Naoyuki Amemiya

Yusuke Sogabe

Yamano Satoshi

Sakamoto Hisaki
email: hisaki.h1.sakamoto@furukawaelectric.com

Satoshi Yamano
email: syamano@superpower-inc.com

Hisaki 629x

Sakamoto

Electromagnetic coupling between filaments in multifilament coated conductors with finite transverse conductance -Comparison between numerical analyses and experiments in various systems

Keywords: coated conductor, coupling time constant, multifilament, transverse conductance

teaching and research institutions in France or abroad, or from public or private research centers.

Electromagnetic coupling between filaments in multifilament coated conductors with finite transverse conductance -Comparison between numerical analyses

and experiments in various systems -

I. INTRODUCTION

One of the drawbacks of coated conductors is their wide tape shape which generates large shielding-current-induced fields (SCIFs) and large AC losses. The striation of superconductor layer of a coated conductor to form filaments is a measure to decrease its SCIF and AC loss. Multifilament coated conductors can be classified into two types: one in which filaments are insulated electrically one another; one with a certain transverse conductance between filaments. The former is ideal for the reduction of magnetization, i.e. SCIFs and AC losses, but the latter is preferable from the viewpoint of current sharing between filaments, which improves stability and helps protection. In this presentation, we focus on the electromagnetic coupling between filaments of the latter. First, we review our numerical and experimental results from the viewpoint of key issues of numerical electromagnetic field analyses of multifilament coated conductor with finite transverse conductance. Then, we present our latest results mainly on such a conductor composing cables.

II. KEY ISSUES OF NUMERICAL ELECTROMAGNETIC FIELD ANALYSES OF MULTIFILAMENT COATED CONDUCTORS WITH FINITE TRANSVERSE CONDUCTANCE

A. 3D Geometry of Thin Superconductor Layer of Coated Conductor

Even in a pancake coil, which is one of the coil with the simplest geometry, we have to consider its spiral structure as shown in Fig. 1 in its electromagnetic field analysis. Furthermore, in various practical applications, coil geometries are much more complicated. Therefore, the consideration of 3D geometries of thin superconductor layer of coated conductor is required essentially for their electromagnetic field analyses.

B. Finite Electric Field -Current Density Characteristics of Superconductors

When we look at the temporal behaviors of SCIF, the finite electric field (E) -current density (J) characteristics of superconductors at low electric field region are important. Such E -J characteristics at low E region cannot be obtained by transport measurements.

C. Transverse Conductance between Filaments

The transverse conductance between filaments is the critical parameter, which dominates the coupling time constant, but it is difficult to measure directly.

III. VARIOUS SYSTEM FOR ANALYSES AND EXPERIMENTS

A. Single Short Pieces of Conductor and Their Stack

We measured and calculated the frequency dependence of magnetization loss at small field amplitude using short pieces of a multifilament coated conductor, whose schematic cross section is shown in Fig. 2. Fig. 3 shows a model for its single short piece for numerical electromagnetic field analyses. When we plot the measured or calculated magnetization loss per cycle against frequency, the coupling time constant can be determined from the frequency which gives the peak of the curve. Through comparison between the coupling time constant obtained experimentally and numerically, we can determine the transverse conductance. Furthermore, measurements and analyses were conducted using stacks of short piece samples in order to study the effect of mutual coupling of coupling current, which may appear between coupling currents in adjacent turns of a pancake coil.

B. Pancake Coils

We wound a pancake coil using a monofilament coated conductor and placed it in a cusp magnetic field to apply the magnetic field normal to the coated conductor. From the temporal change of the magnetic field, which was affected by the shielding current, we could determine the E -J characteristics at low E region: comparing the measured magnetic field with the calculated magnetic fields using various E -J characteristics. Next, a similar experiment and analyses was conducted using a pancake coil wound with a multifilament coated conductor.

C. Cables

We proposed a new concept of cable which consists of multifilament coated conductor with finite transverse conductance. The measured and calculated electromagnetic behaviors of this cable will be presented. Ida, T. Iwashita, and N. Amemiya, "Large-scale electromagnetic field analyses of coils wound with coated conductors using a currentvector-potential formulation with a thin-strip approximation," Supercond. Sci. Technol., Vol. 32, No. 9, Art. No. 094002, 2020.

  N. Amemiya, N. Tominaga, R. Toyomoto, T. Nishimoto, Y. Sogabe, S. Yamano, and H. Sakamoto, "Coupling time constants of striated and copper-plated coated conductors and the potential of striation to reduce shielding-current-induced fields in pnacake coils," Supercond. Sci. Technol., Vol. 31, No. 2, Art. No. 025007, 2018. [2] N. Amemiya, Y. Sogabe, S. Yamano, and H. Sakamoto, "Shielding current in a copper-plated multifilament coated conductor wound into a single pancake coil and exposed to a normal magnetic field," Supercond. Sci. Technol., Vol. 32, No. 11, Art. No.115008, 2019. [3] T. Mifune, N. Tominaga, Y. Sogabe, Y. Mizobata, M. Yasunaga, A.

Fig. 1 .

 1 Fig. 1. Mutual coupling between the sections of the coupling current path in different turns of a pancake coil [1].

Fig. 5 .

 5 Fig. 5. Single pancake coil placed between two double pancake coils, which generate cusp magnetic field, with iron rings for field shaping [2]

This work was supported in part by JST-Mirai Program Grant Number JPMJMI19E1, Japan.


Huadong Yong
email: yonghd@lzu.edu.cn

Youhe Zhou
email: zhouyh@lzu.edu.cn

Mengdie Niu

Jing Xia

Mechanical behavior of HTS coils in high field magnets based on the electromechanical coupling

Keywords: HTS coil, electromechanical coupling, stress, strain

With the increase of both current-carrying capability and mechanical performance, 2G HTS tape is gradually becoming the dominant materials in the high field magnet. HTS coil co-wound by the REBCO coated conductor and insulated material is the insert coils, while the LTS coils usually provide a high background field. Both LTS coils and HTS coils together contribute to a high central magnetic field to meet the design requirements. However, due to the shielding current effect in coated conductor, the central field contributed by the HTS coils deviates from the design field. Moreover, HTS coils in normal operation would withstand the large electromagnetic force. Under the combined effect of electromagnetic force and thermal strain, the coated conductor may experience a large deformation. Investigations of the electro-mechanical characteristics in REBCO coated conductor is vital as the large strain in superconducting layer would cause the degradation of critical current. Thus, 2D electromechanical coupling numerical model is adopted, in which the Ic degradation caused by the strain is taken into account. The profiles of magnetic field and current density in HTS coils are presented for the cases with and without the electromechanical coupling effect. Then, the effect of shielding current on the distributions of magnetic field in HTS coils is discussed. The profiles of stress and strain of HTS coils in high field are calculated. The maximum stresses in the coil for different cases are also analyzed.

I. INTRODUCTION

HTS tapes are promising for the applications of superconducting magnet. The HTS coil can used as the insert coil in high field magnet [START_REF] Weijers | Progress in the development and construction of a 32-T superconducting magnet[END_REF]. The mechanical behavior is of significance in the design and operation of superconducting magnet. Firstly, the interaction of magnetic field and current can generate the Lorentz force. When the HTS coil is subjected to the high Lorentz force, the coil will experience the large mechanical deformation. With the increasing of applied field, the remarkable stress and strain can also affect the superconducting properties. It is well known that the critical current of HTS is strain sensitive [START_REF] Van Der Laan | Large intrinsic effect of axial strain on the critical current of hightemperature superconductors for electric power applications[END_REF][START_REF] Shin | Establishing a test procedure for evaluating the electromechanical properties of practical REBCO coated conductor tapes by the uniaxial tension test at 77K[END_REF][START_REF] Osamura | Mechanism for the uniaxial strain dependence of the critical current in practical REBCO tapes[END_REF]. Recently, the degradation of critical current has attracted many attentions. Then, the reduction of critical current may change the distribution of magnetic field and current density, which can lead to the variation of central field. During the simulation of the mechanical behavior of HTS coil, the strain sensitivity is usually neglected [START_REF] Xia | Stress and strain analysis of a REBCO high field coil based on the distribution of shielding current[END_REF][START_REF] Liu | Thermal stability and mechanical behavior in no-insulation high-temperature superconducting pancake coils[END_REF][START_REF] Li | Magnetization and screening current in an 800 MHz (18.8 T) REBCO nuclear magnetic resonance insert magnet: experimental results and numerical analysis[END_REF]. However, the degradation of critical current will become more important for the magnet with higher field. Thus, it is vital to consider the electromechanical coupling during the design of superconducting magnet.

II. BASIC EQUATIONS

Consider a superconducting magnet, which composed of HTS coil and LTS coil. The HTS coil co-wound by the REBCO coated conductor is the insert coil. The external current is applied in LTS coil to provide the high background field. In order to consider the mechanical behavior in the HTS coil, it is necessary to obtain the magnetic field and current density distribution firstly. The procedure for evaluating electromagnetic field is described in the following part. The Maxwell's equations are =t

   B E (1) = HJ (2)
The Maxwell's equations are solve with the T-A formulation [START_REF] Zhang | An efficient 3D finite element method model based on the T-A formulation for superconducting coated conductors[END_REF]. The magnetic field and current can generate the Lorentz

force f =  f J B (3)
With the Lorentz force, the mechanical equilibrium equation can be given by 0

ij j i f   , ( 4
)
During the simulation, the electromechanical coupling is considered. The critical current density is dependent on the magnetic field. However, the stress or strain can lead to the degradation of critical current. Here, the critical current density is given by

  =, cc JJB ε (5)
The above equations can be solve numerically with Comsol Multiphysics software [9] based on the electromagnetic coupling effect.

III. DISCUSSION AND CONCLUSIONS

In this section, several influencing factors including winding tension, the overband and turn-to-turn friction are investigated on the stress and strain state of HTS coils under normal operating conditions. Meanwhile, the effect of those factors on the current carrying capability of local REBCO coated conductor is analyzed. According to the numerical simulation, the Ic decline induced by local large strain directly affects the distributions of magnetic field and current density in magnets. We also conclude that the stress-strain state analysis of HTS coils is indeed necessary for the correction of deviations in electromagnetic field calculation of magnets. Besides, the overband made of stainless steel strip is found to have a significant impact on suppressing the peak hoop tensile stress in superconducting layers.

ACKNOWLEDGMENT

The authors acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11872195, 11472120, and 11802036).


Konstantin Kovalev

Nicolay Ivanov
email: n.s.ivanov88@gmail.com

Vladimir Podguzov
email: podguzov.volodia@yandex.ru

Sergey Zanegin

Modeling and experimental research of machine with annular HTS winding

Keywords: superconducting electrical machine, HTS, FEM modeling, modeling of HTS electrical machines

Finite element modeling is an important part of modern development of different devices including HTS electric machines and other superconducting devices. At the same time modern HTS 2G superconductors possess nonlinear magnetic, thermal and mechanical properties. In this case finite element modeling in these devices should be verified with experimental data. The paper is devoted to the comparison of the results of finite element modeling and experimental research of the electrical machine with annular AC HTS windings. The obtained results are discussed concerning AC losses and future applications of these HTS machines in onland power systems and transport.

