Z K,(p) µ(K)•(1 + p Z K,(p)
), and a fortiori o p (η), satisfies the above inequalities.

Proof.

-Put η = θ ν , with θ ∈ Z K . The congruence is equivalent to θ k = ζ ν k +Λ p, where Λ ∈ Z K \ {0} (because η / ∈ µ(K)).
Taking a suitable conjugate of this equality, we can suppose |Λ| ≥ 1. Thus

|Λ| p = |θ k -ζ ν k | ≤ |θ| k + ν k giving |θ| k + ν k ≥ p; so, using a conjugate θ 0 such that |θ 0 | = max σ∈G (|θ σ |), we have a fortiori |θ 0 | k + ν k ≥ p, with |θ 0 | ≥ 1 since θ ∈ Z K . 7 (i) If ν ≥ 2, then p ≤ |θ 0 | k + ν k ≤ 2 max(|θ 0 | k , ν k ) giving the result. (ii) The case ν = 1, used in [1, Lemme 6.2], gives |θ 0 | k ≥ p -1, hence the upper bound k ≥ log(p -1) log(c 0 (η)) since |θ 0 | = c 0 (η) > 1 (because η / ∈ µ(K)).
Under the assumptions of Theorem 2.1 we have the following result.

Corollary 4.1.

-Suppose to simplify that η ∈ Z K ; let p be unramified of residue degree n p such that for some

δ | n p , o p (η) = r • d, d | D np,δ (p), r = r δ | Φ δ (p) (cf. Remark 2.1). Then r ≥ log(p -1) log(c 0 (η D np ,δ (s) ))
, where s generates any decomposition group of p.

In the previous example of Section 3, for p ≈ 10 In this section, we examine some probabilistic aspects concerning the orders modulo p | p of an η ∈ K × . For any p, unramified in K/Q, recall that g p is the number of prime ideals p | p and n p the common residue degree of these ideals. Let Z K be the ring of integers of K; the residue fields F p = Z K /p are isomorphic to F p np .

5.1. Densities. -It is assumed in this short subsection that p is fixed and that η ∈ K × is a variable modulo p, prime to the given p; in other words, η varies in the group (Z K,(p) /p Z K,(p) ) × of invertible elements of the quotient Z K,(p) /p Z K,(p) , where Z K,(p) is the ring of p-integers of K, so that we have

(Z K,(p) /p Z K,(p) ) × ≃ p|p F × p (p unramified).
For each prime ideal p | p, let η p ∈ F × p be the residue image of η at p. The density of numbers η, whose diagonal image is given in

p|p F × p , is 1 (p np -1) gp because the map η (mod p) → (η p ) p|p ∈ p|p F ×
p yields an isomorphism (from chinese remainder theorem) and, in some sense, the g p conditions on the η p , p | p, are independent as η varies (the notion of density is purely algebraic and the previous Archimedean obstructions of Sections 2 & 4 do not exist). Thus the orders o p (η) and o p (η) have canonical densities (see § 5.4).

5.2. Probabilities and Independence. -We shall speak of probabilities when, on the contrary, η ∈ K × \ µ(K) is fixed and when p → ∞ is the variable; but to avoid trivial cases giving obvious obstructions (as η ∈ Q × for which o p (η) | p -1 for any p regardless of the residue degree of p; see § 5.3 for more examples), we must put some assumptions on η so that o p (η) can have any possible value dividing p np -1 (by reference to Theorem 2.1, Remark 2.1, and Theorem 4.1 giving moreover theoretical limitations for the orders, so that the true probabilities are significantly lower).

Let H be the decomposition group of a prime ideal p 0 | p, p unramified in K/Q.

F × p = σ∈G/H σF × p 0 where σF × p 0 = F × p σ 0
for all σ ∈ G/H (using additive notation for convenience).

In the same way, the representation η G can be written

η G = σ∈G/H σ η H ,
where η H is the multiplicative Z[H]-module generated by η. So, for natural congruential reasons (that must be valid regardless of the prime p) concerning the map η (mod p) → (η p ) p|p , the representation η G must be induced by the H-representation η H , i.e., we must have

η G = σ∈G/H σ η H
(otherwise, any nontrivial Z-relation between the conjugates of η will give nonindependent variables η p in a probabilistic point of view). Since any cyclic subgroup H of G is realizable as a decomposition group when p varies, the above must work for any H; taking H = 1, we obtain that η G = σ∈G η σ Z which is equivalent for η G to be of Z-rank n, giving the following heuristic in relation with the properties of the normalized p-adic regulator of η studied in [START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF].

Heuristic 5.1. -Let K/Q be Galois of degree n, of Galois group G. Consider η ∈ K × and, for any prime number p ≫ 0, unramified in K/Q and prime to η, let (η p ) p|p be the diagonal image of η in p|p F × p . The components η p are independent, in the meaning that for given (i) This implies that η is not in a strict subfield L of K; otherwise, if H is a non-trivial cyclic subgroup of G (hence of order h > 1) such that L ⊆ K H , for any unramified prime p such that H is the decomposition group of p | p with Frobenius s, o p (η) is not a random divisor of p h -1 but a divisor of p -1, the residue field of K H at p being F p for infinitely many p.

a p ∈ F × p , Prob η p = a p , ∀ p | p = p|p Prob η p = a p , if and only if η generates a multiplicative Z[G]-module of Z-rank n.
(ii) In the same way, η cannot be an element of

K × of relative norm 1 in K/K H , H = 1, because of the relation N K/K H (η) = 1 giving η p h-1 +•••+p+1 ≡ 1 (mod p).
For the unit η = 2 √ 2 + 3 and any p inert in Q( √ 2), we obtain η p+1 ≡ 1 (mod p) (i.e., o p (η) | p + 1), giving infinitely many p such that o p (η) < p:

(p, o p (η)) =
(29, 10), (59, 20), (179, 36), (197, 18), (227, 76), (229, 46), (251, 84), (269, 30), (293, 98), (379, 76), (389, 78), (419, 140), (443, 148), . . .

