S Hadjeras

J J Prince Agbodjan

C Albea Sanchez

G Garcia
email: garcia@laas.fr

J J Prince

Hybrid adaptive control for the half-bridge inverter

Keywords: Half-bridge inverter, hybrid dynamical system, adaptive control, singular perturbation analysis

In this work, we propose a hybrid adaptive control law for the half-bridge inverter subjected to the common problem of unknown load. This controller ensures the system robustness with respect to the output, taking into account the real nature of the signals, which means the continuous-time variables, (voltage and current signals) and the discrete-time variables, (switching signals). The adaptation is accomplished using a state observer and assuming that all states are measurable. Then, stability properties can be ensured using hybrid dynamical system theory and singular perturbation analysis. Finally, the proposed hybrid adaptive controller is validated in simulation.

I. Introduction

Nowadays, power converter control is a subject of more and more research topics. This is mainly due to the fact that power converters have a very wide range of applications, for example, the half and full bridge inverters, which are used in Adjustable Speed Drives (ASDs), Uninterruptible Power Supplies (UPSs), static var compensator, active filters, Flexible AC Transmission Systems (FACTSs) and voltage compensator [1]. Frequently, their connected loads can change or suffer perturbations. This problem is usually solved by using adaptation mechanisms, guarantying a robust output signal with respect to these load changes. Hence, in [START_REF] Lucia | Load-Adaptive Control Algorithm of Half-Bridge Series Resonant Inverter for Domestic Induction Heating[END_REF], a load adaptive control algorithm is designed to cover a variety of loads largely inductive using square wave, asymmetrical duty cycle, and pulse density modulation. Likewise, in [START_REF] Kwon | Half-Bridge Series Resonant Inverter for Induction Heating Applications with Load-Adaptive PFM Control Strategy[END_REF] a load adaptive tuned frequency tracking control strategy using PLL (Phase Locked Loop) is used to control a half bridge series resonant inverter for induction heating operations. And in [START_REF] Kissaoui | Output-Feedback Nonlinear Adaptive Control of Single-Phase Half-Bridge Ups Systems[END_REF], a multiloop nonlinear adaptive control is designed by using a backstepping technique. It is important to emphasize that all these control laws are based on averaged models, losing the properties of the discrete signals.

In this work, we aim to use a more accurate model of half-bridge inverters, considering the real nature of the signals, which means, continuous (voltages and currents) and discrete (on-off state of the switches) signals. To this end, we extend the results presented in [START_REF] Albea | Hybrid control scheme for a half-bridge inverter[END_REF], where a hybrid control is designed for the half-bridge inverter considering a constant and known load.

The novelty of this work lies in the design of a hybrid adaptive control for a half-bridge inverter regarding an unknown or perturbed load. A similar problem is considered in [START_REF] Hadjeras | Hybrid adaptative control of the Boost Converter[END_REF], where the authors stabilize the output of a DC-DC converter to a desired reference, considering a regulation problem and an unknown load. However, here the problem is different, because we are dealing with a tracking problem. To this end, we transform the problem in an output regulation problem [START_REF] Francis | The linear multivariable regulator problem[END_REF][8] and we propose an indirect adaptation mechanism in the hybrid dynamical scheme, in order to adapt the load variations. More precisely, we consider an adaptive law fed by a state observer by assuming that all states are measurable. This method is simpler than the one presented in [START_REF] Hadjeras | Hybrid adaptative control of the Boost Converter[END_REF]. Then, uniformly locally asymptotically stability is ensured by applying the scale-time separation and by using a singular perturbation analysis.

The paper is organized as follows: in Section II, a model is defined under some assumptions and the problem is stated in Section III. Section IV proposes a hybrid adaptive control that achieves the objectives of our problem. Section V deals with the proof of our main result. Some simulations are performed in Section VI. Section VII draws the conclusion and future works.

Notation: Throughout the paper, R denotes the set of real numbers, R n the n-dimensional Euclidean space and R n×m the set of all real m × n matrices. The set S n denotes the set of symmetric positive definite matrices of matrices R n×n . Re(ς) is the real part of a complex number ς.

II. system modeling A. Inverter model

We consider a half-bridge inverter presented in Fig. 1, which is fed by a DC input, 2V in , and generates an AC output. This inverter is composed by a load filter, L, C 0 , a purely resistive load, R 0 . The parasite resistance R LS models not only the switching energy dissipated, but also the resistive component of the inductance. The dynamic behavior of this system can then be described by the following model:

d dt i L (t) v C (t) = -R LS L -1 L 1 C0 -1 R0C0 i L (t) v C (t) + Vin L 0 u, (1)
where i L is the inductance current, v C is the capacitor voltage and these two states are considered as the continuous-time state variables. u:= U 1 -U 2 is the control input and it is assumed to be a discrete variable:

u = -1, if U 1 = OFF and U 2 = ON. u = 1, if U 1 = ON and U 2 = OFF. R LS L i L C 0 R 0 v c + - 2V in U 1 U 2 C 1 C 2 V in V in + + - + - Fig. 1. Half-bridge inverter.
Assumption 1: We assume that,

• current and voltage are measurable.

• All the components are ideal.

• The load R 0 is an unknown constant and can suffer

disturbances in the interval [R m 0 , R M 0 ]. let define β := 1 R0 ∈ [β m , β M ],
which belongs to the following polytope

Ω := λ β,m β m + λ β,M β M , ∀ λ β,m , λ β,M ∈ [0, 1], with λ β,m + λ β,M = 1.
Using definition of β, the system (1) is rewritten as

follows ẋ = A(β)x + Bu, y = x, (2)
with x = [x 1 x 2 ] T = [i L v C ] T
, and A(β) and B are easily deduced from (1). Note that A(β) is Hurwitz for all β > 0. Moreover, we assume that the following property is verified. Property 1: Consider matrix A(β) in (2), with β ∈ [β m , β M ], and a chosen matrix Q ∈ S 2 . Then there exists a matrix P ∈ S 2 satisfying the two following Lyapunov inequalities:

A T (β m )P + P A(β m ) + Q < 0, A T (β M )P + P A(β M ) + Q < 0.
This property imposes finally a common Lyapunov function for the polytopic model [START_REF] Lucia | Load-Adaptive Control Algorithm of Half-Bridge Series Resonant Inverter for Domestic Induction Heating[END_REF]. In the next subsection, we propose a desired tracking trajectory, described by a linear time-invariant model.

B. A reference model

The desired trajectory for the voltage and the current can be modeled by the following equations:

x 2e := V C d (t) = V max sin(wt), x 1e := i L d (t) = C 0 wV max cos(wt) + βV max sin(wt), (3)
where V max and ω are the desired amplitude and frequency of the voltage, respectively, applied to the load R 0 . In order to impose such behavior, let us define the exogenous input z ∈ R 2 produced by this time-invariant exosystem:

ż = 0 -w w 0 z = Θz, z(0) = 0 V max . (4)
From (3), and considering that β is the estimated value of β, we have

V C d (t) i L d (t) = Π( β)z, with Π( β) = wC 0 β 0 1 . ( 5
)
Remark 1: Note that from (4), it is simple to see that

z 2 1 (t) + z 2 2 (t) = V 2 max . Consequently, let define the following compact set Φ = {(z 1 , z 2 ) ∈ R 2 , z 2 1 (t) + z 2 2 (t) = V 2 max }.
Firstly, define Γ( β), such that, the following algebraic equation is verified:

A(β)Π( β) + BΓ( β) = Π( β)Θ. (6)
A simple calculation shows that, Γ( β) does not depend on β and can be written as:

Γ( β) = -wL β Vin -wR LS C0 Vin 1 L -C 0 w 2 + R LS β L L Vin .
The dynamics of the overall system is defined by

ẋ = A(β)x + Bu, ż = Θz, e = Cx + Dz = x -Π( β)z, (7)
where e = [e 1 e 2 ] T ∈ R 2 is the tracking error, C ∈ R 2 is the identity matrix and D = -Π( β).

Note that, equation ( 5) and ( 6) are the well known "regulator equations" [START_REF] Francis | The linear multivariable regulator problem[END_REF] [START_REF] Serrani | Semi-global nonlinear output regulation with adaptive internal model[END_REF]. Then, by using ( 6)-( 7), the tracking error dynamic is given by

ė = ẋ -Π( β) ż -βΠ ( β)z = A(β)e + Bd( β, β, z), with d( β, β, z) := v( β, z) - L V in βz 2 , (8)
v := u -Γ( β)z, (9)
where Π := dΠ d β . In order to present the control law, consider the following model:

ė = A(β)e + Bd( β, β, z), ż = Θz, ( 10
)
where v is the available input given in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] and it is composed of a continuous-time signal Γ( β)z and a switching signal u. The following assumption must be hold to ensure the existence of signal u, making e = 0.

Assumption 2: There exist two functions of time λ 1 , λ -1 ∈ [0, 1] satisfying λ 1 + λ -1 = 1, such that the following equation holds:

λ 1 -λ -1 -Γ( β)z = 0. ( 11
)
Note that this assumption provides the existence of e = 0 with β = β, allowing to be v = 0 through a convex combination of the two operating modes of u as follows

v = λ 1 -λ -1 -Γ( β)z. Remark 2:
It is important to understand that in Assumption 2, the solution e = 0 with β = β, is obtained in sense of Filoppov solution. Indeed, the signal u = λ 1 -λ -1 , is a periodic sequence of arbitrarily small period T , spending a time equal to λ 1 T in mode u = 1, and λ -1 T in mode u = -1.

III. Problem statement

This paper focuses on the design of a switching signal u ∈ {-1, 1} and an adaptive law β, which guarantee the two following properties:

• a suitable trajectory tracking properties of the voltage x 2 (t) to a desired trajectory x 2e (t), ensuring that, the error e = x -x e with x e = [x e1 x e2 ] T , converges to the equilibrium e = 0 in the Filippov sense.

• The convergence of the estimation of the load β to its real value of β.