I. INTRODUCTION

Development of superconducting (SC) electrical machines is a very important scientific and engineering problem. It is known that superconductors have nonlinear magnetic, mechanical and thermal properties [START_REF] Kozub | HTS racetrack coils for electrical machines[END_REF], [START_REF] Kovalev | Multidisciplinary Approach to the Design of Superconducting Electrical Machines[END_REF]. Even if a lot of work has been done on the characterization of superconducting elements (pellets, tapes, etc.), the transition from the element to the system is not trivial, because the behavior of an isolated element can be quite different when it is integrated into a system [START_REF] Statra | A volume integral approach for the modelling and design of HTS coils[END_REF]. Characterization at the system scale is thus necessary which makes the determination of properties of HTS windings during development and producing machine crucial. In this case conventional methods of calculating, modeling and design of electrical machines should be modified and take into account properties of applied SC. Due to this modeling of superconducting devices is rather complicated and needs experimental verification.

Machine with annular high temperature superconducting (HTS) stator windings was developed. During the project number of finite element (FE) models and experimental researches were produced. It gave us opportunity to verify approaches of calculating HTS electrical machines. In this paper results of static and transient modeling of phase current and voltage and their comparison with experimental research are described. Besides results of AC loss measurement is shown in this paper.

II. MACHINE SCHEME AND MAIN PARAMETERS

Considered machine is synchronous non-salient pole machine. Its scheme is shown on Fig. 1. Permanent magnets are mounted on the rotor with ferromagnetic core. The stator part is an annular magnetic core, on which HTS coils are mounted. Scheme and view of a coil are shown on Fig. 2 and Fig. 3. Transient simulation was produced to calculate the EMF of one phase, consisting of 8 coils. As a result (see Fig. 4), rms value of EMF and its waveform for different values of the relative magnetic permeability of the HTS coils were obtained. It could be seen that as μ decreases, the waveform of the EMF curve is distorted. Moreover, its appearance is similar to the type of EMF in the presence of a teeth structure of the stator. Thus, we can say that the appearance of "virtual" stator teeth is possible due to diamagnetic properties of HTS coils. Moreover, the magnitude of the dip in the EMF curve is greater, the smaller the value of µ. This means that the magnetic properties of HTS coils can be estimated using the shape of the EMF curve obtained experimentally. The effective value of the EMF decreases with decreasing µ: 78 V at µ=1, 72 V at µ=0.5, 67 V at µ=0.2.

Fig. 4. EMF waveform

It is known that magnitude and direction of the magnetic flux density of the external field affects the magnitude of AC losses in the HTS winding. In this regard, we determined the maximum values of the parallel and perpendicular components of the magnetic induction vector in the field of coils intime change (Fig 5). It can be seen from the figures that the perpendicular component is in all cases higher than the parallel one. In addition, it is seen that with decreasing µ of the coil, the magnitude of induction decreases and the difference between the maxima of the two components decreases. . We can see a good coincidence of the curves at different frequencies for the generator mode, therefore, the nature of the losses is hysteresis, since it does not depend on the frequency. The level of losses with a movable rotor is much higher. The increase in losses is associated with additional hysteresis losses in the HTS layers of the winding tapes, which occur in an alternating external magnetic field.

  Fig.1. Schematics of the machine

  Fig. 5. Maximum values of the parallel and perpendicular components of the magnetic induction vector in the field of coils in-time change

Fig. 7 .

 7 Fig. 7. AC losses dependencies for static and dinamic regimes


Fernando Mendez

Jan Srpčič

Mark Ainslie
email: mark.ainslie@eng.cam.ac.uk

Mark Ainslie Modelling

Fernando Perez Mendez

Modelling AC

Keywords: AC losses, demagnetization, bulk HTS, HTS machine, numerical modelling, rotating machine

I. INTRODUCTION

There is a worldwide interest in utilizing bulk superconductors as trapped field magnets, because they can provide significantly higher magnetic flux density than permanent magnets. Applications for their use include portable high-field magnet systems, ultra-light electrical machines, and compact desktop NMR/MRI systems [START_REF] Durrell | Bulk superconductors: a roadmap to applications[END_REF].

In the case of electrical machines, the trapped field in bulk superconductors can decay due to time-varying magnetic fields that are produced by the stator windings. The incident alternating fields penetrate the superconductor, inducing a shielding current that interacts with the pinned magnetic flux vortices (and hence supercurrent) producing the trapped field. If the Lorentz force exceeds the pinning force, then flux vortices leave their pinning sites and a redistribution of currents occur in the volume where the alternating magnetic fields have penetrated and where the flux vortices have been displaced, as discussed in [START_REF] Srpčič | Penetration depth of shielding currents due to crossed magnetic fields in bulk (RE)-Ba-Cu-O superconductors[END_REF]. These effects due to AC fields in the superconductor have been widely studied with experiments, but often these have been done with fields acting either in parallel or perpendicular to the superconductor. However, this is far from the real operating conditions a bulk superconductor would experience in a machine and while attempts have been tried [START_REF] Yamagishi | Rotation Test of a Superconducting Bulk Rotor Shielded With Superconducting Rings[END_REF], this is still an area of opportunity to carry out more realistic studies and gain a further understanding of the phenomena and its impact on machine performance, as well as use numerical modelling tools to create better strategies to reduce trapped field decay. Our numerical model and experimental setup aim to fulfil such purpose.

II. EXPERIMENTAL SETUP

The experimental setup, representing a small-scale hightemperature bulk superconducting axial-flux synchronous machine, consists of six triangular-shaped GdBCO bulks that are fixed in a copper rotor to help thermal conductivity from the LN2 bath acting as a coolant. To produce the rotating magnetic field, the machine has a double stator arrangement with six solenoid copper coils above and below the rotor of the machine. The housing and shaft of the motor are made from stainless steel, with some holes in the housing to allow the flow of the coolant in the LN2 bath. A picture of the device is shown in Fig. 1 and a detailed summary of the components is provided in Table 1. The parameter values given for the bulks, which were fabricated by the top seeded melt growth method [START_REF] Shi | The use of buffer pellets to pseudo hot seed (RE)-Ba-Cu-O-(Ag) single grain bulk superconductors[END_REF][START_REF] Zhou | Significant improvement of trapped flux in bulk Gd-Ba-Cu-O grains fabricated by a modified top-seeded melt growth process[END_REF], are based on an average estimate and may vary slightly between samples. The triangular shape of the bulk was selected because this shape maximizes the volume to be occupied by the superconducting material in the rotor and likewise this geometry should produce a higher force as a consequence of the Lorentz force produced by the radially flowing supercurrent and the incident perpendicular field from the stator coils. This geometry should be more representative of practical trapped field magnets in future electrical machines.

To measure the magnetic field during the operation of the machine, one Hall sensor is installed at the centroid on the top surface of one bulk, while another bulk has three Hall sensors installed in a line along the top surface to study the trapped field profile during operation. Finally, another Hall sensor is located at the centre at the end of one of the stator (solenoid) coils. The bulk at magnetized by field cooling as described in [START_REF] Srpčič | Penetration depth of shielding currents due to crossed magnetic fields in bulk (RE)-Ba-Cu-O superconductors[END_REF], and the stator coils are energized with a signal produced by an Agilent DSO-X 2004A that is amplified using a KEPCO bipolar operational power supply. The speed of the rotor was kept constant using an external DC motor connected to the shaft.

III. MODELLING FRAMEWORK

The numerical model is implemented using the finite element method in COMSOL Multiphysics 5.4. Only one sixth of the motor geometry is modelled by exploiting the symmetry of the geometry, as shown in Fig. 2. For the different parts of the model, different formulations of Maxwell's equations were used, as described in [START_REF] Brambilla | A Finite-Element Method Framework for Modeling Rotating Machines With Superconducting Windings[END_REF]. In our model, we solved the domains in the geometry with three different formulations. First, the magnetic scalar potential formulation solves the domains separating the stationary and rotational parts of the model, as this improves its convergence. The H-formulation is used in the domain with the HTS bulk and its surrounding domain. The A-formulation is used in the domains where the stator coils and the surrounding volume are located, as well as the domains surrounding the H-formulation domains and the interface between the H-formulation and the magnetic scalar potential domains. To exploit the symmetry, the periodic boundary conditions were imposed in the Aformulation domain surrounding the H-formulation, thus reducing the size of the model. The coupling between the A and H formulations works by imposing the electric field from A to H, and the magnetic field from H to A. For clarity Figs.

IV. PRELIMINARY RESULTS & DISCUSSION

In our preliminary run of the model, it can be observed that demagnetization effects are lower when the external magnetic field is rotating with respect of the superconductor, contrary to the stationary case. This is to be expected as when the bulk experiences the Lorentz force from the stator, it is free to rotate which reduces the experienced Lorentz force on the pinning sites until the system reaches an equilibrium. This trend can be observed in Fig. 3. The modelling framework will allow us to investigate the the impact of loads, frequencies ranging from 8 to 50 Hz and different magnetic field amplitudes from the stator from 25 to 100 mT, to compare with experimental results. Likewise, in the future, these analyses will guide the appropriate operating strategies to reduce AC losses.

  Fig. 1. (a) Small-scale high-temperature bulk superconducting axial-flux synchronous machine outside the LN2 bath. (b) Top view of the rotor plate where the triangular-shaped GdBCO bulks are fixed.

  2 (a) and (b) show the different domains coloured according to the formulation used in the model.

Fig. 2 .

 2 Fig. 2. (a) Geometry of the high-temperature bulk superconducting axial-gap synchronous machine finite-element model reduced to one sixth of the geometry by exploiting symmetry. (b) Side view with the A-formulation domain around the H-formulation domain hidden, (c) The top of the machine with the top vector and scalar potential domains removed for clarity, the vector potential to the right and left side of the H-formulation domain have the symmetric boundary conditions imposed. The A-formulation domains are shown in red, the magnetic scalar potential domains are yellow and the Hformulation domains are black.

Fig. 3 .

 3 Fig. 3. Results for the simulated trapped field decay: (a) when the bulk experiences an oscillating AC field with a fixed stationary rotor (orange dashed line), (b) when the rotor is rotating at its synchronous speed (green dot-dashed line).

TABLE I .

 I GENERAL PARAMETERS FOR THE SYNCHRONOUS MACHINE

ACKNOWLEDGMENTS FPM would like to acknowledge the support of the Consejo Nacional de Ciencia y Tecnología and of the Secretaría de Energía de México, reference: 541016/439167. MA is supported by the EPSRC Early Career Fellowship, EP/P020313/1.

HTS bulks


Stuart Wimbush
email: stuart.wimbush@vuw.ac.nz

Nick Strickland
email: nick.strickland@vuw.ac.nz

Utilising full angle-dependent critical current data in the electromagnetic modelling of HTS coils

Keywords: electromagnetic modelling, critical current data, anisotropy, hybrid windings, coil orientation, inclined substrate deposition

A methodology is presented for modelling HTS coils using full angle-dependent critical current data. The results are contrasted with those obtained using more common techniques such as a minimum critical current method or a field magnitude-dependent approximation. Several interesting design consequences that emerge only when the full anisotropy of the wire is taken into account are outlined and discussed.