(iii) Let K = Q(j, 3 √ 2)
, where j is a primitive third root of unity, and let

η = 3 √ 2-1 (a unit of Q( 3 √ 2
)); for the same reason with H = Gal(K/Q(j)), from η s 2 +s+1 = 1, we get, for any prime p inert in K/Q(j), η p 2 +p+1 ≡ 1 (mod p) (for p = 7, η is of order 19 modulo p and we have infinitely many p such that o p (η) | p 2 + p + 1).

In such a non-Abelian case, some relations of dependence can also occur in a specific component

F × p , p | p. Since η = 3 √ 2 -1 ∈ Q( 3 √ 2), for any p inert in K/Q( 3 √ 2) (i.e., K H = Q( 3 √ 2), in which case, p splits in K/Q(j)), there exists a rational a such that 3 √ 2 ≡ a (mod p), 3 √ 2 ≡ a j 2 (mod p s ), 3 √ 2 ≡ a j (mod p s 2 ). So η ≡ a -1 (mod p) and o p (η) | p -1 which is not necessary true for o p (η): for p = 5 we have 3 √ 2 ≡ 3 (mod p), 3 √ 2 ≡ 3j 2 (mod p s ), 3 √ 2 ≡ 3j (mod p s 2 )
. Then η ≡ 2 (mod p) is of order 4 modulo p, but η ≡ 3j 2 -1 (mod p s ) is of order 8 modulo p s . So o p (η) = 8 but we have some constraints on the η p .

5.4. Probabilities for the order of η modulo p. -Now we suppose that the multiplicative Z[G]-module η G is of Z-rank n.

Remark 5.1. -From Theorem 2.1, we know that o p (η) ∤ D np,δ (p) for all δ | n p , when p → ∞; in particular, o p (η) ∤ p -1 if we assume n p > 1. For this, the hypothesis on the Z-rank of η G is fundamental. In other words, the probability of some (unbounded) orders is zero. This is strengthened by Remark 2.1. Moreover, Theorem 4.1 gives obstructions for very small orders, which decreases the probabilities of small orders; the total defect of probabilities is less than O(log(p)) and is to be distributed among all orders, which is negligible. Thus, this favors large orders which are more probable; this goes in the good direction because we shall study probabilities of orders o p (η) less than p when n p > 1.

Although the theoretical values of the probabilities are rather intricate, in a first approach, we can neglect these aspects and give some results in an heuristic point of view corresponding to the case where η is considered as a variable (so that probabilities coincide with known densities) and we use the heuristic that when η is fixed once for all, probabilities are much lower than densities as p → ∞ as explained in § 5.2. Furthermore, we shall use rough majorations (except in the quadratic case and n p = 2, where densities are exact).

If D | p np -1, o p (η) | D is equivalent to η D p = 1 for all p | p. So we obtain Prob o p (η) = D ≤ Prob o p (η) | D = p|p Prob η D p = 1 (cf. Heuristic 5.1).
Since F × p is cyclic of order p np -1, we get

Prob η D p = 1 = d|D φ(d) p np -1 = D p np -1 ,
where φ is the Euler function, and we obtain, for any

D | p np -1, Prob o p (η) = D ≤ D p np -1 gp .
If g p = 1, then n p = n, and we can replace this inequality by

Prob o p (η) = D ≤ Density o p (η) = D = φ(D) p n -1 .
When g p > 1, the exact expression is more complicate since o p (η) = D if and only if o p 0 (η p 0 ) = D for at least one p 0 | p and o p (η p ) | D for all p | p, p = p 0 , but we shall not need it.

Probabilities of orders o p (η) < p

Suppose p ≫ 0, non totally split in K/Q. In [START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF], the number η is a fixed integer of K × and we have to consider the set

I p (η) := 1, [η] p , . . . , [η k ] p , , . . . , [η p-1 ] p ,
where [ • ] p denotes a suitable residue modulo p Z K . We need that I p (η) be a set with p distinct elements, to obtain valuable statistical results on the "local regulators ∆ θ p (z)", z ∈ I p (η), to strengthen some important heuristics; this condition is equivalent to η k ≡ 1 (mod p) for all k = 1, . . . , p -1, hence to o p (η) > p.