Inspired by [START_REF] Hadjeras | Hybrid adaptative control of the Boost Converter[END_REF], we extend the work presented in [START_REF] Albea | Hybrid control scheme for a half-bridge inverter[END_REF], for model [START_REF] Lucia | Load-Adaptive Control Algorithm of Half-Bridge Series Resonant Inverter for Domestic Induction Heating[END_REF] with an unknown parameter β.

Problem 1: The idea represented by Fig. 2 is to design a hybrid adaptive controller that considers the continuous-time dynamics, x 1 , x 2 , and the discrete-time dynamic, u, estimating β in continuous-time, at the same time that x 1 and x 2 converge to a sinusoidal references given by (3). In order to solve Problem 1, we propose firstly an adaptation control law with the aim of estimating the unknown constant β. Then, we design a hybrid adaptive controller that considers both, the continuous and the discrete dynamics.

IV. Hybrid model and proposed control law

Consider that ê2 and β are the estimated states of e 2 and β, respectively. Then, the system (10) and the estimated variables mentioned before, can be modeled as a hybrid dynamical system, following the paradigm given in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF], wherein

• the continuous-time behavior encompasses the evolution of e, z, ê2 and β and, • the discrete-time behavior captures the evolution of the input signal u, (through v in ( 9)). We characterize the overall dynamics by the following hybrid model:

H :                                        ė ż v ė2 β       = f (ξ), ξ ∈ C,       e + z + v + ê+ 2 β+       ∈ G(ξ), ξ ∈ D, (12)
where ξ = [e z v ê2 β], f is the flow map and and G is the jump map containing the evolution of the state after jumps

f (ξ) :=        A(β)e + Bd(ξ), Θz -Γ( β)Θz 1 C0 (e 1 -βe 2 ) + α(e 2 -ê2 ) -γe2(e2-ê2) C0        , G(ξ) :=           x z     argmin v=u-Γ(β)z u∈[-1,1] ẽT P (A( β)e + Bd(ξ))     -Γ( β)z ê2 β           , (13)
being

d(ξ) = v( β, z) + γLe 2 (e 2 -ê2 ) V in C 0 z 2 , (14)
and with α, γ >0 are design parameters. The error ẽ = [ẽ 1 ẽ2 ] is defined as follows

ẽ1 = e 1 ẽ2 = e 2 + (e 2 -ê2 ). ( 15
)
Inspired by [START_REF] Albea | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF] and as proposed by [START_REF] Hadjeras | Hybrid adaptative control of the Boost Converter[END_REF], we select the so-called flow and jump sets

C :={ξ : ẽT P (A( β)e + Bd(ξ)) ≤ -ηẽ T Qẽ}, (16)
D :={ξ : ẽT P (A( β)e + Bd(ξ)) ≥ -ηẽ T Qẽ}, (17)
with η ∈ (0, 1) is a tunned parameter.

Remark 3: To achieve the control objectives, i.e. reach the desired equilibrium, e = 0, the proposed control generates arbitrary a fast switching in the steady-state, due to the fact that this desired equilibrium is achieved in Filippov sense. But from a practical point of view, the switching frequency must present a minimal dwell-time between consecutive switches. Note that the switching frequency is reduced if η → 0 and, it is increased if η → 1.

Proposition 1: The hybrid system ( 12) -(17) satisfies the basic hybrid conditions given in [9, Assumption 6.5]:

• sets C and D given in ( 16) and (17) respectively, are closed. • f is a continuous function, thus is therefore outer semicontinuous and locally bounded. Moreover, f is convex for all ξ ∈ C. • G is closed, therefore it is outer semicontinuous [9, Lemma 5.1] and locally bounded. Then, we can conclude that the hybrid system ( 12)-( 17) is well-posed.

Next, we invoke Lemma in [10, Lemma 1], for setting up stability properties.

Lemma 1: Consider matrices P, Q ∈ S 2 satisfying Property 1. Then for each e ∈ R 2 , min ẽT

P v=u-Γ( β)z u∈[-1,1] (A(β)e + Bd(ξ)) ≤ -ẽ T Qẽ.
Remark 4: We note here that for η ∈ (0, 1) and if ẽ = 0, then -ẽ T Qẽ < -ηẽ T Qẽ. Thereafter, using the Hybrid Dynamical System (HDS) theory, we will establish stability properties of the given compact set A := {ξ : e = 0, z ∈ Φ, u ∈ {-1, 1}, ê2 = e 2 , β = β}. (18) Theorem 1: Consider Assumption 1, 2 and matrices P, Q ∈ S 2 satisfying Property 1. Then attractor (18) is Uniformly Locally Asymptotically Stable (ULAS) for hybrid system (12)-( 17). The proof of this Theorem is given in the next Section. Here we will comment the choice of matrices P and Q. These matrices are selected following some optimization criteria given in (34) for a hybrid system ( 12)-( 17). Specifically, we use [5, Theorem 2], where some LQ performance level is guaranteed, then the following bound holds along any solution of our hybrid system

J ≤ η -1 ẽT P ẽ (19) defined in a compact hybrid time domain [9, Definition 2.3].
On the other hand, we find that the suboptimal-level corresponds to a high frequency switches, which can increase the dissipated energy. Therefore, the choice of η decides a trade-off between any LQ performance level and switching frequency as noted in Remark (3).

This next section is devoted to prove the stability of hybrid system ( 12)-(17).

V. Proof of theorem 1

Let us now remind the observer and adaptation dynamics given as

ė2 = 1 C 0 (e 1 -βe 2 ) + α(e 2 -ê2 ), (20) β
= - γe 2 (e 2 -ê2 ) C 0 . ( 21
)
Lemma 2: Considering Assumption 1, then ê2 = e 2 = 0, e 1 = 0 and β = β is not an equilibrium of the differential equation ( 1), ( 20) and (21).

Proof: Consider that β is unknown and constant. Note that (5) is not accomplished, i.e. if β = β, then the reference Π( β)z is different to the desired trajectory (3), which directly depends on the real value of β. Therefore, ė = 0, ė2 = 0 and β = 0 Lemma 3: Regard system (10), Lemma 2 and Assumption 1 and assume that its solutions are bounded. The extended observer (20)-( 21) has the following properties:

i) The estimated states ê2 , β are bounded.

ii) lim t→∞ ê2 (t) = e 2 (t).

iii) lim t→∞ β(t) = β. Proof: We can guarantee that the solutions of ( 10) are bounded from [5, Theorem 1]. Let define the following error variables:

e 2 := e 2 -ê2 , β := β -β. ( 22
)
and from Assumption 1, we have β = -β. Next, from (II-B), ( 20) and ( 21) the errors dynamics are given by ė2 = -β

C 0 e 2 -αe 2 (23) β = γe 2 e 2 C 0 . ( 24
)
To ensure that the errors converge to 0, we propose to design a suitable candidate Lyapunov function, given by

W = 1 2 e 2 2 + β2 γ . ( 25
)
The derivative of W along the trajectories ( 23)-(24) gives:

< ∇W (e 2 , β), f (e 2 , e 2 , β) > = -αe 2 2 ,
and from standard Lyapunov arguments, it follows that the error variables e 2 and β are bounded. In addition, by LaSalle invariant principle, we easily conclude that e 2 → 0, which implies from (24) that β → 0. Likewise, from (23), and concluding from e 2 → 0 and Lemma 2, we get β → 0.

To prove the stability of a hybrid system ( 12)-( 17), let us use the fact that we have two distinct behaviors. In one hand, there are slow time-continuous variables ξ r := (e, z, v), in other hand, ξ f := (ê 2 , β) represents the fast time-continuous variables. Using this last assumption and by considering these variables change, ν := 1 α , γ := γ C0 , we will rewrite the complete system in a singular perturbation form given as follows:

H p :                                        ė ż v ν ė2 ν β       = f p (ξ), ξ ∈ C,       e + z + v + ê+ 2 β+       ∈ G p (ξ), ξ ∈ D, (26)
f p (ξ) :=       A(β)e + Bd(ξ) Θz -Γ( β)Θz ν C0 (e 1 -βe 2 ) + (e 2 -ê2 ) -ν γe 2 (e 2 -ê2 )       , G p (ξ) :=           x z     argmin v=u-Γ( β)z u∈[-1,1] ẽT P (A( β)e + Bd(ξ))     -Γ( β)z ê2 β           (27) where ξ = [e z v ê2 β].
Remark 5: In order to ensure a singular perturbation form, we need to establish that the response time of the slow subsystem, t R (A(β)), is larger than the fast subsystem time response. In this aim, we select γ = C 0 α and

α >> max 1 t R (A(β)) for β ∈ [β m , β M ].
Note that, in the hybrid scheme (26)-( 27), the fast variables directly impact the stability of the slow variables. However, the jumps do not affect the fast variables, because they do not present any jump (ê + 2 = ê2 and β+ = β).

In order to fulfill a singular perturbation analysis, we will check the assumptions given in [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF] for the model (26):

A. Regularity of system's data Regularity of system's data comes directly from Proposition 1.

B. Regularity of "manifold"

The "manifold", which corresponds to the quasisteady-state equilibrium manifold of classical singular perturbation theory [START_REF] Khalil | Nonlinear Systems[END_REF], that means when ν → 0 + is

e 2 -ê2 = 0 β -β = 0. ( 28
)
Note that β -β = 0 comes from Lemma 2,3. As ( 28) is continuous, we can consider that the manifold is empty outside of C, letting take the following set-valued:

M(e 2 ) :=    e 2 β e 2 ∈ C 0 e 2 / ∈ C,
for a given β. Note that M is outer semi-continuous, locally bounded and nonempty.

C. Stability for reduced system

The reduced system is the system ( 12)-( 13) in the manifold M, which is

H r :                  ė ż v  = f r (ξ r ), ξ r ∈ C(M),   e + z + v +   ∈ G r (ξ r ), ξ r ∈ D(M), (29)
f r (ξ r ) :=   A(β)e + Bv Θz -Γ( β)Θz   , G r (ξ r ) :=       e z     argmin ẽT P v=u-Γ( β)z u∈[-1,1] . (A( β)e + Bv)     -Γ( β)z       , (30)
being ξ r = (e, z, v). Note that, the reduced system ignores ê2 and β when determining jumps. Then, we can guarantee asymptotic stability from [5, Theorem 1]. Indeed the attractor 18 is ULAS for the reduced system.