I. INTRODUCTION

Common approaches to the modelling of HTS coils involve either assuming a minimum critical current (I c ) value for the wire or adopting a field dependent I c (B) characteristic, which conveniently lends itself to an approximate functional form. Often the underlying values are taken from real wire data measured at the desired operating temperature (T) and possibly for fields applied both parallel and perpendicular to the wire surface to account in a minimalistic (and unrealistic) manner for the true I c anisotropy of the wire. Less common is the use of full angle-dependent critical current data I c (T,B,θ) due to issues of both the availability of such data and the perceived difficulty of modelling using real data tables in place of functional approximations.

Here we discuss both aspects of the data-driven approach and highlight the benefits that can be obtained in the design of real devices when actual wire data is utilised. In particular this leads towards several novel approaches to device design and construction that are worthy of deeper investigation.

II. MODELLING METHOD

A. Data Sources

One data source that allows convenient investigation of the implementation and benefits of these methods is the freely accessible Robinson HTS wire database [START_REF] Wimbush | A public database of hightemperature superconductor critical current data[END_REF], and we will use example data from this database to illustrate the modelling approach. Of course, at the point of real device construction it is betterif not essentialto re-run the models on the basis of sample data acquired on the actual wire to be used, and internally we typically adopt this approach. However, for educational purposes, any available dataset can be utilised to good effect.

B. Modelling Method

To demonstrate the method, we restrict ourselves to a simple two-dimensional axisymmetric electromagnetic model formulated in the open-source software FEMM [START_REF] Meeker | Finite Element Method Magnetics[END_REF]. Again, it is recognised that this is not the cutting edge of electromagnetic modelling, but it serves to exemplify the method, provides a no-barrier route to access and experimentation, and it is anticipated that all the techniques described can as easily be implemented in more advanced tools, including three-dimensional simulation packages.

The method is straightforward enough. For each element in the mesh that lies within the conductor, the field magnitude and orientation at that point is determined. For an axisymmetric two-dimensional model, we benefit from the convenience that the field components B r and B z fall naturally in the commonly measured out-of-plane (maximum Lorentz force) geometry relative to the wire. In a threedimensional model, it will be necessary to decompose the field vector into an in-plane and an out-of-plane component and to then choose to neglect any in-plane variation.

Once this is done, it is a simple computational step to interpolate the I c (T,B,θ) dataset to obtain the I c of the wire under the particular field conditions (magnitude and orientation) pertaining to the given element. The coil I c is then dictated by the minimum I c element (with whatever engineering margin is desired). For a given operating current, the result can trivially be replotted as a percentage of the wire I c at the operating temperature. By way of example, a simple model of a coil pack comprising ten double-pancakes is shown in Fig. 1, operating at two different temperatures while generating the same 2.5 T central field. It is seen that the weakest point of the coil can vary from the point of greatest field magnitude (as at 65 K) to the point at which the field lies perpendicular to the plane of the wire and through intermediate points where field magnitude and orientation combine to become least favourable (as at 35 K). It depends entirely on the complex behaviour of the wire and is not static but varies in dependence on the operating temperature.

The explanation for this seemingly unusual behaviour is apparent upon inspection of the angle-dependent I c curves of this wire at the two example temperatures (Fig. 2). Such a behaviour will never be predicted by a mathematical model based on a monotonic decay of I c (B) at all temperatures.

C. Comparison with Other Approaches

The comparison with the minimum I c model is trivial. We simply identify the highest field at any point on the conductor and note its minimum I c value at that field. These values are labelled on Fig. 2, although note that in practice the I c value at the perpendicular field angle of 0° is often taken as a proxy for I c min , and the data for 65 K shows clearly how faulty this assumption can be. In the 35 K case where we have a simple anisotropy with close-to-minimum I c for fields perpendicular to the wire and maximum I c for fields inplane, we can obtain a substantial performance benefit by taking advantage of the enhanced in-plane performance of the wire, in this case operating at 325 A vs 299 A, a gain of almost 10%. Of course, this is heavily dependent on the exact features of the wire and the device design as exemplified by the behaviour at 65 K where the wire performance is insufficient to support the desired current. However, if a 0° I c min value were to be assumed here, this would overrate performance by a disastrous 30%.

Any attempt to account in some superficial way for the anisotropy of the wire using a geometrical combination of parallel and perpendicular field dependences is clearly going to fare no better. Only by properly accounting for the real anisotropy of the wire do we reach an optimal device design.

III. NOVEL DEVICE DESIGNS

These considerations suggest several avenues towards the development of novel device designs taking advantage of the unique characteristics of coated conductor wires.

A. Hybrid Windings

Once the critical region of a given coil pack is accurately identified, it becomes an obvious option to substitute a higher-performance wire for the most critical coils, or indeed to simply arrange the coils on hand in such a way as to optimise available wire usage. Often, as in the 35 K case here, there can be a significant difference in the performance of wire required in, for example, the end coils (I c min = 335 A) and the mid coils (I c min = 420 A). Such an arrangement has been termed a hybrid winding [START_REF] Song | AC loss calculation on a 6.5 MVA/25 kV HTS traction transformer with hybrid winding structure[END_REF].

B. Flipped Coils

Likewise, when the full anisotropy of the wire is properly taken into account, it can often be found that an otherwise overlooked asymmetry between field directions either side of the wire normal (not strongly evident in the data presented here) can lead to performance differences. In this case, simply taking care of the orientation of the coils as they are stacked can produce significantly enhanced performance [START_REF] Jiang | 15% reduction in AC loss of a 3-phase 1 MVA HTS transformer by exploiting asymmetric conductor critical current[END_REF].

C. Inclined Planarity Coated Conductors

Coated conductors prepared by a method such as inclined substrate deposition [START_REF] Lao | Critical current anisotropy of GdBCO tapes grown on ISD-MgO buffered substrate[END_REF] which results in a strongly offset inplane peak I c offer a unique opportunity to accommodate conventionally unfavourable field orientations. Together with point B above, coils formed from such wires have the potential to entirely obviate the issues associated with field divergence at the end coils. To our knowledge, no such attempt has been reported to date. Such a dataset [START_REF] Wimbush | Critical current characterisation of THEVA Pro-Line 2G HTS superconducting wire[END_REF] provides an excellent case study as a starting point for investigation.

IV. CONCLUSION

Modelling utilising actual wire data and in particular full angle-dependent critical current characterisation offers many opportunities to optimise devices and exploit features of real wires that are missed when mathematical approximations to their real-world behaviour are adopted. Existing techniques can be relatively easily adapted to make use of such data as it becomes more widely available.

Fig. 1 .

 1 Fig.1. Critical current maps of a ten double-pancake coil pack operated at different temperatures but generating the same 2.5 T central field. The bore is to the left. Notice that the point of lowest Ic is at the end of the bore when operated at 35 K, but shifts to the centre of the bore at 65 K.

Fig. 2 .

 2 Fig.2. Field angle dependences of the critical current of the wire used[START_REF] Wimbush | Critical current characterisation of Shanghai Superconductor PA1212 2G HTS superconducting wire[END_REF] for modelling the coil shown in Fig.1at the two operating temperatures and the highest field (3 T) experienced by the coils.


Yusuke Sogabe
email: sogabe.yusuke.6s@kyoto-u.ac.jp

Yoshiyuki Iwata
email: iwata.yoshiyuki@qst.go.jp

Naoyuki Amemiya
email: amemiya.naoyuki.6a@kyoto-u.ac.jp

Mitigation of shielding-current-induced field of a dipole magnet wound with coated conductors during pattern-operation and evaluation of beam orbit

Keywords: coated conductor, electromagnetic field analysis, shielding-current-induced fields, mitigation

We have been developing large-scale electromagnetic field analysis model of magnets composed of coated conductors to predict their field quality. We applied the model to a dipole magnet wound with coated conductors and calculated shielding-current-induced field during patternoperation. Moreover, to mitigate the influence of the shieldingcurrent-induced field on the field quality of the dipole magnet, we suggested two solutions: current control for dipole component and combination of the dipole magnet and corrections coils for sextupole component. Also, we conducted beam orbit simulations and particle distribution estimation with/without the mitigation methods.

I. INTRODUCTION

The precise prediction of shielding-current-induced fields (SCIFs) of magnets wound with coated conductors by electromagnetic field analyses is quite important to optimize their design and operation [START_REF] Nii | Three-dimensional model for numerical electromagnetic field analyses of coated superconductors and its application to Roebel cables[END_REF]- [START_REF] Ueda | Spatial and temporal behavior of magnetic field distribution due to shielding current in HTS coil for cyclotron application[END_REF]. However, the modelling of the magnets wound with coated conductors is quite difficult and complicated due to the strong nonlinear electric property of the coated conductors and very high aspect ratio of the cross-section of superconductor layer in the coated conductors. In addition, the influence of the shieldingcurrent-induced fields on the beam characteristics We developed large-scale electromagnetic field analysis model to visualize electromagnetic phenomena in the magnets wound with the coated conductors. We apply the developed model to a dipole magnet wound with coated conductors which is designed and operated based on specifications of magnets in a rotating gantry for carbon cancer therapy. To discuss the influence of the SCIF, stability and reproducibility of the integrated magnetic field along beam axis are evaluated by using time-evolution of multipole components of the magnetic field. Then, mitigation methods of dipole and sextupole components of the magnetic filed are discussed. In addition, because the distribution of multipole components of the magnetic field along beam axis can be cause of beam loss and undesired change of beam shape, we conducted beam orbit simulations with and without mitigation methods by using WinAGILE. In the simulations, the distribution of magnetic field along beam axis is considered.

II. ELECTROMAGNETIC FIELD ANALYSIS

The equation in the electromagnetic field analyses is derived from Faraday's law, Biot-Savart's law, Ohm's law, and definition of current vector potential. And also, the thinstrip approximation is applied to the model. The equation to be solved in the analyses is as follows [START_REF] Nii | Three-dimensional model for numerical electromagnetic field analyses of coated superconductors and its application to Roebel cables[END_REF], [START_REF] Mifune | Large-scale electromagnetic field analyses of coils wound with coated conductors using a currentvector-potential formulation with a thin-strip approximation[END_REF]:

         . d 4 1 ext 3 s 0 0                            n B n r n n n S r T t t T S     
In our software, a preconditioning method based on the AMG method is applied to the BiCGSTAB method to improve the convergence property of the linear solver. Furthermore, to address the huge required memory consumption and computation time due to dense matrix B, we applied hierarchical matrices.

III. ANALYSIS RESULTS OF A DIPOLE MAGNET WOUND WITH COATED CONDUCTORS

We apply the developed model to a dipole magnet wound with coated conductors shown in Fig. 1, whose design and operation pattern shown in Fig. 2 are decided from actual magnets in a rotating gantry in HIMAC (Heavy Ion Medical Accelerator in Chiba). The numbers of degree of freedoms of the model is more than 1.5 million; the memory consumption by hierarchical matrices is 177 GB, while that by a dense matrix had been 16.7 TB.

The time-evolution of the multipole components of SCIF is shown in Fig. 2. Here, the relative multipole components of SCIF bln (n = 1, 3, 5, ...) against designed dipole components are plotted against time.