So we are mainely interested by the computation of Prob o p (η) < p when n p > 1 and we intend to give an upper bound for this probability. As we know from Theorem 2.1, taking the example of quadratic fields we have, for

η G of Z-rank 2, o p (η) ∤ p -1 & o p (η) ∤ p + 1, for p → ∞;
but o p (η) < p remains possible for small divisors D of p 2 -1 (e.g., η = 5 + √ -1 for which p = 19 is inert in Q( √ -1) and o 19 (η) = 3 • 5 whereas p -1 = 18 and p + 1 = 20). Suppose that K = Q and that the residue degree of p is n p > 1. Let

D p := {D : D | p np -1, D < p, D ∤ D np,δ (p) ∀ δ | n p }.
Then we consider that we have, for all p ≫ 0 of residue degree n p , the following heuristic inequality (from Theorems 2.1, 4. for which the application of the Borel-Cantelli heuristic supposes the inequality g p (n p -1) ≥ 3, giving possible obstructions for quadratic or cubic fields with p inert, and quartic fields with n p = 2. Of course, if g p (n p -1) increases, the heuristic becomes trivial and we can replace Prob o p (η) < p by Prob o p (η) < p κ , for some κ > 1 (see Remark 6.1 (i)). But we can remove the obstructions concerning the cubic and quartic cases using an analytic argument suggested by G. Tenenbaum: Theorem 6.1. -Let K/Q be Galois of degree n ≥ 2, of Galois group G, and let η ∈ K × be such that the multiplicative Z[G]-module generated by η is of Z-rank n.

For any prime number p, let g p be the number of prime ideals p | p and let n p be the residue degree of p in K/Q. Then, under the above heuristic inequality (6.1), for all unramified p ≫ 0 such that n p > 1, we have (where (m) .

log 2 = log • log) Prob o p (η) < p ≤ 1 p gp (np-1)-ε , with ε = O 1 log 2 (p) .
Taking c = 1 and m = p np -1 < p np , this leads to S p < p gp+ np log 2 (p np -1) for all p ≫ 0. Thus

Prob(o p (η) < p) ≤ S p (p np -1) gp ≤ 1 (p np -1) gp • p -gp- np log 2 (p np -1) = 1 p gp (np-1)-O 1 log 2 (p)
.

To apply the Borel-Cantelli heuristic giving the finiteness of primes p such that o p (η) < p, we must have the inequality g p (n p -1) ≥ ε + 1, hence g p (n p -1) ≥ 2. Otherwise, we get g p = 1 & n p = 2, not sufficient to conclude for quadratic fields with p inert since, in this case,

Prob(o p (η) < p) ≤ 1 p 1-ε with ε = O 1 log 2 (p) . Remarks 6.1. -(i) Still when g p (n p -1) ≥ 2, we can replace the previous inequality Prob o p (η) < p ≤ 1 p gp (np-1)-ε by Prob o p (η) < p κ ≤ 1 p gp (np-1)-ε 12 which is true for any real κ such that 1 ≤ κ ≤ n p -1+ε
gp , in which case the Borel-Cantelli heuristic applies and may have some interest for large n p ; for instance, if K = Q r is the subfield of degree ℓ r (ℓ a prime, r ≥ 1), of the cyclotomic Z ℓ -extension of Q, and if we take primes p totally inert in K/Q, one can take κ = ℓ r -2 (if ℓ r = 2) for any η as usual.

(ii) Note that the proof of the theorem does not take into account the conditions o p (η) ∤ D np,δ (p) and it should be interesting to improve this aspect. But this theorem is a first step, and in the next sections, we intend to use explicitely the set D p for numerical computations and for a detailed study of the more ambiguous quadratic fields case. Indeed, in this case, we have to estimate the more precise upper bound 1

p 2 -1
D∈Dp φ(D) and a numerical experiment with the following PARI program (from [START_REF]PARI/GP version 2.9.0[END_REF]) shows a great dispersion of the number N of such divisors:

(F1[,1],F2[,1]); E=concat(F1[,2],F2[,2]); forvec(v=vectorv(# E,i,[0,E[i]]), my(d=factorback(P,v)); if(d>=p-1, next); if((p-1)%d!=0 && (p+1)%d!=0, N=N+1)); print(p," ", N))}
giving for instance (depending on the factorizations of p -1 and p + 1):

p = 100237, N = 3, where (p -1) • (p + 1) = (2 2 • 3 • 8353) • (2 • 50119), p = 100673, N = 489, where (p -1) • (p + 1) = (2 6 • 11 2 • 13) • (2 • 3 2 • 7 • 17 • 47).
We shall return more precisely to the quadratic case in § 8.4.

We can state to conclude this section:

Conjecture 6.1. -Let K/Q be Galois of degree n ≥ 3, of Galois group G. Let η ∈ K × be such that the multiplicative Z[G]-module generated by η is of Z-rank n.
For any unramified prime p, prime to η, let o p (η) be the order of η modulo p. Then o p (η) > p, for all p non totally split in K/Q, except a finite number. More generaly, o p (η) > p (np-1)+1-np n •(1+ε) , for all p such that n p > 1, except a finite number.

Numerical evidences for the above conjecture

This section is independent of any η ∈ K × and any number field K but depends only on given and fixed integer parameters denoted by abuse (n p , g p ). For n p > 1, we explicitely compute, for any p, S p := D∈Dp D gp and the upper bound S p (p np -1) gp of Prob o p (η) < p , using the program described below. Recall that The favourable cases for the Borel-Cantelli principle are those with

D p := {D : D | p np -1, D < p, D ∤ D np,δ (p) ∀ δ | n p }.
C B b > 1, but the inequalities C B b ≥ C ∞ b := Inf p∈[b,∞] C(p) do not mean that the Borel-Cantelli NUMERICAL EVIDENCES FOR THE ABOVE CONJECTURE principle applies since we ignore if C ∞ b > 1 or not for b ≫ 0, because C ∞ b is an increasing function of b.
In the applications given below, n p is a prime number, for which D np,δ (p) is in {p -1, p np-1 + • • • + p + 1}; for more general values of n p , one must first compute the set D p as defined in Theorem 2.1. {b=10^6; B=10^7; gp=1; np=2; CC= gp*(np-1)+1; C=CC; V=vector(B,i,i^gp); forprime(p=b, B, my(S=0, M=p^np-1, m=p-1, mm=M/m, i); fordiv (M,d,if(d>p, break)

; if(m%d!=0 && mm%d!=0, S+=V[d])); if(S!=0, C=(gp*log(M)-log(S))/log(p)); if(C<CC, CC=C)); print(CC)}
The initial CC := g p (n p -1) + 1 ≥ 2 is an obvious upper bound for C B b .