D. Stability of the boundary layer

The boundary layer, is given by

H bl :=      ξr = 0 ė2 = ν C0 (e 1 -βe 2 ) + (e 2 -ê2 ) β = -ν γe 2 (e 2 -ê2 ) ξ ∈ C ∩ rB
being rB a closed ball of radius r. Note that the boundary layer system ignores the jumps, and during flows ξ r remains constant.

In order to evaluate the stability of the boundary layer, let consider the error equations of H bl and let take as particular solution ě2 bl ∈ R equal to a constant value. and re-scale time t to τ = (t -t 0 )/ν, getting

d dτ ȇ2 = - ν C 0 βê 2 bl -ȇ2 (31)
d dτ β = ν γê 2 bl ȇ2 . (32)
Then, the stability property of (31)-( 32) can be established using the Lyapunov function given in (25). Note that, in this case, the derivative of W along the trajectories in the boundary layer is relative to τ , instead of to ordinary time t.

d dτ z = Jz,
with z = [ȇ β] T , and

J = -1 -ν C0 ê2 bl -ν C0 γê 2 bl 0 z.
Therefore, we can define the next property: Property 2: By replacing ν := 1/α, then the real part of the eigenvalues of J, for ê2 bl ∈ {R\{0}} are all strictly negative, i.e.

λ 1 = Re (-1 + 1 -4 γ α 2 C 2 0 ê2 2 bl /2 < 0 λ 2 = Re -1 -1 -4 γ α 2 C 2 0 ê2 2 bl /2 < 0
Proof of theorem 1: From the analysis given in V-A, V-B, V-C and V-D for the model (26) as ν → 0 + , we prove ULAS of attractor (18) by applying [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF]Theorem 1].

VI. Simulations on half-bridge inverter

In this section, we perform some simulations in MAT-LAB/Simulink by exploiting the HyEQ Toolbox [START_REF] Sanfelice | A toolbox for simulation of hybrid systems in Matlab/Simulink: Hybrid equations (HyEQ) toolbox[END_REF] to verify the properties of the closed loop ( 12)-( 17). Let us consider for these simulations, the parameters given in Table I. We assume that the load can vary in the following set:

R 0 = 240Ω ∈ [120, 360]Ω ⇔ β = 0.0042 ∈ [0.0028, 0.0083], (33)
which corresponds to ±50% of variation with respect to the nominal value of R 0 .

The desired trajectories are given following

x e = v Ce i Le = 220 √ 2 sin(100πt) 19.5 sin(100πt + 86 • )
.

Considering [10, theorem 2], let take the following cost function to get any LQ performance level.

J = min u k∈dom j (ξ) t k+1 t k ρ R 0 (vc(τ, k)-v Cd ) 2 +R(i L (τ, k)-i Ld ) 2 dτ (34)
with ρ = 1000, and

P = 21.6862 0.1721 0.1721 0.0888 , Q = R LS 0 0 ρ R0 .

It satisfies Property 1 and constraint given in (19).

Finally, in order to avoid arbitrary fast switching, as mentioned in Remark 3, we introduce practically a sampling time T s = 10 -6 s. Moreover, from (1), (3) and the convex combination u = λ 1 -λ -1 with λ 1 + λ -1 = 1, we get to stabilize e in 0 in Filippov sense, noted in Remark 2, with λ 1 = 0.5 + 0.036 sin(100πt) + 0.32 cos(100πt), satisfying the condition [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF]. Furthermore, we take η = 0.1 which corresponds to a sub-optimal value that guarantees a trade-off between performance level and switching frequency, as shown in [START_REF] Albea | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF].

We select the convergence speed of the observer state, α, according to Remark 5. Thus, we need to satisfy max {30, 40} << α. For this issue, we choose α = 400 and α = 4000. Moreover, γ = C 0 α.

Finally, we force two load changes, in the transient time at t = 0.001s and in the steady state at t = 0.03s, changing β of ±50%. From β = 42 Figures 3 and4 show the evolution of the voltage and current for α = 400 and for α = 4000, respectively. In both cases, the states converge to their corresponding references. On the other hand, Fig. 5 and Fig. 6 show the evolution of β and ě2 , under the load change of β in the transient time (t=0.001s) and in the steady state (t=0.03s), respectively. The simulations in Fig. VII. Conclusions and future works In this article, we have proposed a hybrid adaptive control for a half-bridge inverter with unknown variable load, which guarantees the robustness of the convergence of the states toward the desired trajectories.

• 10 -3 (R 0 = 240Ω) to β = 83 • 10 -3 (R 0 = 360Ω) at t = 0.001s and from β = 83 • 10 -3 (R 0 = 360Ω) to β = 28 • 10 -3 (R 0 = 120Ω) at t = 0.03s.
The interest of such approach is the use of a hybrid control scheme that considers the continuous-time dynamics as well as the discrete-time dynamics, avoiding the use an average control signal. An indirect adaptive control is proposed to estimate the unknown resistive load. Finally, ULAS of the full system is proven by using a standard singular perturbation analysis.

A future work is to reduce the switching in the steadystate by considering a time-or space-regulation in order to generate a minimum dwell-time.

Fig. 2 .

 2 Fig. 2. Hybrid adaptive control scheme.

Fig. 3 .

 3 Fig. 3. Evolution of voltage and current for α = 400.

Fig. 4 .Fig. 5 .

 45 Figures3 and 4show the evolution of the voltage and current for α = 400 and for α = 4000, respectively. In both cases, the states converge to their corresponding references. On the other hand, Fig.5and Fig.6show the evolution of β and ě2 , under the load change of β in the transient time (t=0.001s) and in the steady state (t=0.03s), respectively. The simulations in Fig.5a), Fig.6a), Fig.5 b) and Fig.6 b) are performed for a value of α = 400 and Fig.5 c), Fig.6 c), Fig.5 d) and , Fig.6 d) are done using α = 4000. Note as α larger is, the convergence of β to β faster is. Similar performance is obtained with the error ě2 .Therefore, these figures show Theorem 1 statement.

Fig. 6 .

 6 Fig.6. Evolution of the estimation of β and the error ě2 in the steady state for α = 400 and for α = 400.

TABLE I

 I

	Simulation parameters
	Parameter	Convention	Value	Units
	DC input voltage Reference peak voltage	V in Vmax	96 220 √	2	V V
	Nominal angular	ω	100π	rad/s
	frequency
	Nominal load resistance	R 0	240		Ω
	Estimated series	R LS	2		Ω
	resistance
	Inductor	L	50		mH
	Output capacitor	C 0	200		µF


Denys Dutykh

Jean-Louis Verger-Gaugry

On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials

Keywords: Lacunary polynomial, Newman polynomial, Almost Newman polynomial, Reducibility, Lenticular root MSC: [2010] 11C08 (primary), 65H04 (secondary) PACS: [2010] 02.70.Uu (primary), 02.60.Cb (secondary)

The class B of lacunary polynomials f (x) := -1

called an almost Newman polynomial. A general theorem of factorization of the almost Newman polynomials of the class B is obtained. Such polynomials possess lenticular roots in the open unit disk off the unit circle in the small angular sector -π/18 arg z π/18 and their nonreciprocal parts are always irreducible. The existence of lenticuli of roots is a peculiarity of the class B . By comparison with the Odlyzko-Poonen Conjecture and its variant Conjecture, an Asymptotic Reducibility Conjecture is formulated aiming at establishing the proportion of irreducible polynomials in this class.

This proportion is conjectured to be 3/4 and estimated using Monte-Carlo methods. The numerical approximate value ≈ 0.756 is obtained. The results extend those on trinomials (Selmer) and quadrinomials (Ljunggren, Mills, Finch and Jones).

Introduction

In this note, for n 3 , we study the factorization of the polynomials

f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s , (1.1)
where s 0 , m 1 -n n -1 , m q + 1 -m q n -1 for 1 q < s . Denote by B the class of such polynomials, and by B n those whose third monomial is exactly x n , so that

B = n 2 B n .
The case s = 0 corresponds to the trinomials G n (z) := -1 + z + z n studied by Selmer [START_REF] Selmer | On the irreducibility of certain trinomials[END_REF]. Let θ n be the unique root of the trinomial G n (z) = 0 in (0, 1) . The algebraic integers θ -1 n > 1 are Perron numbers. The sequence (θ -1 n ) n 2 tends to 1 if n tends to + ∞ .

Theorem 1 (Selmer [START_REF] Selmer | On the irreducibility of certain trinomials[END_REF]). Let n 2 . The trinomials G n (x) are irreducible if n ≡ 5 (mod 6) , and, for n ≡ 5 (mod 6) , are reducible as product of two irreducible factors whose one is the cyclotomic factor x 2 -x + 1 , the other factor (-1 + x + x n ) / (x 2x + 1) being nonreciprocal of degree n -2 .

Theorem 2 (Verger-Gaugry [START_REF] Verger-Gaugry | On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions[END_REF]). Let n 2 . The real root θ n = D (θ n ) + tl (θ n ) ∈ (0, 1) of the trinomial G n admits the following asymptotic expansion:

D (θ n ) = 1 - Log n n × × 1 - n -Log n n Log n + n -Log n Log Log n -n Log 1 - Log n n -Log n , (1.2)
and

tl (θ n ) = 1 n • O Log Log n Log n 2 , (1.3)
with the constant 1 2 involved in O (•) . Remark 1. A simplified form of expression (1.2) is the following:

D (θ n ) = 1 - 1 n Log n -Log Log n + Log Log n Log n . (1.4)
By definition a Newman polynomial is an integer polynomial having all its coefficients in { 0, 1 } . A polynomial having its coefficients in { 0, 1 } except its constant coefficient equal to -1 is called an almost Newman polynomial. It is not difficult to see that polynomials f ∈ B are almost Newman polynomials. The following irreducibility Conjecture (called Odlyzko-Poonen (OP)) holds for the asymptotics of the factorization of Newman polynomials.