The calculated time-changing SCIF help optimization of magnet design and its operation. the influence of SCIF on the dipole and sextupole components need to be mitigated for application because their change during operation is more than required field quality which is less than 1  10 -3 . Therefore,. we need to consider mitigation methods of them numerically by the developed model.

In addition, we conducted beam orbit simulations on single magnet and series of the magnets to compose the rotating gantry considering the distribution of magnetic field including the SCIF to evaluate the influence of the SCIFs on beam characteristics such as beam orbits and distribution of particles.

IV. MITIGATION OF INFLUENCE OF SHIELDING-CURRENT-INDUCED FIELD ON FIELD QUALITY

The mitigation method for the integrated dipole component is current control. Because the dipole magnet is designed to generated only dipole component of magnetic field, control of transport current can change generated dipole component of magnetic field directly. Note that the dipole magnetic field is not linear to the transport current due to influence of shielding-current-induced field. Therefore, we conduct electromagnetic field analyses several times with controlled transport current until we obtain enough stability of dipole component during operation.

The integrated sextupole component of the magnetic field cannot be mitigated by current control because the cause of sextupole component is not transport current but threedimensional distribution of shielding current in the magnet. Therefore, we need to change magnet design to mitigate the influence of SCIF on the sextupole component. To mitigate the sextupole component of the SCIF, we investigated the design of the correction coils. We designed sextupole correction coils wound with coated conductors into racetrack geometry. The correction coils are wound with same coated conductor as the dipole coil and placed at the end of the dipole magnet as shown in Fig. 3. The operation pattern of the correction coils are decided from Fig. 2.

And also, we conducted beam orbit simulations on single dipole magnet with HTS sextupole correction coils and series of the magnets with the correction coils to compose the rotating gantry considering the distribution of magnetic field including the SCIF to evaluate the influence of the SCIFs on beam characteristics. We discuss the influence of local strong sextupole component of the magnetic field by the correction coils on the beam characteristics.

Fig. 2 .

 2 Fig. 2. Saddle-shaped coil of a cosine-theta dipole magnet for a rotating gantry for carbon cancer therapy [2].

Fig. 3 .

 3 Fig. 3. Current profile in the analysis and calculated influence of the SCIF on the dipole, sextupole, and decapole components.

Fig. 1 .

 1 Fig. 1. Combination of the dipole magnet and the correctoin coils for mitigation of sextupole component of the SCIF.

ACKNOWLEDGMENT

This work was supported in part by JSPS KAKENHI Grant Numbers JP16H02326 and JP16J07799.


THE N-MEASURE FOR ALGEBRAIC INTEGERS HAVING ALL THEIR CONJUGATES IN A SECTOR

Let α be a nonzero algebraic integer of degree d whose all conjugates α 1 = α, α 2 , . . . , α d lie in a sector | arg z| ≤ θ, 0 ≤ θ ≤ 90 • . We define the N-measure of α by N(α) =

and the absolute N-measure of α by ν(α) = N(α) 1/ deg (α) . Firstly, we consider the case θ = 0. We prove that N(α) ∈ N and that, if α is a reciprocal algebraic integer, N(α) is a square. Then, we study the set N of the quantities ν(α). We prove that there exists a number l such that N is dense in (l, ∞). Finally, using the method of auxiliary functions, we find the seven smallest points of N in (2, l). In case of 0 < θ ≤ 90 • , we compute the greatest lower bound c(θ) of the absolute N-measure of α, for α belonging to eight subintervals of ]0, θ[. Among these subintervals, two are complete. These computations are also done by using auxiliary functions. The polynomials involved in the auxiliary functions are found by our recursive algorithm.

Introduction

Let α be a nonzero algebraic integer of degree d. We define the N-measure of α by:

N(α) = d i=1 (|α i | + 1/|α i |).
The absolute N-measure of α is the quantity ν(α) = N(α) 1/d . If P is the minimal polynomial of α, then N(P ) = N(α) and ν(P ) = ν(α).

It is obvious that ν(α) ≥ 2 and the equality holds if and only if α is a root of unity. Indeed, if for i = 1, . . . , d, we have |α i | + 1/|α i | = 2 then for all i = 1, . . . , d, |α i | = 1. By the theorem of Kronecker, we deduce that α is a root of unity.

The case θ = 0

Suppose first that θ = 0. It means α is a totally positive algebraic integer : α and all its conjugates are positive real numbers.

Theorem 1. For all nonzero totally positive algebraic integers α whose minimal polynomial P satisfies |P (0)| = 1, N(α) ∈ N.

Moreover, if α is reciprocal i.e., α and 1/α are conjugates, then N(α) is a square.

In order to study the structure of the set N of the quantities ν(α), we show the following Theorem 2. N is dense in (l, ∞) where l = lim n→∞ ν(β 2 n ) = 3.513182 . . ..

The β 2

n were defined by C.J. Smyth [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF] as follows:

β 2 0 = 1 β 2 n = β 2 n+1 + β -2 n+1 -2 β 2
n is a totally positive algebraic integer of degree 2 n .

Towards determining the structure of N in the gap (2, l), we prove the following Theorem 3. If α is a nonzero totally positive algebraic integer whose minimal polynomial has a constant term of modulus 1 and is different from x -1, x 2 -3x + 1, x 4 -7x 3 + 13x 2 -7x + 1, x 8 -15x 7 +83x 6 -220x 5 +303x 4 -220x 3 +83x 2 -15x+1, x 6 -11x 5 +41x 4 -63x 3 +41x 2 -11x+1, x 8 -15x 7 + 84x 6 -225x 5 + 311x 4 -225x 3 + 84x 2 -15x + 1 and x 16 -31x 15 + 413x 14 -3141x 13 + 15261x 12 -50187x 11 + 115410x 10 -189036x 9 + 222621x 8 -189036x 7 + 115410x 6 -50187x 5 + 15261x 4 -3141x 3 + 413x 2 -31x + 1 then we have:

ν(α) ≥ 3.483671.
Therefore, the points of N in (2, l) are:

2 = ν(x -1) = ν(β 2 0 ), 3 = ν(x 2 -3x + 1) = ν(β 2 1 ), 3.316624 . . . = ν ( x 4 -7x 3 + 13x 2 -7x + 1) = ν(β 2 2 ), 3.433631 . . . = ν(x 8 -15x 7 + 83x 6 -220x 5 + 303x 4 -220x 3 + 83x 2 -15x + 1) = ν(β 2 3 ), 3.448217 . . . = ν(x 3 -5x 2 + 6x -1) = ν(x 3 -6x 2 + 5x -1), 3.470100 . . . = ν(x 4 -7x 3 + 14x 2 -8x + 1) = ν(x 4 -8x 3 + 14x 2 -7x + 1), 3.479784 . . . = ν(x 16 -31x 15 + 413x 14 -3141x 13 + 15261x 12 -50187x 11 + 115410x 10 -189036x 9 + 222621x 8 -189036x 7 + 115410x 6 -50187x 5 + 15261x 4 -3141x 3 + 413x 2 -31x + 1) = ν(β 2 4 ). 1.2 The case 0 < θ < 90 •
The study of "small points" leads us to extend the search to algebraic integers α whose conjugates lie in a sector S θ = {z ∈ C such that | arg z| ≤ θ} with 0 < θ < 90 • . We follow here the work of M. Langevin [La] on the absolute Mahler measure Ω(α) = ( d i=1 max(1, |α i |)) 1/d of algebraic integers α having all their conjugates in a sector. He proved that there exists a function c(θ) on [0, 180 • ), always > 1, such that if α = 0 is not a root of unity, whose conjugates all lie in S θ , then Ω(α) ≥ c(θ). Note that this result is not numerical. G. Rhin and C. Smyth [RS] succeeded in finding the exact value of c(θ) for θ in nine subintervals of [0, 120 • ] and conjectured that c(θ) is a "staircase" function of θ, which is constant except for finitely many left discontinuities in any closed subinterval of [0, 180 • ). Their method uses explicit auxiliary polynomials A(X) = X a R(X) where R is a reciprocal polynomial in Z Z[X], found by heuristic methods. In 2004, thanks to Wu's algorithm [Wu], G. Rhin and Q. Wu [RW] gave the exact value of c(θ) for four new subintervals of [0, 140 • ] and extended four existing subintervals. In 2013, the author and G. Rhin [FR] found for the first time a complete subinterval and a fourteenth subinterval. A complete subinterval is an interval on which the function c(θ) describing the minimum on the sector | arg z| ≤ θ is constant, with jump discontinuities at each end. These improvements are due to our recursive algorithm.

Here, we prove the following result: Theorem 4. There exist a left discontinuous, strictly positive, staircase function g on [0, 90 • ) and a positive, continuous, monotonically decreasing function f on [0, 90 • ) such that:

for θ ∈ [0, 90 • ), if α is a nonzero algebraic integer whose conjugates all lie in S θ then

N(α) ≥ min(f (θ), g(θ)).
Moreover, the exact value of c(θ) = inf

| arg α|≤θ N(α) is known on eight subintervals of [0, 90 • ).
These intervals are given in Table 1.

The function g(θ) is a decreasing staircase function having left discontinuities. The function g(θ)

is the smallest value of ν(α) that could be found for α having all its conjugates in | arg z| ≤ θ.

The function f (θ) is given by f (θ) = max 1≤i≤8 (f i (θ)), and the functions f i (θ) are defined as follows:

f i (θ) = min z∈S θ   log(|z| + 1/|z|) - 1≤j≤J c ij log |Q ij (z)|  
where the polynomials Q ij and the real numbers c ij are read off from Table 3.

x 4 -x 3 + 4x 2 -2x + 3
In Section 2, we prove Theorem 1. Section 3 deals with the denseness of the set N . In Section 4, we describe the method of explicit auxiliary functions. We link these functions with the integer transfinite diameter. Then, we explain the recursive algorithm which enables us to obtain the constant of Theorem 3. In Section 5, we deal with the case 0 < θ ≤ 90 • . All the computations were done on a MacBookPro with the languages Pascal and Pari.

Proof of Theorem 1

Remember that, if

P ∈ Z[x] of degree d, the reciprocal polynomial of P is P * = x d • P ( 1 x ) and P is reciprocal if P * = P .
Let α be a nonzero totally positive algebraic integer of degree d and let P =

d i=1 (x -α i ) be its minimal polynomial satisfying |P (0)| = 1. Put Q(x) = P (x) • P * (x). Then, Q is a reciprocal polynomial of degree 2d in which we do the change of variable y = x + 1/x -2. We get Q(x) = x d • R(y) where R ∈ Z[y] with deg(R)=d. The fact that |P (0)| = 1 implies that the polynomial R is monic. The roots of R are the β i = α i + 1/α i -2 for i = 1, . . . , d. Thus, N(P ) = d i=1 |β i + 2| = |R(-2)| because R is monic and |R(-2)| ∈ N. Now, if P is a reciprocal polynomial of degree 2d, then P = x d .R(y) and N(P ) = 2d i=1 |α i + 1/α i | = d i=1 (α i + 1/α i ) 2 = (R(-2)) 2 . 3 Denseness of the set N 3.1 Study of the sequence (ν(β 2 n )) n≥1
Remember that the sequence (β 2 n ) n≥0 satisfies:

β 2 0 = 1 β 2 n = β 2 n+1 + β -2 n+1 -2
We first prove the following Lemma 5.