7.2. Application to quadratic fields with n p = 2. -We have g p = 1. We obtain C ≈ 0.56402... for 10 6 ≤ p ≤ 10 7 , then C ≈ 0.58341... for 10 7 ≤ p ≤ 10 8 , and C ≈ 0.58326... for 10 8 ≤ p ≤ 10 9 . For larger primes p it seems that the constant C stabilizes. If we replace D by φ(D) the result is a bit better (e.g., C ≈ 0.64766... instead of 0.56402... for 10 6 ≤ p ≤ 10 7 ).

The local extremum of C are obtained by primes p, like 166676399, 604929599, 1368987049, 1758415231, for which p 2 -1 is "friable" (product of small primes; see the computations that we shall give in § 8.4).

Numerical examples with fixed η and p → ∞

The above computations are of a density nature and the upper bound 1 p C is much higher than the true probability. So we intend to take a fixed η ∈ K × , restrict ourselves to primes p with suitable residue degree n p , and compute the order of η modulo p to find the solutions p of the inequality o p (η) < p.

The programs verify that η generates a multiplicative Z[G]-module of rank n. In the studied cases, K/Q is Abelian (G = C 2 , C 3 , C 4 ) and the condition on the rank is equivalent to η e = 1 in η G ⊗ Q, for all rational idempotents e of Q[G].

8.1. Cubic cyclic fields. -We then consider the following program with the polynomial P = X 3 + X 2 -2 X -1 (see data in Section 3). Put η = ax 2 + bx + c; then a is fixed and to expect more solutions, b, c vary in [-10, 10] and p in [3, 10 5 ]: {P=x^3+x^2-2*x-1; x0=Mod(x,P); Borne=10^5;[START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF]10,10,Eta0=a*x0^2+b*x0+c;Eta1=a*x1^2+b*x1+c;Eta2=a*x2^2+b*x2+c;N=norm(Eta0) Borne,if(p%N!=0,T=Mod(p,[START_REF] Tenenbaum | Introduction à la Théorie Analytique et Probabiliste des Nombres. 4 e édition revue et augmentée[END_REF]^2; if(T!=1, A=Mod(a,p); B=Mod(b,p); C=Mod(c,p); Y=Mod(A*x^2+B*x+C, P); my(m=p-1, mm=p^2+p+1); fordiv (m*mm, d, if(d>p, break); Z=Y^d; if(Z==1, print(a," ",b," ",c," ",p," ",d)))))))))}

; R1=Eta0*Eta1*Eta2; R2=Eta0^2*Eta1^-1*Eta2^-1; if(R1!=1 & R2 !=1 & R1!=-1 & R2 !=-1, forprime(p=1,
No solution is obtained except the following triples (the eventual multiples of o p (η) are not written):

(a, b, c, p, o p (η)) = (1, -7, 7, 137, 56), (1, -3, 3, 37, 28), [START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF][START_REF] Moree | Artin's Primitive Root Conjecture -A Survey[END_REF][START_REF]Introduction to Cyclotomic Fields[END_REF]47,37), (1, 6, -10, 31, 18). We have here an example (η = x 2 + 4x + 8, p = 47) where o p (η) = 37 divides p 2 + p + 1 = 37 • 61; this can be possible because p is too small regarding η s 2 +s+1 = 1 + 8 p = 377 (see Lemma 1.1).

8.2. Quartic cyclic fields. -We consider the quartic cyclic field K defined by the polynomial P No solution is obtained except the following ones, where we consider at most a solution (p, o p (η)) for a given p (other solutions may be given by conjugates of η and/or by η ′ ≡ η (mod p); many solutions with p = 19 and the orders 12 and 15); we eliminate also the solutions (p, λ o p (η)), λ > 1):

= X 4 -X 3 -6X 2 + X + 1 of discriminant 34 2 . The quadratic subfield of K is k = Q( √ 17
(a, b, c, d, p, o p (η)) = (1, -10, 2, -10, 19, 12), (1, -10, 5, -9, 19, 15), (1, -9, 6, 9, 43, 33), (1, -7, -2, -6, 19, 8), (1, -7, 2, -8, 19, 10), (1, -8, 7, 7, 461, 276), (1, -4, 1, 8, 1549, 1395), (1, -3, 0, -6, 223, 64), (1, -1, -6, -10, 229, 184), (1, -1, 3, -2, 59, 40), (1, 3, -8, 6, 53, 9), (1, 3, -5, 10, 83, 21), (1, 9, -7, 5, 43, 22).