Conjecture 1 ). Let P d, + := {1 + d j = 1 a j x j a j ∈ {0, 1} , a d = 1} denote the set of all Newman polynomials of degree d . We introduce also the class P + := d 1 P d, + . Then, in P + , almost all polynomials are irreducible. More precisely, if Ω d denotes the number of irreducible polynomials in P d, + , then

lim d → ∞ Ω d 2 d -1 = lim d → ∞ #{f ∈ P d, + f is irreducible } 2 d -1 = 1 .
The best account of the Conjecture is given by Konyagin [START_REF] Konyagin | On the number of irreducible polynomials with 0, 1 coefficients[END_REF]:

Ω d ≫ 2 d Log d .
Replacing the constant coefficients 1 by -1 gives the variant Conjecture (called "variant OP") for almost Newman polynomials.

Conjecture 2. (Variant OP) Let P d, -:= {-1 + d j = 1 a j x j a j = 0 or 1 , a d = 1} denote the set of all almost Newman polynomials of degree d . Denote

P -= d 1 P d, -.
Then, in P -, almost all polynomials are irreducible. More precisely,

lim d → ∞ #{f ∈ P d, -f is irreducible} 2 d -1 = 1 .
There is a numerical evidence that the OP Conjecture and the variant OP Conjecture are true (cf. Table 1, Sect. §6).

The objectives of this note consist in [START_REF] Borst | Irreducibility of Random Polynomials[END_REF] Establishing the type of factorization of the polynomials f of the class B (Theorem 3), in the context of Schinzel' s and Filaseta' s theorems on the factorization of lacunary polynomials [START_REF] Filaseta | On the Factorization of Polynomials with Small Euclidean Norm[END_REF]13,[START_REF] Schinzel | Reducibility of lacunary polynomials III[END_REF]. [START_REF] Dobrowolski | On a question of Lehmer and the number of irreducible factors of a polynomial[END_REF] Characterizing the geometry of the zeroes of the polynomials f of the class B , in particular in proving the existence of lenticuli of zeroes in the angular sector -π/18 arg z π/18 inside the open unit disk in Solomyak' s fractal (with numerical examples to illustrate Theorem 4).

(3) Estimating the probability for a polynomial f ∈ B to be irreducible (Heuristics called "Asymptotic Reducibility Conjecture") by comparison with the variant OP Conjecture.

Notations used in the sequel.

If P (x) = r j = 0 a j x r ∈ Z [x]
, we refer to the reciprocal polynomial of P (x) as

P * (x) = r j = 0 a r -j x r . The Euclidean norm || P || of P (x) = r j = 0 a j x r ∈ Z [x] is || P || := r j = 0 a 2 j 1 2 . If α is an algebraic number, P α (x)
denotes its minimal polynomial; if P α (x) is reciprocal we say that α is reciprocal. A Perron number α is either 1 or a real algebraic integer > 1 such that its conjugates α (i) are strictly less than α in modulus. The integer n is called the dynamical degree of the real algebraic integer β > 1 if 1/β denotes the unique real zero of f (x) = -1 + x + x n + s q = 1 x m q ∈ B . Let T denote the unit circle in the complex plane. Theorem 3. For any f ∈ B n , n 3 , denote by

f (x) = A (x) • B (x) • C (x) = -1 + x + x n + x m 1 + x m 2 + . . . + x m s , where s 1 , m 1 -n n -1 , m j + 1 -m j n -1 for 1 j < s ,
) such that 1/β is the unique real root of f in (θ n -1 , θ n ) , the nonreciprocal polynomial -C * (x) of C (x)
is the minimal polynomial of β , and β is a nonreciprocal algebraic integer.

Remark 2. For all polynomials f , as described in Theorem 3, we observe numerically the following lower bound on the degree of the nonreciprocal part C :

deg (C) ⌊ m s -1 2 ⌋ ,
At the current stage this minoration is a conjecture.

Let us now define the lenticular roots of an f of the class B . In the case s = 0 , i.e. for the trinomials G n (x) , from [START_REF] Verger-Gaugry | Uniform Distribution of the Galois Conjugates and Beta-Conjugates of a Parry Number Near the Unit Circle and Dichotomy of Perron Numbers[END_REF]Proposition 3.7], the roots of modulus < 1 of G n all lie in the angular sector -π/3 < arg z < π/3 . The set of these "internal" roots has the form of a lenticulus, justifying the terminology (Figure 1(a) for n = 37); they are called lenticular roots. For extending the notion of "lenticulus of roots" to general polynomials f of the class B , with s 1 , we view

f (x) = -1 + x + x n + x m 1 + x m 2 + . . . + x m s = G n (x) + x m 1 + x m 2 + . . . + x m s , (where n 3 , s 1 , m 1 -n n -1 , m j + 1 -m j n -1 for 1 j < s) as a perturbation of G n (x) by x m 1 + x m 2 + . . . + x m s .
The lenticulus of roots of f is then a deformation of the lenticulus of roots of G n (Figure 1(b)). In this deformation process, the aisles of the lenticulus may present important displacements, in particular towards the unit circle, whereas the central part remains approximately identical. Therefore it is hopeless to define the lenticulus of roots of f in the full angular sector -π/3 < arg ω < π/3 . From the structure of the asymptotic expansions of the roots of G n (x) [START_REF] Verger-Gaugry | On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions[END_REF] it is natural to restrict to the angular sector to -π/18 < arg ω < π/18 . More precisely,

Theorem 4 ([21]). Let n

260 . There exist two positive constants c n and c A, n , c A, n < c n , such that the roots of f ∈ B n ,

f (x) = -1 + x + x n + x m 1 + x m 2 + . . . + x m s , where s 1 , m 1 -n n -1 , m j + 1 -m j n -1 for 1 j < s , lying in -π/18 < arg z < π/18 either belong to z | | z | -1 | < c A, n n , or to z | | z | -1 | c n n .
The lenticulus of zeroes of f is then defined as

L β := ω | ω | < 1 , - π 18 < arg ω < π 18 , | | ω | -1 | c n n ,
where 1/β is the positive real zero of f . The proof of Theorem 4 requires the structure of the asymptotic expansions of the roots of G n and is given in [START_REF] Verger-Gaugry | A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus[END_REF].

A typical example of lenticularity of roots with n = 481 is given in Figure 8, in which

f (x) = -1 + x + x 481 + x 985 + x 1502 .
Let κ = 0.171573 . . . be the maximum of the function

y -→ 1 -exp - π y 2 exp π y - 1
on (0, + ∞) . The following formulation for c n is given in [START_REF] Verger-Gaugry | A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus[END_REF]:

c n = -1 + 1 n • Log κ + 1 n • O Log Log n Log n 2 ,
with c n ≃ -Log κ ≈ 1.76274 . . . to the first-order. In the present note Theorem 4 is only examplified. Namely, in Section 3 we show that the statement of this Theorem also holds on examples, in particular pentanomials, for dynamical degrees n less than 260 . Concerning the asymptotic probability of irreducibility of the polynomials of the class B at large degrees, our numerical results shown in Figure 7, using the Monte-Carlo method (see the pseudo-code 1), suggest the following ) which are irreducible is given by the limit, assumed to exist,

lim N → ∞ #{ f ∈ B (N ) f irreducible } #{ f ∈ B (N ) } = 3 4 .

Quadrinomials (s = 1)

Since every f ∈ B is nonreciprocal and such that f (1) = 0 , f is never divisible by the cyclotomic nonreciprocal polynomial -1 + x . When f ∈ B is a quadrinomial, the following Theorems provide all the possible factorizations of f . Theorem 5 (Ljunggren [8]). If f ∈ B , as

f (x) = -1 + x + x n + x m 1 ,
has no zeroes which are roots of unity, then f (x) is irreducible. If f (x) has exactly q such zeroes, then f (x) can be decomposed into two rational factors, one of which is cyclotomic of degree q with all these roots of unity as zeroes, while the other is irreducible (and nonreciprocal).

Ljunggren' s Theorem 5 is not completely correct. Mills corrected it (Theorem 8). Finch and Jones completed the results (Theorem 9).

Theorem 6 (Ljunggren [8]). If f ∈ B , with s = 1 , as

f (x) = -1 + x + x n + x m 1 ,
with e 1 = gcd (m 1 , n -1) , e 2 = gcd (n, m 1 -1) , then all possible roots of unity of f (x) are simple zeroes, which are to be found among the zeroes of

x e 1 = ± 1 , x e 2 = ± 1 , x = -1 . Theorem 7 (Ljunggren [8]). If f ∈ B , with s = 1 , as f (x) = -1 + x + x n + x m 1 ,
is such that both n and m 1 are odd integers, then f (x) is irreducible.

Theorem 8 (Mills [11]). Let f ∈ B , with s = 1 ,

f (x) = -1 + x + x n + x m 1 decomposed as f (x) = A (x)•B (x)
where every root of A (x) and no root of B (x) is a root of unity. Then A (x) is the greatest common divisor of f (x) and f * (x) = x m 1 • f (1/x) , then reciprocal cyclotomic, and the second factor B (x) is irreducible, then nonreciprocal, except when f (x) has the following form:

-1 + x r + x 7 r + x 8 r = (x 2 r + 1) • (x 3 r + x 2 r -1) • (x 3 r -x r + 1) .
In the last case, the factors x 3 r + x 2 r -1 and x 3 r -x r + 1 are (nonreciprocal) irreducible.

Theorem 9 (Finch -Jones, [START_REF] Finch | On the Irreducibility of {-1,0,1} -Quadrinomials[END_REF]). Let f ∈ B , with s = 1 ,

f (x) = -1 + x + x n + x m 1 .
Let e 1 = gcd (m 1 , n -1) , e 2 = gcd (n, m 1 -1) . The quadrinomial f (x) is irreducible over Q if and only if m 1 ≡ 0 ( mod 2 e 1 ) , n ≡ 0 ( mod 2 e 2 ) .