ν(β 2 n ) = 2 n-1 i=0 (1 + λ i ) 1/2 i where λ 0 = 1 2 and λ i+1 = λ i (1 + λ i ) 2 for i ≥ 0. Proof For n ≥ 0, we put γ n = β 2 n , so γ n = γ n+1 + γ -1 n+1 -2.
Therefore, we can write: N(

β 2 n ) = N(γ n ) = 2 n i=1 (γ n,i + γ -1 n,i ) = 2 n-1 i=1 (γ n,i + γ -1 n,i ) 2 because γ n,i = γ 2 n-1 +i for 1 ≤ i ≤ 2 n-1 . (For 1 ≤ i ≤ 2 n , γ n,i denote the conjugates of γ n ).
Then, we have:

N(β 2 n ) = 2 n-1 i=1 γ 2 n,i + γ -2 n,i + 2 = 2 n-1 i=1 (2 + γ n-1,i ) 2 because γ 2 n+1 + γ -2 n+1 = γ 2 n + 4γ n + 2. Finally, we have N(β 2 n ) = 2 2 n-1 2 n-1 i=1 (1 + 1/2γ n-1,i ) 2 .
We recall here a more general lemma that we proved in [F]:

Lemma 6. The notations are the same as previously. Then we have

2 n i=1 (1 + λ 0 γ n,i ) = n i=0 (1 + λ i ) 1/2 i 2 n where λ 0 = 1 2 and λ i+1 = λ i (1 + λ i ) 2 for i ≥ 0. Now, we can conclude that ν(β 2 n ) = 2 n-1 i=0 (1 + λ i ) 1/2 i .
A short program, that can be read off from Annex below, allows us to compute with Pari the limit l of the sequence ν(β 2 n ) and we obtain l = 3.513182 . . .. Note that l gives an upper bound for the first accumulation point of N .

Proof of Theorem 2

The proof and notations follow those of C.J. Smyth in [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF]. For a given function g

: [0, ∞) → R, put M(g) the set of all means M g (α) = 1 d d i=1 g(|α i |)
for α a totally real algebraic integer, i.e., all its conjugates

α 1 = α, • • • , α d are real numbers. When the limits exist, put a(g) = lim n→∞ M g (β n ) and c(g) = lim n→∞ M g (2 cos(2π/n)).
Here a convenient choice for g is

g : x → log(x 2 + 1 x 2 ) (3.1) because then M g (α) = log ν(α 2 ).
The proof consists in two parts.

First step of the proof

C.J. Smyth [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF] proved the following Theorem 7. Let g : R + → R + be a monotonic increasing function, zero on [0, 1]such that

lim x→∞ g(x + 1)/g(x) = 1
and the values of log 2 g(2k + 1) mod 1 (k=0,1,2,. . .) are everywhere dense in (0, 1).

Then the limit a(g) exists and M(g) is dense in (a(g), ∞).

We replace the function g defined by (3.1) by the function g * which satisfies the hypothesis of Theorem 7:

g * (x) =    g(x) if x > 1 0 if 0 ≤ x ≤ 1 (3.2) Since M g (β n ) = M g * (β n )
, the existence of a(g * ) implies those of a(g) and a(g * ) = a(g).

It is easy to see that g * satisfies the first hypothesis of Theorem 7. So, it is sufficient to study the denseness of the set F = {log 2 g(2k + 1) mod 1,k ∈ N}.

Let t ∈ [0, 1] and > 0. Does there exist f ∈ F such that |f (k) -t| < ? We search for n and k satisfying:

| log 2 g * (2k + 1) -t -n| < i.e., | log g * (2k + 1) -t -n log 2| < (3.3)
The uniform continuity of the function log on [1, ∞) gives :

∀ > 0, ∃ η( ) such that ∀x, y > 0, |x -y| < η( ) ⇒ | log x -log y| < .
We choose n such that 2 -n < η( ) and k such that

|(2k + 1) -(g * ) -1 (2 n e t )| ≤ 1. As (g * ) is
bounded by 1, the mean value Theorem for g * on (1, ∞) gives:

|g * (2k + 1) -2 n e t | ≤ 1, i.e., |2 -n g * (2k + 1) -e t | ≤ 2 -n < η( )
and the inequality (3.3) follows immediately. Thus, we have proved that It is easy to see that, for our function g defined by (3.1), the Lipschitz condition is satisfied for

M(g) is dense in (a(g * ), ∞) = (a(g), ∞) = (log l, ∞) = (1.256522 . . . , ∞).
B(λ) = 2λ • (λ 2 + 1). Thus, M(g) is dense on (c(g), ∞) = (1.465715 . . . , ∞).

Conclusion

We have shown that M(g) is dense on (min(a(g), c(g)), ∞) = (log l, ∞) which means that N is dense on (l, ∞), where l = lim n→∞ ν(β 2 n ) = 3.513182 . . ..

Proof of Theorem 3

The explicit auxiliary function

The auxiliary function involved in Theorem 3 is of the following type:

for x > 0, f (x) = log(x + 1/x) - 0≤j≤J c j log |Q j (x)|
where the c j are positive real numbers and the polynomials Q j are non zero polynomials in Z Z[x].

Let α be a totally positive algebraic integer with conjugates α 1 = α,. . .,α d . Then we have

d i=1 f (α i ) ≥ md,
where m denotes the minimum of the function f . Thus,

log N(α) ≥ md + 1≤j≤J c j log | d i=1 Q j (α i )|.
We assume that P does not divide any Q j , then

d i=1 Q j (α i ) is a nonzero integer because it is the resultant of P and Q j .
Therefore, if α is not a root of Q j , we have

ν(α) ≥ e m .
To avoid convergence problems with Pascal, we put y = x + 1/x -2. Then, f (x) becomes for y > 0, g(y) = log(y + 2) -1≤j≤J c j log |U j (y)| (1). Now, the main problem is to find a good list of polynomials U j which gives a value of m as large as possible. Thus, we link the auxiliary function with the integer transfinite diameter in order to find the polynomials with our recursive algorithm.

Auxiliary functions and integer transfinite diameter

In this section, we shall need the following definition: Let K be a compact subset of C.

If ϕ is a positive function defined on K, the ϕ-integer transfinite diameter of K is defined as

t Z,ϕ (K) = lim inf inf sup |P (y)| 1 n ϕ(y) . n ≥ 1 P ∈ Z[Y ] y ∈ K n → ∞ deg(P ) = n
This weighted version of the integer transfinite diameter was introduced by F. Amoroso [A] and is an important tool in the study of rational approximations of logarithms of rational numbers.

Inside the auxiliary function (1), we replace the numbers c j by rational numbers a j /q where q is a integer such that q.c j is an integer for all 0 ≤ j ≤ J. Then we can write:

for y > 0, g(y) = log(y + 2) - t r log |U (y)| (2)
where

U = J i=0 U a j j ∈ Z Z[Y ] is of degree r = J i=0 a j deg(U j ) and t = J j=0 c j deg(U j
) is a positive real number. We want to get a function whose minimum m is as large as possible. Thus we search a polynomial

U ∈ Z Z[Y ] such that sup y>0 |U (y)| t/r (y + 2) -1 ≤ e -m .
If we suppose that t is fixed, it is clear that we need an effective upper bound for the quantity

t Z,ϕ ((0, ∞)) = lim inf inf sup |P (y)| t r ϕ(y) r ≥ 1 P ∈ Z[Y ] y > 0 r → +∞ deg(P ) = r
where we use the weight ϕ(y) = (y + 2) -1 .

Even if we replace the compact subset K by the infinite interval (0, ∞), the weight ϕ ensures that the quantity t Z,ϕ ((0, ∞)) is finite.

Construction of the auxiliary function

The improvement compared with Wu's algorithm is that our polynomials are obtained by induction. Suppose that we have U 1 , U 2 , ..., U J . Then we use the semi-infinite linear programming (introduced in number theory by C. J. Smyth [START_REF] Smyth | On the measure of totally real algebraic numbers[END_REF]) to optimize g for this set of polynomials (i.e., to get the greatest possible m). We obtain the numbers c 1 , c 2 , ..., c J and g in the form (2)

with t = J i=1 c j deg(U j ).
For several value of k, we seek a polynomial R(y)

= k l=0 a l y l ∈ Z Z[y] such that sup y>0 |U (y)R(y)| t r+k (y + 2) -1 ≤ e -m , i.e., such that sup y>0 |U (y)R(y)|(y + 2) -(r+k)/t
is as small as possible.

We apply the LLL algorithm to the linear forms in a 0 ,. . ., a k

U (y i )R(y i )(y i + 2) -(r+k)/t
where y i are control points uniformly distributed in the interval [0,70], including the points where g has its least local minima. We get a polynomial R whose factors R j are good candidates to enlarge the set of polynomials (U 1 , U 2 , ...,U J ). We only keep the polynomials R j which have a nonzero coefficient c j in the new optimized auxiliary function g. After optimization, some previous polynomials U j may have a zero coefficient and are removed.

In order to get the constant of Theorem 3, we take k from 4 to 20 successively.

The polynomials and the coefficients c j involved in the auxiliary function of Theorem 3 are listed in the Table 2 below.

Proof of Theorem 4

Let α be a nonzero algebraic integer all of whose conjugates lie in a sector S θ = {z ∈ C such that | arg z| ≤ θ} with 0 < θ ≤ 90 • . The auxiliary functions f i , 1 ≤ i ≤ 8 are of the following type:

∀z ∈ S θ , f (z) = log(|z| + 1/|z|) - 1≤j≤J c j log |Q j (z)|,
where the c j are positive real numbers and the polynomials Q j are non zero polynomials in Z Z[z].

The function f is harmonic outside a finite set containing the roots of the polynomials Q j and thus, it takes its minimum on the boundary of

S θ . it is clear that f (z) → ∞ as z → ∞. So, since f (z) = f (z)
, it is sufficient to search the minimum of f on the upper edge of the sector S θ where z = xe iθ with x > 0.

The auxiliary function on the half line

R θ = {z ∈ C, z = xe iθ , x > 0} is f (z) = log(x + 1/x) - 1≤j≤J c j log |Q j (z))|.
Note that, concerning the integer transfinite diameter, we deal here with the compact K replaced by the half line R θ . But the weight ϕ(z) = (|z| + 1/|z|) -1 ensures that the quantity t Z Z,ϕ (R θ ) is a finite number.

Then, for several values of k, we search a polynomial

R(z) = k l=0 a l z l ∈ Z Z[z] such that sup x>0 |Q(z)R(z)| t r+k |x + 1/x| -1 ≤ e -m ,
Where m is the minimum of the function f . We search

sup x>0 |Q(z)R(z)||x + 1/x| -(r+k)/t
is as small as possible.

But, here, R(z) is not a real linear form in the unknown coefficients a i . Thus we replace it by its real part and its imaginary part. Then, we apply LLL to the linear forms

|Q(z n ).Re(R(z n )) • |x n + 1/x n | -(r+k)/t and |Q(z n ) • Im(R(z n )).|x n + 1/x n | -(r+k)/t .
The numbers x n are suitable control points in [0, 50] , including the points wheref has its least local minima. Then we apply our recursive algorithm as described in Section 3.