For the last three cases, the order divides p + 1 for the same reason as above. We have the more exceptional solution (1, -4, 1, 8, 1549, 1395) where (a, b, p, o p (η)) = (1, -10, 79, 65), (1, -10, 101, 75), (1, -10, 967, 847), (1, -10, 20359, 13234), (1, -10, 90149, 72700), (1, -9, 89, 55), (1, -9, 6163, 4623), (1, -9, 29501, 6705), (1, -8, 10711, 2210), (1, -6, 1123, 843), (1, -5, 86969, 81172), (1, -4, 30941, 25785), (1, -9, 41, 15), (1, -9, 1301, 403), (1, -8, 5, 3), (1, -7, 29, 24), (1, -7, 103, 39), (1, -7, 727, 143), (1, -4, 701, 675), (1, -3, 43, 33).

If Conjecture 6.1 is likely for degrees n ≥ 3, the question arises for quadratic fields with n p = 2. We give here supplementary computations with the following simplified program which can be used changing m, a, b: 3), this gives the few solutions (up to p ≤ 10 9 ): (p, o p (η)) = (5, 4), (29, 21), (1063, 944), (32707, 23384), (90401, 68930). (43, 11), (131, 24), (811, 174), (911, 133), (5743, 3168), (2378711, 1486695).

Although this kind of repartition of the solutions looks like the case of Fermat quotients, for which a specific heuristic can be used (see [START_REF] Gras | Étude probabiliste des quotients de Fermat[END_REF]), it seems that we observe more systematic large solutions in the quadratic case with p inert, and we have possibly infinitely many solutions. This should be because the problem is of a different nature and is connected with generalizations of primitive roots problem in number fields (see the extensive survey by P. Moree [START_REF] Moree | Artin's Primitive Root Conjecture -A Survey[END_REF]). So we shall try in the next subsection to give some insights in the opposite direction for quadratic fields (infiniteness of inert primes p with o p (η) < p). Theorem 2.1). We have the inequalities:

1 3 ≤ 1 p -1 D∈Dp φ(D) < c(p) log 2 (p), p → ∞,
where c(p) is probably around O(log 2 (p)).

The majoration p , this suggests the possible infiniteness of inert p such that o p (η) < p for fixed η ∈ K × such that η 1+s and η 1-s are distinct from roots of unity. Indeed, for p ∈ {2, 3, 5, 7, 17}, we get the strict inverse inequality

1 p 2 -1 D∈Dp φ(D) < c(p) • log 2 (p) p + 1 ∼ c(p) • 1 p 1-2•log 2 (p)/log
1 p -1 D∈Dp φ(D) < 1 3
and we have no other examples up to 10 9 . The equality S+= prod(i=1,# v, if(v[i], (P[i]-1)*P[i]^(v[i]-1),1)))); Density=S/(p^2-1.0); Delta=S/(p-1.0)-1/3; C= Density*p/log(p); print(p," ", Density," ", Delta," ",C))}

1 p -1 D∈Dp φ(D) = 1 3 is doubtless equivalent to p -1 = 2 u+2 • 3 v & p + 1 = 2 • ℓ, for some u ≥ 0, v
Then we obtain, for the inequalities (iii) Large values of C are, on the contrary, obtained when p 2 -1 is the product of small primes (friable numbers). This may help to precise the upper bound of C since the local maxima increase slowly. For instance: For p > 1758415231 the running time becomes prohibitive although we may conjecture the infiniteness of these numbers.

166676399 2 -1 = 2

  7 and o p (η) = r • d, d | D 3,δ (p), we find, from the corollary, r ≥ 3 for δ = 1 (i.e., r | p -1), r ≥ 9 for δ = 3 (i.e., r | p 2 + p + 1).5. Densities-Probabilities for o p (η) and o p (η)

5. 3 .

 3 Remarks and examples. -Suppose that η generates a multiplicative Z[G]-module of Z-rank n, which has obvious consequences (apart the fact that η / ∈ µ(K)):

  1 & § 5.4): (6.1) Prob o p (η) < p ≤ Prob o p (η) ∈ D p

  Proof. -Let S p := D∈Dp D gp ; under the two conditions D | p np -1, D < p, we get S p < D|p np -1 p D gp D gp = p gp • τ (p np -1), where τ (m) denotes the number of divisors of the integer m. From [7, Theorem I.5.4], we have, for all c > log(2) and for all m ≫ 0, τ (m) ≤ m c log 2

  {b=10^5; B=b+10^3; forprime(p=b, B, N=0; my(e=kronecker(-4,p)); F1=factor(2*(p-e)); F2=factor((p+e)/2); P=concat

7. 1 .

 1 General program about the divisors D ∈ D p . -It is sufficient to precise the integers n p > 1, g p ≥ 1, and the interval [b, B] of primes p. The program gives the least value C B b of C(p), p ∈ [b, B], where S p (p np -1) gp =: 1 p C(p) .

7. 3 .

 3 Application to cyclic cubic fields with n p = 3. -We use the program with g p = 1, n p = 3. For instance, for 10 6 ≤ p ≤ 10 7 , we get C ≈ 1.5652... > 1 as expected from Theorem 6.1; for 10 7 ≤ p ≤ 10 8 the value of C is 1.5399325... and for 10 8 ≤ p ≤ 3.65717251 • 10 8 , we get C ≈ 1.5809....7.4. Application to quartic fields with n p = 2 and for n p = 4. -For g p = 2, n p = 2, and 10 6 ≤ p ≤ 10 7 , we get C = 1.6103...; for 10 8 ≤ p ≤ 10 9 , the result for C is 1.6186.... Naturally, for n p = 4 we obtain a larger constant C = 2.4596.... But in the case n p = 4 we can test the similar stronger condition Prob o p (η) < p 2 for which one finds C = 1.28442..., giving the conjectural finiteness of totally inert primes p in a Galois quartic field such that o p (η) < p 2 .