Noncyclotomic reciprocal factors

In this Section we investigate the possible irreducible factors, in the factorization of a polynomial f ∈ B n , with n large enough, which vanish on the lenticular zeroes, or a subcollection of them. In Proposition 1 it is proved that the degrees of the noncyclotomic reciprocal factors (if they exist), and therefore the degrees of such f , should be fairly large. Proposition 1 does not say that the degrees of the noncyclotomic reciprocal factors are large. For the sake of simplicity, the value c n (defining the lenticulus of zeroes of f ) is taken to be equal to -Log κ .

Proposition 1. If f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s ∈ B n , s 1 , n 260
, admits a reciprocal noncyclotomic factor in its factorization which has a root of modulus

1 + (1 -c) • -Log κ n + c ( θ -1 n - 1
) , for some 0 c 1 , then the number s + 3 of its monomials satisfies:

s + 3 1 + 1 n Log n c κ (1 -c) n -1 + 1
and its degree has the following lower bound

m s = deg f 1 + 1 n Log n c κ (1 -c) n -1 -1 • (n -1) + 1 . (3.1)
Proof. The Perron number θ -1 n is the dominant root of -1 + x + x n , and

θ -1 n -1 of -1 + x + x n -1 . Since f ∈ B n , s
1 , by Lemma 5.1 (ii) in [6] (cf. Section 5.4), the dominant (positive real) zero of f * (x) lies in the interval (θ -1 n , θ -1 n -1 ) . The (external) lenticulus of zeroes of f * (x) is defined as the image of that of f by z -→ 1/z . The existence of c ∈ [ 0, 1 ] and a reciprocal noncylotomic factor vanishing at the zeroes of the subcollection of the lenticulus of f defined by c , implies that this reciprocal noncylotomic factor also vanishes at the zeroes of the lenticulus of f * , external to the unit disk, in the same proportion.

Lemma 1 (Mignotte -Ştefănescu [START_REF] Mignotte | On the roots of lacunary polynomials[END_REF]). Let

P (x) = x q + a q -k x q -k + . . . + a 1 x + a 0 ∈ Z [x] \ Z .
Then the moduli of the roots of P (x) are bounded by

| a 0 | + | a 1 | + . . . + | a q -k | 1/k . (3.2)
The number of monomials in

f ∈ B n , n 2 , is equal to s + 3 . Then the sum | a 0 | + | a 1 | + . . . + | a q -k | of Proposition 1, applied to P (x) ≡ f (x) with q = m s , is equal to s + 2 , and k is n -1 . If we assume that f contains an irreducible reciprocal noncyclotomic factor B having a root of modulus 1 + (1 -c)• -Log κ n + c•(θ -1 n -1)
, for some 0 c 1 then we should have, by Lemma 1 and by Equation (1.4),

(s + 2) 1/k 1 + 1 n -Log κ (1 -c) + c Log n .
Therefore,

1 k Log (s + 2) Log 1 + 1 n Log n c κ (1 -c) , which implies Log (s + 2) Log 1 + 1 n Log n c κ (1 -c) n -1
and the result. Moreover,

m s = (m s -m s -1 ) + (m s -1 -m s -2 ) + . . . + (m 2 -m 1 ) + (m 1 -n) + (n -1) + 1 (s + 1) • (n -1) + 1 , from which (3.1) is deduced. Example 1. Let f ∈ B n , with n = 400
, for which it is assumed that there exists a reciprocal noncyclotomic factor of f vanishing on the subcollection of roots of the lenticulus of f given by c = 0.95 . Then, by (3.1), the degree m s of f should be above 121 786 .

The case where the summit (real > 1) of the lenticulus of zeroes of f * is a zero of a reciprocal noncyclotomic factor of f never occurs by the following Proposition.

Proposition 2. If f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s ∈ B n , s 1 , n 3 , is factorized as f (x) = A (x) • B (x) • C (x) as in Theorem 3. Then, the unique positive real root of f (x) is a root of the nonreciprocal part C (x) .
Proof. By Descartes' s rule the number of positive real roots of f should be less than the number of sign changes in the sequence of coefficients of the polynomials f . The number of sign changes in f is 1 . If say 1/β is the unique root of f in (0, 1) , and assumed to be a root of a factor of B then β and 1/β = β would be two real roots of f , what is impossible.

4. Lenticuli of zeroes: an example with s = 12 , and various pentanomials (with s = 2)

In this paragraph let us examplify the fact that the roots of any

f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s , where s 1 , m 1 -n n -1 , m q + 1 -m q n -1 for 1 q < s , n 3
, are separated into two parts, those which lie in a narrow annular neighbourhood of the unit circle, and those forming a lenticulus of roots ω inside an angular sector -γ < arg ω < γ with γ say < π/3 off the unit circle. This dichotomy phenomenon becomes particularly visible when n and s are large. This lenticulus is shown to be a deformation of the lenticulus determined by the trinomial -1 + x + x n made of the first three terms of f ; the lenticulus of zeroes of -1 + x + x n is constituted by the zeroes of real part > 1/2 , equivalently which lie in the angular sector -π/3 < arg (z) < π/3 , symmetrically with respect to the real axis, for which the number of roots is equal to

1 + 2 ⌊ n/6 ⌋ [20, Prop. 3.7].
The value of γ is taken equal to π/18 as soon as n is large enough, due to the structure of the asymptotic expansions of the roots of G n [START_REF] Verger-Gaugry | On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions[END_REF], so that the number of roots of the lenticulus of roots of f can be asymptotically defined by the formula

1 + 1 3 n 6 ± 1 . (4.1)
At small values of n , the value of γ = π/18 is also kept as a critical threshold to estimate the number of elements in the lenticulus of roots of f by (4.1). It can be shown [START_REF] Verger-Gaugry | A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus[END_REF] that the lenticulus of roots of f is a set of zeroes of the nonreciprocal irreducible factor in the factorization of f . Even though it seems reasonable to expect many roots of f on the unit circle, it is not the case: all the roots α of the nonreciprocal irreducible component of f (x) are never on the unit circle: | α | = 1 , as proved in Proposition 5.

(i) Example of a polynomial in B 37 with s = 12 : Let the polynomial f (x) de defined as

f (x) := -1 + x + x 37 + x 81 + x 140 + x 184 + x 232 + x 285 + x 350 + x 389 + x 450 + x 590 + x 649 ≡ G 37 (x) + x 81 + x 140 + x 184 + x 232 + x 285 + x 350 + x 389 + x 450 + x 590 + x 649 (4.2)
The zeroes are represented in Figure 1 Denser lenticuli of roots (for n 1 000 for instance) are difficult to visualize graphically for the reason that the lenticuli of roots are extremely close to the unit circle, and apparently become embedded in the annular neighbourhood of the nonlenticular roots.

(

) Dynamical degree n = 5 . Let f 1 (x) = -1 + x + x 5 + x 9 + x 15 . 1
It is reducible and its factorization admits only one irreducible cyclotomic factor, the second factor being irreducible nonreciprocal:

f 1 (x) = (1 + x + x 2 ) • (-1 + 2 x -x 2 -x 3 + 2 x 4 -2 x 6 + 2 x 7 -x 9 + x 10 -x 12 + x 13 ) . Let f 2 (x) = -1 + x + x 5 + x 9 + x 18 .
In the factorization of f 2 (x) two irreducible cyclotomic factors appear and where the third factor is irreducible and nonreciprocal:

f 2 (x) = (1 -x + x 2 ) • (1 + x + x 2 ) • (-1 + x + x 2 -x 3 + x 5 -x 6 + x 8 -x 12 + x 14 ) .
In both cases, the lenticulus of zeroes of f

Factorization of the lacunary polynomials of class B

In a series of papers Schinzel [13-16] has studied the reducibility of lacunary polynomials, their possible factorizations, the asymptotics of their numbers of irreducible factors, reciprocal, nonreciprocal, counted with multiplicities or not, for large degrees. Dobrowolsky [START_REF] Dobrowolski | On a question of Lehmer and the number of irreducible factors of a polynomial[END_REF] has also contributed in this domain in view of understanding the problem of Lehmer. First let us deduce the following Theorem on the class B , from Schinzel' s Theorems.

Theorem 10. Suppose f ∈ B of the form

f (x) = -1 + x + x n + x m 1 + . . . + x m s , n 2 , s 1 .
Then the number ω (f ) , resp. ω 1 (f ) , of irreducible factors, resp. of irreducible noncyclotomic factors, of f (x) counted without multiplicities in both cases, satisfy

• ω (f ) ≪ m s Log (s + 3) Log Log m s , m s → ∞ ,
• for every ε ∈ (0, 1) ,

ω 1 (f ) = o (m ε s ) • Log (s + 3) 1 -ε , m s → ∞ .
Proof. Theorem 1 and Theorem 2, with the "Note added in proof " in Schinzel [16, p. 319]. The lenticulus of roots of f (x) has 7 zeroes.

Cyclotomic parts

Let us first mention some results on the existence of cyclotomic factors in the factorization of the polynomials of the class B . Then, in Proposition 4, we prove the existence of infinitely many polynomials f ∈ B which are divisible by a given cyclotomic polynomial Φ p , for every prime number p 3 .

Lemma 2. Suppose f ∈ B of the form

f (x) = -1 + x + x n + x m 1 + . . . + x m s , n 2 , s 1 ,
and divisible by a cyclotomic polynomial. Then there is an integer m = p q 1 1 • . . . • p q r r having all its prime factors p i s + 3 such that Φ m (x) divides f (x) .

Proof. Lemma 3.2 in [START_REF] Filaseta | On three questions concerning 0, 1-polynomials[END_REF].

The divisibility of f ∈ B by cyclotomic polynomials Φ p (x) , where p are prime numbers, implies a condition on those p ' s by the following Proposition 3.