The polynomials and the coefficients c j involved in the auxiliary function of Theorem 4 are listed in the N(Q j ) c j θ(Q j ) Q j 2.1213203 0.072660766 45.000000

3 -5x 2 + 6x -1 9 0.00091541 x 3 -6x 2 + 5x -1 10 0.01112510 x 4 -7x 3 + 13x 2 -7x + 1 11 0.00424427 x 4 -7x 3 + 14x 2 -8x + 1 12 0.00010144 x 4 -8x 3 + 15x 2 -8x + 1 13 0.00145858 x 5 -9x 4 + 26x 3 -29x 2 + 11x -1 14 0.00050273 x 6 -11x 5 + 42x 4 -67x 3 + 45x 2 -12x + 1 15 0.00055594 x 7 -13x 6 + 61x 5 -131x 4 + 136x 3 -66x 2 + 14x -1 16 0.00041956 x 7 -13x 6 + 61x 5 -132x 4 + 138x 3 -67x 2 + 14x -1 17 0.00129057 x 7 -13x 6 + 62x 5 -137x 4 + 147x 3 -73x 2 + 15x -1 18 0.00287103 x 8 -15x 7 + 83x 6 -220x 5 + 303x 4 -220x 3 + 83x 2 -15x + 1 19 0.00075792 x 8 -15x 7 + 85x 6 -232x 5 + 328x 4 -242x 3 + 91x 2 -16x + 1 20 0.00055993 2x 8 -26x 7 + 128x 6 -308x 5 + 391x 4 -265x 3 + 94x 2 -16x + 1 21 0.00019384 x 9 -17x 8 + 116x 7 -411x 6 + 819x 5 -934x 4 + 599x 3 -206x 2 + 34x -2 Table 3: The auxiliary functions f i , 1 ≤ i ≤ 8 f 1 N(Q j ) c j θ(Q j ) Q j 2.
x 2 -2x + 2 2.2117398 0.027983260 45.309985

x 4 -4x 3 + 8x 2 -7x + 3 2.2613295 0.015502898 45.459748

x 6 -6x 5 + 18x 4 -29x 3 + 28x 2 -15x + 4 2.3694703 0.000822075 47.464280

x 8 -9x 7 + 41x 6 -112x 5 + 200x 4 -233x 3 + 177x 2 -80x + 18 2.3451610 0.000681118 50.081319

x 8 -9x 7 + 40x 6 -106x 5 + 183x 4 -207x 3 + 154x 2 -69x + 16 2.0000000 0.041823516 60.000000

x 2 -x + 1 f 5 N(Q j ) c j θ(Q j ) Q j 2.1213203 0.063690701 45.000000

x 2 -2x + 2 2.2117398 0.007017163 45.309985

x 4 -4x 3 + 8x 2 -7x + 3 2.0000000 0.077288797 60.000000

x 2 -x + 1 f 6 N(Q j ) c j θ(Q j ) Q j 2.0000000 0.074090899 60.000000

x 2 -x + 1 2.1933236 0.002608550 63.681104

x 4 -3x 3 + 7x 2 -7x + 5 2.2473333 0.016313417 68.667414

x 4 -2x 3 + 5x 2 -3x + 2 2.1213203 0.060755551 69.295189

x 2 -x + 2 f 7 N(Q j ) c j θ(Q j ) Q j 2.1213203 0.077121906 69.295189

x 2 -x + 2 2.0000000 0.068834995 90.000000

x 2 + 1 f 8 N(Q j ) c j θ(Q j ) Q j 2.1162455 0.043532810 82.588473

x 4 -x 3 + 4x 2 -2x + 3 2.0000000 0.083777873 90.000000

x 2 + 1

  Let g : R + →: R + be a function such that lim x→∞ g(x) = ∞ and which satisfies a Lipschitz condition |g(x) -g(y)| < B(λ)|x -y| for x, y ∈ [0, λ], for each λ > 0. Then M(g) is dense on (c(g), ∞), where c(g) = 2 π π/2 0 g(2 cos θ)dθ.

Table 1

 1 Intervals [θ i , θ i ] where c(θ) is known exactly. Here c(θ) = c(θ i ) = ν(P ) for θ ∈ [θ i , θ i ].The angles are given in degrees.

	i	c(θ)	θ i	θ i	P
	1	2.5	0	23.15	x -2
	2 2.455652 26.267699	27.23	x 4 -7x 3 + 19x 2 -23x + 11
	3 2.309401	30	37.17	x 2 -3x + 3
	4 2.265128 39.985729 45.309985 x 4 -5x 3 + 11x 2 -11x + 5
	5 2.211739 45.309985	52.6	x 4 -4x 3 + 8x 2 -7x + 3
	6 2.193323 63.681104	67.59	x 4 -3x 3 + 7x 2 -7x + 5
	7 2.12.13.20 69.295189 82.588473 x 2 -x + 2
	8 2.116245 82.588473	89.99

Table 2 :

 2 Table 3 below. The polynomials involved in Theorem 3.

	Annex: The program which computes lim n→∞	ν(β 2 n )
	N=25;
	lambda=vector(N,j,0);
	lambda[1]=2/9; for(k=2,N,lambda[k]=lambda[k-1]/(1+lambda[k-1]) 2 );p=1;for(k=1,N,p=p*(1+lambda[k]) 1/2 k ;print(2*p*(1+1/2)))
	The computation gives:
	3.31662479035540
	3.43363162301252
	3.47978396444099
	3.49878633852976
	3.50685371295751
	3.51035762903346
	3.51190629049069
	3.51260021930633
	3.51291460021634
	3.51305831890415
	3.51312451476539
	3.51315519816299
	3.51316949809199
	3.51317619392804
	3.51317934210340
	3.51318082764469
	3.51318153088927
	3.51318186475827
	3.51318202367455
	3.51318209949318
	3.51318213574302
	3.51318215310816
	3.51318216144157
	3.51318216544726
	3.51318216737561

  -10x 7 + 47x 6 -131x 5 + 236x 4 -281x 3 + 218x 2 -102x + 23 f 4

		0000000 0.27726826	0	x -1
		2.5000000 0.13135075	0	x -2
		2.6321199 0.00099184	23.749627	x 4 -8x 3 + 25x 2 -35x + 19
		2.4556520 0.00681346	26.267699	x 4 -7x 3 + 19x 2 -23x + 11
		2.3094011 0.02113516	30.000000	x 2 -3x + 3
	f 2	N(Q j )	c j	θ(Q j )	Q j
		2.0000000 0.28694004	0	x -1
		2.4556520 0.01101937	26.267699	x 4 -7x 3 + 19x 2 -23x + 11
		2.6832816 0.01253213	26.565051	x 2 -4x + 5
		2.5057826 0.00273581	27.277283	x 4 -7x 3 + 19x 2 -22x + 10
		2.4304356 0.00124547	29.171773	x 8 -13x 7 + 76x 6 -258x 5 + 554x 4 -769x 3 + 675x 2 -344x + 79
		2.3094011 0.04293284	30.000000	x 2 -3x + 3
	f 3	N(Q j )	c j	θ(Q j )	Q j
		2.0000000 0.24253217	0	x -1
		2.3094011 0.01928768	30.000000	x 2 -3x + 3
		2.3420541 0.00307938	37.011218	x 8 -11x 7 + 56x 6 -168x 5 + 323x 4 -406x 3 + 327x 2 -155x + 34
		2.3167326 0.00438622	37.168675	x 6 -8x 5 + 29x 4 -58x 3 + 68x 2 -44x + 13
		2.2651285 0.01631844	39.985729	x 4 -5x 3 + 11x 2 -11x + 5
		2.1213203 0.02800549	45.000000	x 2 -2x + 2
		2.2804413 0.00209584	46.139190	x 8


Valérie Flammang

UPPER BOUNDS FOR THE USUAL MEASURES OF TOTALLY POSITIVE ALGEBRAIC INTEGERS WITH HOUSE LESS THAN 5.8

Upper bounds for the usual measures of totally positive algebraic integers with house less than 5.8

Introduction

Let α be a totally positive algebraic integer,i.e., its conjugates α 1 = α, . . . , α d are positive real numbers. First we recall the definitions of the usual measures for α.

The trace of α is Tr(α) = d i=1 α i and tr(α) = 1 d Tr(α) denotes the absolute trace of α. The well known "Schur-Siegel-Smyth trace problem"is the following: fix ρ < 2, then show that all but finitely many totally positive algebraic integers α have tr(α) > ρ. Many authors studied this problem throughout the years. The results obtained after those of I. Schur [Schu] and C.L. Siegel [Si] are all based on the method of explicit auxiliary functions with heuristic search of polynomials until 2009 ([F2]). For more details, see for example [S1], [ABP], [START_REF] Aguirre | The Trace Problem for Totally Positive Algebraic Integers, Number Theory and Polynomials[END_REF], [START_REF] Aguirre | The integer Chebyshev constant of Farey intervals[END_REF], [START_REF] Aguirre | The trace problem for totally positive algebraic integers. With an appendix by Jean-Pierre Serre[END_REF], [LW], [DW], [F1]. In 2016, thanks to a slightly change in the use of our recursive algorithm developed in [F2], we solved it for ρ < 1.792812 which is the better result at our knowledge [F1]. The recursive algorithm substitutes the heuristic search by a systematic search by induction of suitable polynomials. On the other hand, J.P. Serre [START_REF] Aguirre | The trace problem for totally positive algebraic integers. With an appendix by Jean-Pierre Serre[END_REF] showed that this method does not give such an inequality for any ρ larger than 1.8983021... . Therefore this method cannot be used to prove that 2 is the smallest limit point of the set of quantities {tr(α), α totally positive algebraic integer}.

If P = x d + . . . + a d = a 0 (x -α 1 ) . . . (x -α d ) denotes the minimal polynomial of α then the

length of α is L(α) = d i=0
|a i | and l(α) = L(α) 1/d is absolute length of α. In 1994, we studied the structure of the set L of the quantities {l(α), α totally positive algebraic integer} [F3]. Using the principle of auxiliary functions with heuristic search of polynomials, we found the fifth smallest points of L in the interval (2, 2.361101]. At last, we proved that L is dense in [2.376841 . . . , ∞). Again, thanks to our recursive algorithm, we proved [F4] that all but finitely many totally positive algebraic integers α have l(α) ≥ 2.365827. This constant improves those of Q. Mu and Q. Wu [MW](2013) and is the best constant to our knowledge.

1

The Mahler measure of α is defined as M(α) = The smallest known value is due to Lehmer himself and is M(α) = 1.176280 . . . where α is the algebraic integer whose minimal polynomial is P (z) = z 10 + z 9 -z 7 -z 6 -z 5 -z 4 -z 3 + z + 1. In 1973, A. Schinzel [Schi] showed that all totally positive algebraic integers α, different from 0 and 1, satisfy Ω(α

) ≥ 1 + √ 5 2
and the equality holds if α is a root of the polynomial

x 2 -3x + 1. It means that 1 + √ 5 2
is the smallest element of the set M of the quantities {Ω(α), α totally positive algebraic integer}. In 1981, C. Smyth [S2] proved that all but finitely many totally positive algebraic integers satisfy Ω(α) ≥ 1.717177. He found the three following points of M. He also showed that M is dense in [1.727305 . . . , ∞). In 1994, with the same method and thanks to numerical improvements, we obtained [F3] that all but finitely many totally positive algebraic integers have Ω(α) ≥ 1.720678 . . .. This lower bound gives the two following points of M. In [F5], thanks to our recursive algorithm, we got that all but finitely many totally positive algebraic integers verify Ω(α) ≥ 1.722069. This constant improves those of Q. Mu and Q. Wu [MW] (2013) and is the best constant to our knowledge.