  ) and K = k (17 + √ 17)/2 . The program is analogous to the previous one with the parameters n p = g p = 2. Put η = ax 3 + bx 2 + cx + d; then b, c, d vary in [-10, 10], and p in [3, 10 5 ]:

8 . 3 .

 83 1395 = 9 • 5 • 31 with 9 | p -1 and 5 • 31 | p + 1. Quadratic fields. -We consider the field K defined by the polynomial P = X 2 -3 and the following program with η = a √ 3 + b, a = 1, b ∈ [-10, 0].

3

 3 we obtain the solutions (up to p ≤ 10 9 ): (p, o p (η)) =(7, 6), (29, 21), (137, 92), (7498769, 5927335), (39208553, 31070928).The large solution (p = 39208553, o p (η) = 31070928) (where p 2 -1 is friable) is a bad indication for finiteness. (iii) Consider K = Q( √ -1) with p ≡ 3 (mod 4) up to p ≤ 10 9 . For η = √ -1 + 4 (N(η) = 17), we obtain the solutions: (p, o p (η)) = (49139, 19593), (25646167, 22440397). For η = √ -1 + 2 (N(η) = 5), we obtain the solution: (p, o p (η)) = (9384251, 6173850). For η = 3 √ -1 + 11 (N(η) = 130), we obtain the solutions: (p, o p (η)) = (3, 2),

8. 4 .

 4 Analysis of the quadratic case. -Starting from the formula Prob o p (η) < p ≤ Density o p (η) < p = 1 p 2 -1 D∈Dp φ(D) of Remark 6.1 (iii), we study the right member of the normalized equality (p + 1) • Density(o p (η) < p) = 1 p-1 D∈Dp φ(D), remembering that it is an upperbound of the probability. From numerical experiments, we can state: Conjecture 8.1. -Let D p be the set of divisors D of p 2 -1 such that D < p, D ∤ p -1, D ∤ p + 1 (see

  (p) is to be compared with the upper bound 1 p 1-ε with ε = O 1 log 2 (p) of Theorem 6.1, but the sets of divisors D | p 2 -1 are not the same and this information is only experimental. On the contrary, the minoration 1 p -1 D∈Dp φ(D) ≥ 1 3 seems exact (except very few cases), and although the density (≥ probability) is O(1)

1

 1 D∈Dp φ(D) < c(p) • log 2 (p), the following data, showing their great dispersion, first for some small prime numbers, then for some larger ones, where • Density ≪ c(p) • log(p):

( i )

 i For p = 1000000012345678910111213141516172457 above, we have:C ≈ 0.004086, C/log(p) ≈ 4.930 • 10 -5 , p -1 = 2 3 • 3 2 • 389 • 62528362319 • 571006238831466292903, p + 1 = 2 • 8131511 • 61489187701134445376216864339. (ii) For p = 10123456789123456789125887, we obtain ∆ ≈ 5.0641 • 10 -23 , C ≈ 0.005789, C/log(p) ≈ 10.054 • 10 -5 , p -1 = 2 • 5061728394561728394562943, p + 1 = 2 8 • 3 • 13181584360837834360841.

  Theorem 2.1. -Let K/Q be Galois of degree n, of Galois group G. Let h | n be a possible residue degree in K/Q. Let µ(K) be the group of roots of unity contained in K. Let η ∈ K × be such that the multiplicative Z[G]-module generated by η is of Z-rank n. Then for all (unramified) prime number p ≫ 0, with residue degree n p = h, and for any prime ideal p | p, the least integer k ≥ 1 for which there exists ζ ∈ µ(K) such that η k ≡ ζ (mod p) is a divisor of p h -1 which does not divide any of the integers

	D h,δ (p) :=	p h -1 Φ δ (p)	, δ | h,
	where Φ δ (X) is the δth cyclotomic polynomial.
	Hence o p (η) and a fortiori o p (η) (cf. Definition 1.1), do not divide any of the
	D h,δ (p).

  1 3 •gcd (o p (η), p-1)•gcd (o p (η), p 2 + p + 1):With the same data, the least values of gcd (o p (η), p -1) are: 1 (for p = 2, 3, 17), 2 (for p = 101), 3 (for p = 13669, for wich we get o p (η) = 560565693), 4 (for p = 5, 317), 9 (for p = 19, 73).

	(iv) p ≡ 5 (mod 7):
	p gcd (o 5	4	31
	19		9	381
	47 (i) p ≡ 2 (mod 7): 61	23 10	2257 1261
	89	11	8011
	p 103 131	gcd (o p (η), p -1) 102 65	10713 gcd (o p (η), p 2 + p + 1) 17293
	2 173	1 172	1 30103
	23	11	553
	37	36	201
	79	78	6321
	107	53	11557
	149	37	22351
	163	54	26733
	191	190	36673
	3		1	13
	17 *		1	307
	31		15	993
	59		58	3541
	73		9	5403
	101 * *		2	10303
	157		26	8269
	199		198	39801
	(iii) p ≡ 4 (mod 7):
	p	gcd (o p (η), p -1)	gcd (o p (η), p 2 + p + 1)
	11	10	133
	53	26	2863
	67	33	4557
	109	27	11991
	137	136	18907
	151	75	22953
	179	89	32221
	193	192	37443