Lemma 3 (Boyd). Let p be a prime number. Suppose f ∈ B of the form

f (x) = m s j = 0 a j x j = -1 + x + x n + x m 1 + . . . + x m s , n 2 , s 1 .
Denote

c i = k ≡ i (p) a k . Then, Φ p (x) f (x) ⇐⇒ c 0 = c 1 = . . . = c p -1 .
Proof. Φ p (x) divides f (x) if and only if (x p -1) divides (x -1) • f (x) .

Proposition 3. Suppose f (x) ∈ B of the form

f (x) = m s j = 0 a j x j = -1 + x + x n + x m 1 + . . . + x m s , n 2 , s 1
and that Φ p (x) f (x) for some prime number p . Then

p (s + 1) . Proof. Using Lemma 3, since f (1) = s + 1 = k a k = p -1 i = 0 k ≡ i (p) a k = p • c 0 , we deduce the claim.
A necessary condition for f (x) to be divisible by Φ p (x) is that s should be congruent to -1 modulo p . Proposition 4. Let p 3 be a prime number. Let n 2 . There exist infinitely many f ∈ B n such that Φ p (x) f (x) .

Proof. Let ζ p denote the primitive root of unity e 2 i π / p . Let us assume that f (x) ∈ B n vanishes at ζ p , as

f (ζ p ) = -1 + ζ p + ζ n p + ζ m 1 p + . . . + ζ m s p = 0 .
We consider the residues modulo p of the (s + 1)-tuple (n, m 1 , m 2 , . . . , m s ) so that f (ζ p ) can be written

f (ζ p ) = c 0 + c 1 ζ p + c 2 ζ 2 p + . . . + c p -1 ζ p -1 p = 0 , c 0 , c 1 , . . . , c p -1 ∈ Z . The polynomial Φ p (x) = 1 + x + x 2 + . . . + x p -1 = (x p -1) / (x -1)
is the minimal polynomial of ζ p . Then, if c 0 or c p -1 is equal to 0 , then all the coefficients c i should be equal to

0 since {1, ζ p , ζ 2 p , . . . , ζ p -2 p } is a free system over Z . If c 0 • c p -1 = 0 then the equalities c 0 = c 1 = c 2 = . . . = c p -1 = 0
should hold since the polynomial p -1 j = 0 c j x j vanishes at ζ p and is of the same degree as Φ p (x) . The common value can be arbitrarily large. In both cases we have the condition

c 0 = c 1 = c 2 = . . . = c p -1 .
It means that the distribution of the exponents n , m 1 , m 2 , . . ., m s by class of congruence modulo p should be identical in each class.

Then, if p n , the constant term -1 "belongs to" the class " ≡ 0 mod p ", and ζ p to the class " ≡ 1 mod p ". The term ζ n p may belong to another class " ≡ i mod p " with i = 0, 1 or to one of the classes " ≡ 0 mod p " or " ≡ 1 mod p ". If p > n then the term ζ n p belongs to another class " ≡ i mod p" with i = 0 , 1 . In both cases we can complete the classes by suitably adding terms "ζ m i p ". We now chose s 1 and m 1 , m 2 , . . ., m s sequentially such that the distribution of the residues modulo p m 1 mod p , m 2 mod p , . . . , m s mod p in the respective classes " ≡ i mod p ", with i = 0, 1, . . . , p -1 , is equal.

If one solution (m 1 , . . . , m s ) is found, then Φ p (x) divides f (x) . Another solution f ♯ ∈ B n is now found with s ♯ = s + p and a suitable choice of the exponents

m s + 1 , . . . , m s + p f ♯ (x) = -1 + x + x n + x m 1 + . . . + x m s + x m s + 1 + . . . x m s + p so that f ♯ (ζ p ) = c ♯ 0 + c ♯ 1 ζ p + . . . + c ♯ p -1 ζ p -1 p ,
where the p residues modulo p of m s + 1 , . . . , m s + p are all distinct, satisfying

c ♯ 0 = c ♯ 1 = . . . = c ♯ p -1 = c 0 + 1 . Then Φ p (x) also divides f ♯ (x)
. Iterating this process we deduce the claim. For studying the irreducibility of the nonreciprocal parts of the polynomials f ∈ B , we will follow the method introduced by Ljunggren [8], used by Schinzel [13,[START_REF] Schinzel | Reducibility of lacunary polynomials III[END_REF] and Filaseta [START_REF] Filaseta | On the Factorization of Polynomials with Small Euclidean Norm[END_REF].

Nonreciprocal parts

Lemma 4 (Ljunggren [8]). Let P (x) ∈ Z [ x ] , deg (P ) 2 , P (0) = 0 . The nonreciprocal part of P (x) is reducible if and only if there exists w (x) ∈ Z [ x ] different from ± P (x) and ± P * (x) such that w (x) • w * (x) = P (x) • P * (x) .
Proof. Let us assume that the nonreciprocal part of P (x) is reducible. Then, there exists two nonreciprocal polynomials u (x) and v (x) such that P (x) = u (x) • v (x) . Let w (x) = u (x) • v * (x) . We have:

w (x) • w * (x) = u (x) • v * (x) • u * (x) • v (x) = P (x) • P * (x) .
Conversely, let us assume that the nonreciprocal part c (x) of P (x) is irreducible and that there exists w (x) different of ± P (x) and ± P * (x) such that w (x)•w * (x) = P (x)•P * (x) . Let P (x) = a (x) • c (x) be the factorization of P where every irreducible factor in a is reciprocal. Then, Proof. Let us assume that C is reducible, and apply Lemma 4. Then, there should exist

P (x) • P * (x) = a 2 (x) • c (x) • c * (x) = w (x) • w * (x) . We deduce w (x) = ± a (x) • c (x) = ± P (x) or w (x) = ± a (x) • c * (x) = ± P * (x) . Contradiction. Proposition 6. For any f ∈ B n , n 3 , denote by f (x) = A (x) • B (x) • C (x) = -1 + x + x n + x m 1 + x m 2 + . . . + x m s , where s 1 , m 1 -n n -1 , m j + 1 -m j n -1 for 1 j < s ,
w (x) different of ± f (x) and ± f * (x) such that w (x) • w * (x) = f (x) • f * (x) . For short, we write f (x) = r j = 0 a j x d j and w (x) = q j = 0 b j x k j ,
where the coefficients a j and the exponents d j are given, and the b j ' s and the k j ' s are unkown integers, with

| b j | 1 , 0 j q , a 0 = -1 , a 1 = a 2 = . . . = a r = 1 , 0 = d 0 < d 1 = 1 < d 2 = n < d 3 = m 1 < . . . < d r -1 = m s -1 < d r = m s , 0 = k 0 < k 1 < k 2 < . . . < k q -1 < k q . The relation w (x) • w * (x) = f (x) • f * (x)
implies the equality: 2 k q = 2 d r ; expanding it and considering the terms of degree k q = d r , we deduce

|| f || 2 = || w || 2 = r + 1 which is equal to s + 3 . Since f * (1) = f (1)
and that w * (1) = w (1) , it also implies f (1) 2 = w (1) 2 and b 0 • b q = -1 . Then we have two equations

r -1 = q -1 j = 1 b 2 j , (r -1) 2 = q -1 j = 1 b j 2 .
We will show that they admit no solution except the solution w (x) = ± f (x) or w (x) = ± f * (x) . Since all | b j | ' s are 1 , the inequality q r necessarily holds. If q = r , then the b j ' s should all be equal to -1 or 1 , what corresponds to ± f (x) or to ± f * (x) . If 2 q < r , the maximal value taken by a coefficient b 2 j is equal to the largest square less than or equal to r -q + 1 , so that | b j | √ r -q + 1 . Therefore, there is no solution for the cases "q = r -1" and "q = r -2". If q = r -3 all b 2 j ' s are equal to 1 except one equal to 4 , and

r -1 = r -4 j = 1 b 2 j , (r -1) 2 > r -4 j = 1 b j 2 .
This means that the case "q = r -3" is impossible. The two cases "q = r -4" and "q = r -5" are impossible since, for m = 5 and 6 , r -m j = 1 b 2 j cannot be equal to r -1 . This is general. For q r -3 at least one of the | b j | ' s is equal to 2 ; in this case we would have

r -1 = ± q -1 j = 1 b j q -1 j = 1 | b j | < q -1 j = 1 b 2 j = r -1 .
Contradiction.

Thickness of the annular neighbourhoods of | z | = 1 containing the nonlenticular roots

Let n 3 , and δ n be a real number > 0 , smaller than 1 . Let

D n, δ n := z | z | < 1 , δ n < | G n (z) | } .
We now characterize the geometry of the zeroes, in D n, δ n , of a given

f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s ∈ B n ,
where s > 0 , m 1 -n n -1 , m q + 1 -m q n -1 for 1 q < s . Obviously a zero x ∈ D n, δ n of f is = 0 and is not a zero of the trinomial -1 + x + x n . Moreover,

δ n < | -1 + x + x n | = | x m 1 + x m 2 + . . . + x m s | < | x | m 1 + | x | m 2 + . . . + | x | m s . (5.1)
This inequality implies that 1 -| x | is necessarily small. Indeed the function Y : u -→ s j = 1 u m j is increasing, with increasing derivative, on ( 0, 1 ] , so that the unique real value 0 < r < 1 which satisfies Y (r) = δ n admits the upper bound e sup < 1 given by s -

δ n = Y ′ (1) • (1 -e sup ) =
e inf = δ 1/(2 n -1) n r .
As soon as the assumption lim sup n → ∞ Log δ n n = 0 is satisfied, then e inf tends to 1 as n tends to infinity. This assumption means that the domain D n, δ n should avoid small disks centered at the lenticular roots of G n . If s 2 , using the inequalities m q + 1 -m q n -1 , 1 q < s , we deduce, from Equation (5.1),

δ n < | x | m 1 + | x | m 2 + . . . + | x | m s | x | 2 n -1 1 -| x | (n -1)•(s -1) 1 -| x | n -1 + | x | m s . Putting H = | x | n -1
, we are now bound to solve the following equation in H

δ n = H 2 • 1 -H s -1 1 -H + H m s -1
to find e inf , for H < 1 close to one, of the form 1 -ε . It is easy to check that the expression of ε , at the first-order, is

ε = 2 n s -δ n n + δ n -s n s 2 + n s -s 2 + 2 m s -2 n -s ,
leading to

e inf = 1 -2 n s -δ n n + δ n -s n s 2 + n s -s 2 + 2 m s -2 n -s 1 / (n -1)
.