All these lower bounds were obtained with an explicit auxiliary function of the type:

∀x > 0, f (x) = g(x) - 1≤j≤J c j log |Q j (x)|
where g(x) = x for the trace, g(x) = log(x + 1) for the length and g(x) = log max(1, x) for the Mahler measure. The c j are positive real numbers and the polynomials Q j are non zero polynomials in Z Z[x].

In this work, in order to use the method of explicit auxiliary functions to get upper bounds this time, we have to suppose that x belongs to an interval I of R. We don't choose it arbitrary. We study the localization of the roots of the totally positive polynomials involved in our auxiliary functions (there are more than one hundred) and we notice that, mostly, they are bounded by the value 5.8. Therefore, we focus our attention to the set of totally positive algebraic integers α having all their conjugates in I = (0, 5.8). It's equivalent to say that the house of α defined by α = max 1≤i≤d |α i |, is bounded by 5.8. Then, an explicit auxiliary function of the type:

∀x ∈ I, f (x) = c -g(x) - 1≤j≤J c j log |Q j (x)|
where c is a real number allows us to get upper bounds for the three usual measures.

More precisely, we obtain:

Theorem 1. If α is a totally positive algebraic integer such that α ≤ 5.8 then tr(α) ≤ 4.0779.

Remark: In the following table, we list ten algebraic integers, whose minimal polynomials appear in the auxiliary function involved in Theorem 1, that have an absolute trace equal to 4. It proves that our above upper bound is quite good.

tr(α) α Minimal polynomial of α 4 4 x -4 4 5.6920215 x 3 -12x 2 + 41x -29 4 5.7320508 x 2 -8x + 13 4 5.7491175 x 4 -16x 3 + 90x 2 -204x + 146 4 5.7606389 x 6 -24x 5 + 231x 4 -1130x 3 + 2920x 2 -3694x + 1719 4 5.7675038 x 9 -36x 8 + 562x 7 -4972x 6 + 27314x 5 -95852x 4 + 212323x 3 -280802x 2 + 194166x -49279 4 5.7881137 x 9 -36x 8 + 562x 7 -4972x 6 + 27312x 5 -95806x 4 + 211907x 3 -278957x 2 + 190161x -45878 4 5.7905003 x 7 -28x 6 + 325x 5 -2010x 4 + 7060x 3 -13768x 2 + 13185x -4227 4 5.7927026 x 10 -40x 9 + 706x 8 -7222x 7 + 47266x 6 -205975x 5 + 602162x 4 -1158217x 3 + 1389569x 2 -926043x + 254366 4 5.7956925 x 12 -48x 11 + 1038x 10 -13346x 9 + 113359x 8 -668138x 7 + 2791595x 6 -8290714x 5 + 17257804x 4 -24333909x 3 + 21771184x 2 -10870656x + 2212927
Theorem 2. If α is a totally positive algebraic integer such that α ≤ 5.8 and with minimal polynomial different from x-4, x-5, x 2 -8x+14, x 2 -9x+19 and x 4 -17x 3 +103x 2 -257x+211 then l(α) ≤ 4.78746.

Remark:

The totally positive algebraic integer whose house is 5.6964949 and with minimal polynomial x 5 -20x 4 + 155x 3 -579x 2 + 1037x -709 (appeared in the auxiliary function involved in Theorem 2) has an absolute length equal to 4.782145. It proves again that the above upper bound is quite good.

Theorem 3. If α is a totally positive algebraic integer such that α ≤ 5.8 and with minimal polynomial different from x -4, x -5, x 2 -8x + 14, x 2 -9x + 19, x 4 -17x 3 + 103x 2 -257x + 211,

x 5 -20x 4 +155x 3 -579x 2 +1037x-709 and x 11 -44x 10 +868x 9 -10124x 8 +77490x 7 -408213x 6 + 1508359x 5 -3904032x 4 + 6926534x 3 -8010726x 2 + 5426958x -1629014 then Ω(α) ≤ 3.713593.

Remark:

The totally positive algebraic integer whose house is 5.761517 and with minimal polynomial x 8 -32x 7 +440x 6 -3390x 5 +15979x 4 -47095x 3 +84583x 2 -84452x+35809 (appeared in the auxiliary function involved in Theorem 3) has an absolute Mahler measure equal to 3.7089332. It proves again that the above upper bound is quite good.

The principle of auxiliary functions

Remember that I = (0, 5.8). As mentioned in the introduction, the auxiliary functions used in our theorems are of the type:

∀x ∈ I, f (x) = c -g(x) - j∈J c j log |Q j (x)|
where c is a real number, the c j s are positive real numbers and the

Q j s are non zero polynomials in Z Z[x].
Suppose that the constant c is chosen such that the minimum m of the function f is > 0.

Consider now a totally positive algebraic integer with conjugates α 1 = α, . . . , α d , whose house is bounded by 5.8 and whose minimal polynomial P does not divide any Q j . We have

for i = 1, . . . , d, f (α i ) = c -g(α i ) - 1≤j≤J c j log |Q j (α i )| ≥ m > 0, i.e., d i=1 f (α i ) = dc - d i=1 g(α i ) - 1≤j≤J c j log | d i=1 Q j (α i )| > 0. But | d i=1 Q j (α i )| is a nonzero integer because it is the resultant of P and Q j . Hence, if α is not a root of Q j , we have c ≥ 1 d d i=1 g(α i ).
If we take, g(x) = x, we get c ≥ tr(α).

If we take g(x) = log(x+1), we get e c ≥ l(α) because we can prove by induction that the minimal polynomial P has a sequence of coefficients with alternate signs so, L(α

) = |P (-1)| = d i=1 (1+α i ).
If we take g(x) = log max(1, x), we get e c ≥ Ω(α).

In order to reduce the size of the coefficients of the polynomials Q j which will ensure a better precision in our calculus, we do the following transformation: y = x -5. Finally, we work with for y ∈ I = (-5, 0.8), h(y) = g(y + 5) -

1≤j≤J c j log |U j (y)| (1).
3 Link between auxiliary functions and generalized integer transfinite diameter

Let K be a compact subset of C. The transfinite diameter of K is defined by

t(K) = lim inf inf |P | 1 n ∞,K n ≥ 1 P ∈ C[X] n → ∞ P monic deg(P ) = n where |P | ∞,K = sup z∈K |P (z)| for P ∈ C[X].
We define the integer transfinite diameter of K by

t Z Z (K) = lim inf inf |P | 1 n ∞,K n ≥ 1 P ∈ Z Z[X] n → ∞ deg(P ) = n
Finally, if ϕ is a positive function defined on K, the ϕ-generalized integer transfinite diameter of K is defined by

t Z,ϕ (K) = lim inf inf sup |P (z)| 1 n ϕ(z) . n ≥ 1 P ∈ Z[X] z ∈ K n → ∞ deg(P ) = n
In the auxiliary function (1), we replace the coefficients c j by rational numbers a j /q where q is a positive integer such that q.c j is an integer for all 1 ≤ j ≤ J. Then we can write:

for y ∈ I , h(y) = c -g(y + 5) - t r log |U (y)| ≥ m (2)
where U = If we suppose that t is fixed, it is equivalent to find an effective upper bound for the weighted integer transfinite diameter over the interval I with the weight ϕ(y) = e -(c-g(y+5)) :

t Z Z,ϕ (I ) = lim inf inf sup |P (y)| t r ϕ(y) r ≥ 1 P ∈ Z Z[y] y ∈ I r → ∞ deg(P ) = r

Construction of the auxiliary functions

The problem is to find a set of "good "polynomials U j ,i.e., which gives the best value possible for the minimum m of the function h. We use here our recursive algorithm. This algorithm is based on an algorithm developed by Q. Wu [Wu] in 2003, which substitues the heuristic search of good polynomials by a systematic search of them. In 2009, we made two improvements to this previous algorithm in the use of the LLL algorithm. The idea is to get the polynomials U j by induction. For more details, see [F2]. Note that the auxiliary functions were then used to get lower bounds for the usual measures.

Here, we start with an arbitrary value of c (for example, c=5 for the trace) and k=3. Suppose that we have U 1 , U 2 , ..., U J . Then we use the semi-infinite linear programming (introduced in number theory by C. J. Smyth [S1]) to optimize h for this set of polynomials i.e., to get a minimum m > 0. We obtain the numbers c 1 , c 2 , ..., c J and h in the form (2) with t = (c-g(y+5)) ) (r+k)/t is as small as possible.

We apply LLL algorithm to the linear forms in a 0 ,. . ., a k

U (y i )R(y i )(e -(c-g(y i +5)) ) (r+k)/t
where y i are control points uniformly distributed in the interval I', including the points where h has its least local minima. We get a polynomial R whose factors R j are good candidates to enlarge the set of polynomials (U 1 , U 2 , ...,U J ). We only keep the polynomials R j which have a nonzero coefficient c j in the newly optimized auxiliary function h. After optimization, some previous polynomials U j may have a zero coefficient and are removed.

Remember that the upper bound thus obtained is equal to c or e c . So, the smaller the value c is, the better the upper bound will be. Therefore, at this step, we start to decrease the value c until the minimum m > 0 is as close as possible to 0. The process is then repeated with the new value of c i.e., we seek a polynomial R. We increase the value of k when any new good polynomial is found. In order to get the constants of our theorems, we take k from 3 to 20 successively.

The polynomials and the coefficients c j involved in the auxiliary functions are listed in the tables below.