(ii) p ≡ 3 (mod 7):

p gcd (o p (η), p -1) gcd (o p (η), p 2 + p + 1) p (η), p -1) gcd (o p (η), p 2 + p + 1)

Up to p ≤ 10 7 , we have no other solutions for gcd (o p (η), p -1) < 10. For gcd (o p (η), p 2 + p + 1) < 100 we get 1 (for p = 2), 13 (for p = 3), 31 (for p = 5); for gcd (o p (η), p 2 + p + 1) < 1000 we only have the primes p =

  Considering F × p 0 as a H-module (H is generated by the global Frobenius s = s p 0 which by definition makes sense in F p 0 /F p ),

	p|p	F × p is the induced representation
	and we get
	p|p

  For small primes p there are solutions o p (η) | p -1 or o p (η) | p + 1:

	{m=3; P=x^2-m; x0=Mod(x,P); x1=-x0; a=1; Borne=10^5;
	for(b=-10, 10, Eta0=a*x0+b; Eta1=a*x1+b; N=norm(Eta0);
	R1=Eta0*Eta1; R2=Eta0*Eta1^-1; if(R1!=1 & R2 !=1 & R1!=-1 & R2 !=-1,
	forprime(p=1, Borne, if(p%N!=0,
	if(kronecker(m, p)==-1, A=Mod(a,p); B=Mod(b,p); Y=Mod(A*x+B,P);
	my(m=p-1, mm=p+1); fordiv(m*mm, d, if(d>p, break);
	Z=Y^d; if(Z==1, print(a," ",b," ",p," ", d))))))))}

  5 • 3 3 • 5 2 • 7 • 11 • 17 • 19 • 23 • 29 • 31 • 41 • 61 with C ≈ 41.91845 and C/log(p) ≈ 2.21421. 1758415231 2 -1 = 2 8 • 3 4 • 5 • 7 • 11 • 13 • 17 • 19 • 29 • 31 • 37 • 47 • 59 with C ≈ 81.51733 and C/log(p) ≈ 3.82932.The following program computes these successive local maxima:

	{B=10^20; CC=0.0; forprime(p=3, B, my(S=0, e=kronecker(-4,p)); F1=factor(2*(p-e));
	F2=factor((p+e)/2); P=concat(F1[,1],F2[,1]); E=concat(F1[,2],F2[,2]);
	forvec(v=vectorv(#E,i,[0,E[i]]), my(d=factorback(P,v)); if(d>p, next);
	if((p-1)%d!=0 && (p+1)%d!=0, S+= prod(i=1,#v,if(v[i],(P[i]-1)*P[i]^(v[i]-1),1))));
	Pr=S/(p^2-1.0); C=Pr*p/log(p); if(C>CC, CC=C; print(p," ",CC," ",CC/log(p))))}
	p	CC	CC/log(p)
	11	0.1529118768	0.0637692056
	19	0.2867929851	0.0974015719
	29	0.3690965111	0.1096121427
	(......)
	604929599	51.9605419985	2.5696806133
	1368987049	61.6784084466	2.9318543821
	1758415231	81.5173320978	3.8293199014


Self-driving technologies have improved to the point that major industrial players are now investing in, developing and testing self-driving vehicles in various countries. It is touted that fleets of self-driving vehicles will be operational within five years. Besides economic reasons, a major incentive behind self-driving vehicles is safety. Road traffic deaths across the world reached 1.25 million in 2015 and since the driver is responsible for most crashes, it seems natural to strive to design self-driving vehicles. Obviously, an autonomous vehicle is safe if it avoids collision. Now, collisions happen for different reasons: hardware or software failures, perceptual errors that result in an incorrect understanding of the situation (in May 2016, a self-driving vehicle crashed killing its driver because its camera failed to recognise a white truck against a bright sky), and reasoning errors, i.e., a wrong decision is made (in February 2016, a self-driving vehicle was for the first time responsible for a crash because of a wrong decision). Our research focuses on reasoning errors. At a fundamental level, a collision will happen as soon as a vehicle reaches an inevitable collision state (ICS), i.e., a state for which, no matter what the vehicle does, a collision will eventually occur [START_REF] Fraichard | Inevitable collision states. a step towards safer robots?[END_REF].

The insights on collision avoidance resulting from our investigation of the ICS concept are expressed succinctly by the following quote: 'One has a limited time only to make a motion decision, one has to globally reason about the future evolution of the environment and do so over an appropriate time horizon.'

As abstract and general as this quote may seem, it implicitly contains motion safety laws whose violation is likely to cause a collision [START_REF] Fraichard | Will the driver seat ever be empty?[END_REF]. ICS were initially investigated with the aim of designing motion strategies for which collision avoidance could be formally guaranteed. Assuming the availability of an accurate model of the future up to the appropriate time horizon, we were able to design motion strategies with guaranteed absolute motion safety i.e., no collision ever takes place.