For n , δ n and s fixed, the function m s -→ ε is decreasing and then m s -→ e inf is increasing. This means that the thickness of the annular neighbourhood of | z | = 1 containing the nonlenticular roots of f diminishes as the degree m s of f tends to infinity, for a fixed number of monomials s + 3 and a fixed dynamical degree n .

Therefore all the zeroes of f which lie in D n, δ n belong to z e inf < z < 1 .

An example of dependency of e inf with m s is given by Figure 2 > 1 is the sequence of digits of the coefficient vector of G n (x) + 1 (cf. Lothaire [9,Chap. 7]); the digits lie in the alphabet {0, 1} . We have

d θ -1 n (1) = 0 . 1 0 n -2 1 . Similarly d θ -1 n -1 (1) = 0 . 1 0 n -3 1
, where the sequence of digits comes from the coefficient vector of G n -1 (x) + 1 . Let β > 1 denote the real algebraic integer such that the Rényi β-expansion of 1 in base β is exactly the sequence of digits of the coefficient vector of f (x) + 1 . We have:

d β (1) = 0 . 1 0 n -2 1 0 m 1 -n -1 1 0 m 2 -m 1 -1 1 . . . 1 0 m s -m s -1 -1 1 .
Since the two following lexicographical conditions are satisfied: Since -C * is nontrivial, monic, irreducible, nonreciprocal, and vanishes at β , it is the minimal polynomial of β , and β is nonreciprocal.

d θ -1 n (1) = 0 . 1 0 n -2 1 lex d β (1) lex d θ -1 n -1 (1) = 0 . 1 0 n -3 1 . Lemma 5.1 (ii) in Flatto, Lagarias and Poonen [6] implies: θ -1 n < β < θ -1 n -1 ⇐⇒ θ n -1 < 1/β < θ n .

Heuristics on the irreducibility of the polynomials of B

The Monte-Carlo method is used for testing the Odlyzko-Poonen Conjecture ("OP Conjecture") on the Newman polynomials, the variant Conjecture ("variant OP Conjecture") on the almost Newman polynomials and for estimating the proportion of irreducible polynomials in the class B . The Conjectures "OP" and "variant OP" state that the proportion of irreducible polynomials in the class of Newman polynomials, resp. almost Newman polynomials, is one. This value of one is reasonable in the context of the general Conjectures on random polynomials [START_REF] Borst | Irreducibility of Random Polynomials[END_REF].

The probability of f ∈ B to be an irreducible polynomial can be defined asymptotically as follows. Let s 1 , n 2 and N n . Let B (N, s) n denote the set of the polynomials f ∈ B n having s + 3 monomials such that deg (f )

N . Denote s) .

B (N, s) := 2 n N B (N, s) n , B (N ) := s 1 B (N,
Then B = N 2 B (N ) . For s 1 , let B [s] = N 2 B (N, s) . For every s 1 though the two adherence values lim inf N → ∞ #{ f ∈ B (N, s) f irreducible # f ∈ B (N, s) lim sup N → ∞ # f ∈ B (N, s) f irreducible # f ∈ B (N, s) , (6.1)
exist, and, in a similar way,

lim inf N → ∞ # f ∈ B (N ) f irreducible # f ∈ B (N ) lim sup N → ∞ # f ∈ B (N ) f irreducible # f ∈ B (N ) , (6.2)
exist, without being equal a priori, we find that, for s = 1 and s = 2 , and for arbitrary values of s 1 , there is a numerical evidence that the limits exist in both (6.1) and (6.2) (i.e. lim inf = lim sup). Table 1 reports the proportion of irreducible quadrinomials (s = 1), resp. irreducible pentanomials (s = 2), in the class B , with the 90%-confidence interval under the assumption that the limit exists in each case. We find that the proportion of irreducible polynomials in B is lim

N → ∞ # f ∈ B (N ) f irreducible # f ∈ B (N ) = 0.756 ± 0.02235 .
This value justifies the statement of the "Asymptotic Reducibility Conjecture". The reason of this residual reducibility finds its origin in Proposition 4 where cyclotomic polynomials are asymptotically present in the factorizations, though the authors have no proof of it. By Monte-Carlo methods, polynomials of degrees N up to 3 000 are tested (see Figure 7), and the number of monomials s + 3 in each f ∈ B

(N ) n is random in the range of values of s .

In the case "s = 0", n 2 B (N = n, s = 0) denotes the set of trinomials of the type -1 + x + x n , n 2 , whose factorization was studied by Selmer [START_REF] Selmer | On the irreducibility of certain trinomials[END_REF]; the proportion of irreducible trinomials is exact:

lim n → ∞ # f ∈ B (n, s = 0) f irreducible # f ∈ B (n, s = 0) } = 5 6
= 0.8(3) . (6.3) Table 1. Asymptotic proportion of irreducible polynomials in various classes:

Newman polynomials, almost Newman polynomials, B and the subclasses B [ 0 ] , B [ 1 ] , B [ 2 ] of } (Maximal polynomial degree: In this paragraph we first recall the constructions of Solomyak [START_REF] Solomyak | Conjugates of Beta-Numbers and the Zero-Free Domain for a Class of Analytic Functions[END_REF] on the sets of zeroes of the family W of power series having real coefficients in the interval [ 0, 1 ] , in the interior of the unit disk, and Solomyak' s Theorem 11. Then, we will recall how the polynomials of the class B are related to elements of W .

Let

W := h (z) = 1 + + ∞ j = 1
a j z j a j ∈ [ 0, 1 ] be the class of power series defined on | z | < 1 equipped with the topology of uniform convergence on compacts sets of | z | < 1 . The subclass W 0, 1 of W denotes functions whose coefficients are all zeros or ones. The space W is compact and convex. Let

G := λ | λ | < 1 , ∃ h (z) ∈ B such that h (λ) = 0 ⊂ z | z | < 1
be the set of zeroes of the power series belonging to W . The elements of G lie within the unit circle and curves in | z | < 1 given in polar coordinates, close to the unit circle, by [START_REF] Verger-Gaugry | On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions[END_REF]. The domain D (0, 1) \ G is star-convex due to the fact that: h (z) ∈ W =⇒ h (z / r) ∈ W , for any r > 1 (cf. [18, Section §3]). For every φ ∈ (0, 2 π) , there exists λ = r e i φ ∈ G ; the point of minimal modulus with argument φ is denoted λ φ = ρ φ e i φ ∈ G , ρ φ < 1 . A function h ∈ W is called φ-optimal if h (λ φ ) = 0 . Denote by K the subset of (0, π) for which there exists a φ-optimal function belonging to W 0, 1 . Denote by ∂ G S the "spike": -1, 1 2 (1 -√ 5) on the negative real axis.

Theorem 11 (Solomyak).

(1) The union G T ∂ G S is closed, symmetrical with respect to the real axis, has a cusp at z = 1 with logarithmic tangency ( cf.

Conjecture 3 (N 2 B

 32 Asymptotic Reducibility Conjecture). Let n 2 and N n . Let B (N ) n denote the set of the polynomials f ∈ B n such that deg (f ) N . Let B (N ) proportion of polynomials in B = (N

  (b), those of G 37 (x) = -1 + x + x 37 in Figure1(a). The polynomial f is irreducible. The zeroes of f (x) are either lenticular or lie very close to the unit circle. The lenticulus of zeroes of f contains 3 zeroes, compared to 13 for the cardinal of the lenticulus of zeroes of the trinomial G 37 (x) . It is obtained by a slight deformation of the restriction of the lenticulus of zeroes of G 37 (x) to the angular sector | arg z | < π/18 . (ii) Examples of pentanomials (s = 2) The examples show different factorizations of polynomials f ∈ B n for various values of n , having a small number of roots in their lenticulus of roots; in many examples the number of factors is small (one, two or three). The last examples exhibit polynomials f ∈ B having a larger number of zeroes in the lenticuli of roots (5 , 7 and 27).

Figure 1 .

 1 Figure 1. (a) The 37 zeroes of G 37 (x) = -1 + x + x 37 ; (b) The 649 zeroes of f (x) = G 37 (x) + . . . + x 649 given in Equation (4.2). The lenticulus of roots of f (having 3 simple zeroes) is obtained by a very slight deformation of the restriction of the lenticulus of roots of G 37 (x) to the angular sector | arg z | < π/18 , off the unit circle. The other roots (nonlenticular) of f can be found in a narrow annular neighbourhood of | z | = 1 .

Figure 2 .

 2 Figure 2. (a) The 12 zeroes of G 12 (x) ; (b) The 35 simple zeroes of f (x) = -1 + x + x 12 + x 23 + x 35 . By definition, only one root is lenticular, the one on the real axis, though the "complete" lenticulus of roots of -1 + x + x 12 , slightly deformed, can be guessed.(a) the densification of the annular neighbourhood of | z | = 1 by the zeroes of f (x) , (b) the decrease of the thickness of the annular neighbourhood containing the nonlenticular roots of f (x) . This phenomenon is general (cf. Section 5.3). (3) Dynamical degree n = 81 . Let f (x) = -1 + x + x 81 + x 165 + x 250 . It is irreducible. The lenticulus of zeroes of G 81 (x) contains 27 points (Figure 4(a)), while that of f (x) (Figure 4(b)) contains 5 points, in particular the real root 0.9604 . . . , close to the real positive root 0.9608 . . . of G 81 (x) . (4) Dynamical degree n = 121 . Let f (x) = -1 + x + x 121 + x 250 + x 385 . It is irreducible. The lenticulus of zeroes of G 121 (x) contains 41 points (see Figure 5(a)), whereas the lenticulus of roots of f (x) (Figure 5(b)) contains 7 points, in particular the real root 0.9709 . . . , close to the real positive root 0.971128 . . . of G 121 (x) .

Figure 3 .

 3 Figure 3. (a) The 12 zeroes of G 12 (x) ; (b) The 385 zeroes of f (x) = -1 + x + x 12 + x 250 + x 385 . The lenticulus of roots of the trinomial G 12 (x) = -1 + x + x 12 can be guessed, slightly deformed and almost "complete". It is well separated from the other roots, and off the unit circle. Only one root of f is considered as a lenticular zero, the one on the real axis: 0.8525 . . . . The thickness of the annular neighbourhood of | z | = 1 which contains the nonlenticular zeroes of f (x) is much smaller than in Figure 2(b).

Figure 4 .

 4 Figure 4. (a) Zeroes of G 81 (x) ; (b) Zeroes of f (x) = -1 + x + x 81 + x 165 + x 250 . On the right the distribution of the roots of f (x) is zoomed twice in the angular sector -π/18 < arg (z) < π/18 . The number of lenticular roots of f (x) is equal to 5 .

Figure 5 .

 5 Figure 5. (a) Zeroes of G 121 (x) ; (b) Zeroes of f (x) = -1 + x + x 121 + x 250 + x 385 . On the right the distribution of the roots of f (x) is zoomed twice in the angular sector -π/18 < arg (z) < π/18 . The lenticulus of roots of f (x) has 7 zeroes.

Proposition 5 .

 5 If P (z) ∈ Z [z] , P (1) = 0 , is nonreciprocal and irreducible, then P (z) has no root of modulus 1 . Proof. Let P (z) = a d z d + . . . + a 1 z + a 0 , with a 0 • a d = 0 , be irreducible and nonreciprocal. We have gcd (a 0 , . . . , a d ) = 1 . If P (ζ) = 0 for some ζ , | ζ | = 1 , then P ( ζ) = 0 . But ζ = 1/ζ and then P (z) would vanish at 1/ζ . Hence P would be a multiple of the minimal polynomial P * of 1/ζ . Since deg (P ) = deg(P *) there exists λ = 0 , λ ∈ Q , such that P = λ P * . In particular, looking at the dominant and constant terms, a 0 = λ a d and a d = λ a 0 . Hence, a 0 = λ 2 a 0 , implying λ = ± 1 . Therefore P * = ± P . Since P is assumed nonreciprocal, P * = P , implying P * = -P . Since P * (1) = P (1) = -P (1) , we would have P (1) = 0 . Contradiction.

  the factorization of f , where A is the cyclotomic component, B the reciprocal noncyclotomic component, C the nonreciprocal part. Then, C is irreducible.

s j = 1

 1 m j • (1 -e sup ) ; so that r < e sup = 1 -s -δ n s j = 1 m j . Let us now give a lower bound e inf of r , as a function of n , s , δ n and m s . If s = 1 , using m 1 n + (n -1) , the inequality δ n | x | m 1 | x | 2 n -1 implies:

5 . 4 . 3 ( 1 )

 5431 (b) and Figure 3(b): for fixed n = 12 and s = 5 , and varying m s from 35 to 385 . The Monte-Carlo approach allows to compare the thicknesses δ n with numerical values of thicknesses δ (n) obtained by numerical computation of roots for polynomials of B with n 3 000 . The results are reported in Figure 6. Proof of Theorem By Proposition 2 the nonreciprocal part C is nontrivial. By Proposition 6 the nonreciprocal part C is irreducible. By Proposition 5 the irreducible factor C never vanishes on the unit circle. (2) For n 3 the Rényi β-expansion of 1 in base θ -1 n

Figure 6 .

 6 Figure 6. Thickness δ (n) , proportional to 1/n at the first-order, of the annular neighbourhood of the unit circle which contains the nonlenticular roots (of modulus < 1), represented as a function of the dynamical degree n , for the almost Newman polynomials f of the class B . The curve "lenticular roots" represents the distance between 1 and the positive real zero (summit) of the lenticulus of f ∈ B ; this distance is ∼ Log n/n at the first-order. The Monte-Carlo method in B n is used, for n less than 3 000 .

Figure 7 .

 7 Figure 7. Probability to be irreducible for a polynomial of the class B havingdegree less than N . The estimated 90%-confidence intervals are represented. A limit value, as N tends to infinity, is conjectured to exist and its value is conjectured to be the rational number 3/4 .

7 .

 7 Lenticular roots on continuous curves stemming from z = 1 and boundary of Solomyak's fractal

[ 18 ,√ 5 -1 2 , 1 2 (-1 + √ 5 )

 1852125 Figure 1]).[START_REF] Dobrowolski | On a question of Lehmer and the number of irreducible factors of a polynomial[END_REF] The boundary ∂ G is a continuous curve, given by φ-→ | λ φ | on [ 0, π) , taking its values in , with | λ φ | = 1 if and only if φ = 0 . It admits a left-limit at π -, 1 > lim φ → π -| λ φ | > | λ π | = 1 , the left-discontinuity at π corresponding to the extremity of ∂ G S .(3) At all points ρ φ e i φ ∈ G such that φ / π is rational in an open dense subset of (0, 2) , ∂ G is non-smooth. (4) There exists a nonempty subset of transcendental numbers L tr , of Hausdorff dimension zero, such that φ ∈ (0, π) and φ ∈ K π Q ∪ π L tr implies that the boundary curve ∂ G has a tangent at ρ φ e i φ (smooth point).Proof.[START_REF] Solomyak | Conjugates of Beta-Numbers and the Zero-Free Domain for a Class of Analytic Functions[END_REF] Sections §3 and §4].Let β > 1 be a real number and T β : [ 0, 1 ] -→ [ 0, 1 ] , x -→ β x -⌊ β x ⌋ = { β x } be the β-transformation. The i-iterate of T β is denoted by T i β . The orbit (T i β (1)) i 1 of

  the factorization of f where A is the cyclotomic part, B the reciprocal noncyclotomic part, C the nonreciprocal part. Then, (1) the nonreciprocal part C is nontrivial, irreducible and never vanishes on the unit circle, (2) if β > 1 denotes the real algebraic integer uniquely determined by the sequence (n, m 1 , m 2 , . . . , m s

  3 000 , number of Monte-Carlo runs: 4 000)

	Polynomials (class)		Proportion	90%-confidence	Expected
					Interval (estimated)
	OP (Newman)		0.967		0.00930	1 (Conjectured)
	variant OP (almost Newman)	0.968		0.00916	1 (Conjectured)
	Class B		0.756		0.02235	3/4 (Conjectured)
	Trinomials (s = 0)		5/6 = 0.8(3)	-	5/6 exact (Selmer)
	Quadrinomials (s = 1)	0.575		0.02573	unkown
	Pentanomials (s = 2)	0.826		0.01601	unkwon
	0.88
	0.86
	0.84
	0.82
	0.8
	0.78
	0.76
	0.74
	0.72
	0.7
	0	500	1000	1500	2000	2500	3000

Acknowledgements

The authors would like to thank Dr. Bill Allombert (Institut de Mathématiques de Bordeaux, France) for helpful discussions on PARI/GP software.

In this angular sector the other zeroes of f (x) can be found in a thin annular neighbourhood of the unit circle. The real root 1/β > 0 of f (x) is such that β satisfies:

(1) ⌋ , which belong to the alphabet { 0, 1 } and satisfy the conditions of Parry (Lothaire [START_REF] Lothaire | Algebraic Combinatorics on Words[END_REF]Chap. 7]). The Parry Upper function f β (z) at β is defined as the power series having coefficient vector: "-1 t 1 t 2 t 3 . . . ". When the Parry Upper function f β (z) at β is a polynomial, by Lemma 5.1 (ii) in [6], and

the Conditions of Parry are exactly expressed by the defining conditions

The polynomials f of the class B can be viewed as all the polynomial sections of all the Parry Upper functions f β (z) at β for all 1 < β < θ -1 2 . The correspondance β -→ f β (z) is one-to-one [START_REF] Verger-Gaugry | Uniform Distribution of the Galois Conjugates and Beta-Conjugates of a Parry Number Near the Unit Circle and Dichotomy of Perron Numbers[END_REF]. Now the identity

for which the second factor belongs to W . Hence, except the collection of the real zeroes 1/β which are those of the polynomials f ∈ B in [ 0, 1 ] , all the zeroes of the polynomials f ∈ B , of modulus < 1 , lie within Solomyak' s fractal domain G , having boundary described by Theorem 11. By construction the zero locus of the first roots in Figure 9 is included in this boundary. Therefore it has logarithmic tangency at z = 1 . The zero loci of the second roots, third roots, etc., closer to | z | = 1 , in Figure 9, lie within G . In Figure 9 are represented these (universal) curves on which the zeroes of the preceding examples are reported. A complete study of these curves will be performed in the nearest future.

A. Algorithms and programs

The pseudo-code of the employed Monte-Carlo algorithm and the PARI/GP program A used in the present study is given below: Algorithm 1 Pseudo-code of the PARI/GP program used to estimate the probability to find an irreducible polynomial in the class

Require:

⊲ Approximate probability by frequency

The following PARI/GP script estimates the probability of finding a sparse irreducible polynomial with coefficients in {-1, 0, 1} in the class B : / * F i r s t , we i n c r e a s e t he s t a c k s i z e : * / d e f a u l t ( p a r i s i z e , 1 0 7 Var += ( I r r e d [ i ] -Mean ) ^2 ; } ) ; Var = s q r t ( Var / (M -1 ) ) ; Err = 1 . 6 4 5 * Var/ s q r t (M) ; p r i n t f ( " Estimated p r o b a b i l i t y : %1.3 f \n " , Mean ) ; p r i n t f ( " Estimated 90%%-e r r o r : %1.5 f \n " , Err ) ; p r i n t f ( " C o n f i d e n c e i n t e r v a l : [%1 .5 f , %1.5 f ] \ n " ,\ max ( 0 . 0 , Mean -Err ) , min ( Mean + Err , 1 . 0 ) ) ;