In the real world, things are not so rosy since accurate information about the future evolution of the environment is not available. The choice then is between conservative or probabilistic models of the future. In conservative models, each obstacle is assigned its reachable set, i.e., the set of positions it can potentially occupy in the future. Conservative models solve the problem of the discrepancy between the predicted future and the actual future. In theory, they could allow for guaranteed motion safety. In practice however, the growth of the reachable sets as time passes by is such that every position becomes potentially occupied by an obstacle and it is then impossible to find a safe solution. In probabilistic models, the position of an obstacle at any given time is represented by an occupancy probability density function. Such models are well suited to represent the uncertainty that prevails in the real world. However, as sound as the probabilistic framework is, it cannot provide motion safety guarantees that can be established formally, minimising the collision risk is the only thing that can be done then. Today, most, if not all, self-driving vehicles rely upon probabilistic modelling and reasoning to drive themselves. The current paradigm is to test self-driving vehicles in vivo and, should a problem occur, to patch the self-driving system accordingly. In an effort to improve the situation and to provide provable motion safety guarantees, we would like to advocate an alternative approach that can be summarised by the following motto: "Better guarantee less than guarantee nothing."

The idea is to settle for levels of motion safety that are weaker than absolute motion safety but that can be guaranteed. One example of such a weaker level of motion safety guarantees that, if a collision must take place, the selfdriving vehicle will be at rest. This motion safety level has been dubbed

Will the driver Seat Ever be Empty? by Thierry Fraichard (Inria)

Self-driving vehicles are here and they already cause accidents. Now, should road safety be considered in a trial and error perspective or should it be addressed in a formal way? The latter option is at the heart of our research.

passive motion safety and it has been used in several autonomous vehicles. Passive motion safety is interesting for two reasons: (i) it allows provision of at least one form of motion safety guarantee in challenging scenarios (limited field-of-view for the robot, complete lack of knowledge about the future behaviour of the moving obstacles [START_REF] Bouraine | Provably safe navigation for mobile robots with limited field-ofviews in dynamic environments[END_REF]), and (ii) if every moving obstacle in the environment enforces it then no collision will take place at all. We are currently exploring more sophisticated levels of motion safety. We are also studying the relationship between the perceptual capabilities of the self-driving vehicles and the levels of motion safety that can be achieved. The long-term goal of our research is to investigate if and how current selfdriving technologies can be improved to the point that the human driver can safely be removed from the driving loop altogether, paving the way to truly selfdriving vehicles whose motion safety can be formally characterised and guaranteed.

Special theme: Autonomous Vehicles

In our latest research activity, a tractor (type Steyr 6230 CVT) was provided by the Austrian Armed Forces for the purpose of automating agricultural tasks on special areas. In a first step, this mobile machine was prepared in close cooperation with the manufacturer CNH Austria to serve as a mobile research platform. An electrical single-point interface provides access to the various functions of the tractor via actuators and the CAN bus system, which enables a set of additional computers and dedicated software to take over control. The vehicle is equipped with a multi-modal sensor system combining stereo vision, laser scanning, RTK-GPS and IMU to address difficult and varying outdoor conditions. A dedicated AIT stereo vision system mounted behind the windshield at the front and another on the rear of the vehicle provide dense 3D data of the tractor's environment (Figure 1). An important step to further increase efficiency in farming is to develop completely driverless machines which would, for example, enable one operator to monitor and control multiple machines at once. Thus, AIT investigates important computer vision based technologies and methods that improve the abilities of future autonomous vehicles and machines for agricultural applications. Furthermore, research is driven by a specific use-case defined by the Austrian Armed Forces: cultivation (e.g., mulching, mowing) of firebreaks on military training areas endangered by explosive ordnances. The technology developed by AIT allows the operator to safely monitor and control the semiautonomous machine from a safe distance with a high level of situational awareness.

The sensed 3D data of the stereo camera systems [START_REF] Fraichard | Inevitable collision states. a step towards safer robots?[END_REF] and the laser scanner is continuously fused into an 'elevation map' -a 2.5D representation of the terrain. As the vehicle moves, the reconstruction of the vehicle's surroundings becomes more and more complete. An essential task is estimating the ego-motion and the pose of the vehicle. The trajectories of the stereo visual odometries, the orientation of the inertial measurement unit and the position of the RTK-GPS are fused with an extended Kalman filter. The filter models the kinematics of the vehicle and computes a precise estimate of the pose [START_REF] Fraichard | Will the driver seat ever be empty?[END_REF]. Furthermore, the geometry of the elevation map is analysed in order to obtain a traversability map, which denotes drivable and non-drivable areas as well as obstacles around the vehicle. The map is organised using a tile based approach, which allows storing, loading and updating large scale maps. In that way the georeferenced mapped areas can also be accessed via a geographic information system.

The real-time traversability map is used by the path planning module to calculate a collision free trajectory along a predefined mission path. Dynamically appearing obstacles are avoided by calculating a local bypass route using a sampling-based motion planning algo-

Intelligent Environmental Perception for Navigation and operation of Autonomous Mobile Machines

by Robert Rößler, Thomas Kadiofsky, Wolfgang Pointner, Martin Humenberger and Christian Zinner (AIT)

Research at AIT on autonomous land vehicles is focusing on transport systems and mobile machines which operate in unstructured and heavily cluttered environments such as off-road areas. In such environments, conventional technologies used in the field of advanced driver assistance systems (ADAS) and highly automated cars show severe limitations in their applicability. Therefore, more general approaches for environmental perception have to be found to provide adequate information for planning and decision making. As a special challenge, our focus is on vehicles that actively change their environment, e.g., by cutting high plants or manipulating piles of pellet materials. To operate such machines autonomously, novel approaches for autonomous motion planning are required.

Figure1:Aself-drivingvehicleamongfixedandmovingobstacles(left);2Dsliceofthe5Dstatespaceoftheself-drivingvehicle, theblackareasarethecorrespondinginevitablecollisionstatesthatmustbeavoided(right).