Table 1: Polynomials expressed on (-5,0.8) and their coefficient involved in Theorem1

j c j U j 1 0.60788001 x 2 0.01969114 x + 1 3 0.27896186 x 2 + x -1 4 0.07143360 x 2 + 2x -2 5 0.12185385 x 3 + 2x 2 -x -1 6 0.01513001 x 3 + 3x 2 -4x + 1 7 0.12949995 x 4 + 3x 3 -2x 2 -2x + 1 8 0.01274926 x 4 + 3x 3 -x 2 -3x + 1 9 0.01030851
x 4 + 4x 3 -2x 2 -4x + 2 10 0.00347525 x 4 + 4x 3 -4x + 1 11 0.00616329 x 5 + 4x 4 -x 3 -5x 2 + x + 1 12 0.01615593 x 6 + 5x 5 -8x 3 + 4x -1 13 0.00208365 x 6 + 6x 5 + 6x 4 -10x 3 -5x 2 + 6x -1 14 0.00808357 x 7 + 6x 6 + 7x 5 -11x 4 -10x 3 + 9x 2 + x -1 15 0.00465891

x 7 + 7x 6 + 10x 5 -10x 4 -15x 3 + 7x 2 + 5x -2 16 0.00317086 2x 8 + 11x 7 + 7x 6 -27x 5 -8x 4 + 23x 3 -3x 2 -4x + 1 17 0.00183290 x 9 + 8x 8 + 15x 7 -13x 6 -36x 5 + 17x 4 + 22x 3 -11x 2 -2x + 1 18 0.00357286 x 9 + 8x 8 + 17x 7 -5x 6 -35x 5 + 3x 4 + 23x 3 -6x 2 -3x + 1 19 0.00090485 x 9 + 9x 8 + 22x 7 -2x 6 -46x 5 -2x 4 + 33x 3 -7x 2 -4x + 1 20 0.00062738 x 9 + 9x 8 + 22x 7 -2x 6 -48x 5 -6x 4 + 37x 3 -2x 2 -9x + 2 21 0.00093622 x 10 + 10x 9 + 30x 8 + 11x 7 -63x 6 -31x 5 + 51x 4 + 10x 3 -13x 2 -x + 1 22 0.00104114 x 10 + 10x 9 + 31x 8 + 18x 7 -54x 6 -45x 5 + 37x 4 + 23x 3 -11x 2 -3x + 1 23 0.00072434 2x 10 + 15x 9 + 24x 8 -37x 7 -69x 6 + 64x 5 + 44x 4 -50x 3 + 5x 2 + 5x -1 24 0.00001135 x 12 + 11x 11 + 38x 10 + 23x 9 -97x 8 -92x 7 + 121x 6 + 72x 5 -78x 4 -8x 3 + 13x 2 + x -1 25 0.00327455 x 12 + 11x 11 + 41x 10 + 45x 9 -60x 8 -129x 7 + 30x 6 + 116x 5 -15x 4 -43x 3 + 9x 2 + 4x -1 26
0.00128312 x 12 + 12x 11 + 48x 10 + 54x 9 -86x 8 -178x 7 + 65x 6 + 186x 5 -46x 4 -79x 3 + 24x 2 + 9x -3 27 0.00165324 x 13 + 11x 12 + 39x 11 + 29x 10 -97x 9 -125x 8 + 124x 7 + 142x 6 -115x 5 -52x 4 + 56x 3 -5x 2 -5x + 1 28 0.00023279 x 13 + 12x 12 + 47x 11 + 45x 10 -110x 9 -180x 8 + 127x 7 + 211x 6 -116x 5 -91x 4 + 61x 3 + 3x 2 -7x + 1 29 0.00088998 x 14 + 14x 13 + 70x 12 + 127x 11 -62x 10 -416x 9 -100x 8 + 555x 7 + 118x 6 -388x 5 + 121x 3 -20x 2 -12x + 3 30 0.00040193 x 16 + 15x 15 + 82x 14 + 172x 13 -43x 12 -630x 11 -365x 10 + 957x 9 + 663x 8 -873x 7 -426x 6 + 481x 5 + 86x 4 -130x 3 +6x 2 + 12x -2 31 0.00003509 x 16 + 15x 15 + 82x 14 + 173x 13 -33x 12 -597x 11 -341x 10 + 879x 9 + 548x 8 -787x 7 -275x 6 + 400x 5 + 9x 4 -79x 3 +10x 2 + 5x -1 32 0.00074353 x 16 + 15x 15 + 82x 14 + 174x 13 -25x 12 -581x 11 -356x 10 + 824x 9 + 570x 8 -717x 7 -316x 6 + 374x 5 + 38x 4 -88x 3 +11x 2 + 5x -1 33 0.00138686 x 16 + 15x 15 + 83x 14 + 183x 13 -2x 12 -584x 11 -428x 10 + 794x 9 + 660x 8 -674x 7 -372x 6 + 342x 5 + 63x 4 -76x 3 +7x -1 34 0.00062528 x 16 + 15x 15 + 83x 14 + 183x 13 -4x 12 -600x 11 -460x 10 + 818x 9 + 760x 8 -691x 7 -489x 6 + 371x 5 + 115x 4 -99x 3 -x 2 + 8x -1

x 3 + 4x 2 + 2x -2 10 0.00648598 x 4 + 3x 3 -x 2 -3x + 1 11 0.01491318 x 4 + 3x 3 -2x 2 -2x + 1 12 0.00159782 x 4 + 4x 3 + x 2 -5x + 1 13 0.00306567 x 4 + 4x 3 + 2x 2 -3x -1 14 0.00177842 x 5 + 5x 4 + 5x 3 -4x 2 -3x + 1 15 0.00245270 x 6 + 6x 5 + 9x 4 -2x 3 -8x 2 + 1
x 3 + 2x 2 -x -1 9 0.00515778 x 3 + 3x 2 -1 10 0.00953120 x 4 + 3x 3 -x 2 -3x + 1 11 0.01371676 x 4 + 3x 3 -2x 2 -2x + 1 12 0.00422687 x 4 + 4x 3 + x 2 -5x + 1 13 0.00656933 x 4 + 4x 3 + 2x 2 -3x -1 14 0.00541172 x 5 + 5x 4 + 5x 3 -4x 2 -3x + 1 15 0.00133178 x 6 + 6x 5 + 8x 4 -6x 3 -9x 2 + 3x + 1 16 0.00593261 x 6 + 6x 5 + 9x 4 -2x 3 -8x 2 + 1 17 0.00088740 x 7 + 8x 6 + 20x 5 + 11x 4 -16x 3 -11x 2 + 2x + 1 18 0.00032713 x 8 + 7x 7 + 14x 6 -20x 4 -3x 3 + 10x 2 -x -1 19 0.00153253 x 8 + 8x 7 + 19x 6 + 5x 5 -27x 4 -12x 3 + 13x 2 + 2x -1 20 0.00624530 x 8 + 8x 7 + 20x 6 + 10x 5 -21x 4 -15x 3 + 8x 2 + 3x -1 21 0.00038110 2x 8 + 12x 7 + 18x 6 -9x 5 -27x 4 + x 3 + 11x 2 -1 22 0.00063568 x 10 + 7x 9 + 12x 8 -12x 7 -39x 6 + 4x 5 + 39x 4 -x 3 -14x 2 + x + 1 23 0.00054001 x 10 + 8x 9 + 19x 8 + 2x 7 -41x 6 -24x 5 + 31x 4 + 17x 3 -11x 2 -2x + 1 24 0.00054553 x 10 + 9x 9 + 26x 8 + 14x 7 -47x 6 -40x 5 + 40x 4 + 18x 3 -14x 2 -x + 1 25 0.00078408 x 10 + 11x 9 + 44x 8 + 69x 7 -104x 5 -57x 4 + 47x 3 + 31x 2 -7x -4 26
0.00006017 2x 10 + 15x 9 + 33x 8 + 2x 7 -60x 6 -24x 5 + 41x 4 + 12x 3 -12x 2 -x + 1 27 0.00259968 x 11 + 10x 10 + 35x 9 + 42x 8 -23x 7 -79x 6 -17x 5 + 46x 4 + 15x 3 -12x 2 -2x + 1 28 0.00061682 x 11 + 11x 10 + 43x 9 + 60x 8 -27x 7 -127x 6 -31x 5 + 88x 4 + 24x 3 -25x 2 -3x + 2 29 0.00138347 x 11 + 11x 10 + 43x 9 + 61x 8 -20x 7 -113x 6 -31x 5 + 68x 4 + 19x 3 -16x 2 -2x + 1 30 0.00066958 x 11 + 11x 10 + 43x 9 + 61x 8 -20x 7 -113x 6 -31x 5 + 69x 4 + 22x 3 -18x 2 -4x + 2 31 0.00069381 x 11 + 11x 10 + 44x 9 + 68x 8 -7x 7 -118x 6 -57x 5 + 68x 4 + 36x 3 -18x 2 -4x + 1 32 0.00275776 x 12 + 12x 11 + 53x 10 + 95x 9 + 14x 8 -156x 7 -118x 6 + 79x 5 + 83x 4 -15x 3 -19x 2 + x + 1 33 0.00002970 x 14 + 14x 13 + 75x 12 + 177x 11 + 99x 10 -309x 9 -431x 8 + 175x 7 + 436x 6 -62x 5 -187x 4 + 30x 3 + 25x 2 -3x -1 34 0.00002671 x 15 + 15x 14 + 89x 13 + 251x 12 + 265x 11 -253x 10 -800x 9 -229x 8 + 737x 7 + 401x 6 -340x 5 -178x 4 + 82x 3 + 26x 2 -7x -1 35 0.00019284 2x 17 + 31x 16 + 190x 15 + 550x 14 + 564x 13 -761x 12 -2207x 11 -480x 10 + 2622x 9 + 1438x 8 -1608x 7 -1009x 6 + 573x 5 +303x 4 -113x 3 -36x 2 + 9x + 1 36 0.00051457 x 18 + 16x 17 + 102x 16 + 311x 15 + 350x 14 -438x 13 -1501x 12 -566x 11 + 1869x 10 + 1530x 9 -1147x 8 -1251x 7 + 423x 6 +483x 5 -109x 4 -86x 3 + 18x 2 + 5x -1 37 0.00085382 2x 18 + 31x 17 + 190x 16 + 548x 15 + 541x 14 -856x 13 -2346x 12 -385x 11 + 3052x 10 + 1553x 9 -2098x 8 -1213x 7 + 873x 6 +398x 5 -208x 4 -53x 3 + 22x 2 + 3x -1 38 0.00006742 2x 19 + 37x 18 + 285x 17 + 1149x 16 + 2373x 15 + 1292x 14 -4465x 13 -8383x 12 -209x 11 + 10965x 10 + 5593x 9 -7044x 8 -5210x 7 + 2727x 6 + 2169x 5 -715x 4 -431x 3 + 121x 2 + 32x -9

  |α i |) and Ω(α) = M(α) 1/d denotes the absolute Mahler measure of α. If M(α) = 1, then a classical theorem of Kronecker[K] tells us that α is a root of unity. It suggests the question: infα not a root of unity M(α) > 1 ? It is known as the "Lehmer's problem "and it is still open. Another formulation can be given as follows. Does there exist an absolute constant c > 0 such that: if M(α) > 1 then M(α) > 1 + c ?

a

  j deg U j and t = J j=1 c j deg U j (this formulation was introduced by J. P. Serre). Thus we seek a polynomial U ∈ Z Z[y] such that sup y∈I |U (y)| t/r e -(c-g(y+5)) ≤ e -m .

  l ∈ Z Z[y] such that sup y∈I |U (y)R(y)| t r+k (e -(c-g(y+5)) ) ≤ e -m , i.e., such that sup y∈I |U (y)R(y)|(e -

Table 2 :

 2 Polynomials expressed on (-5,0.8) and their coefficient involved in Theorem 2

	j	c j	U j
	1	0.13036736	x
	2	0.02568103	x + 1
	3	0.07351847	x 2 + x -1
	4	0.00563913	x 2 + 2x -1
	5	0.01599146	x 2 + 2x -2
	6	0.00040951	x 2 + 3x + 1
	7	0.03575081	x 3 + 2x 2 -x -1
	8	0.00347327	x 3 + 3x 2 -1
	9	0.00028418

Table 3 :

 3 Polynomials expressed on (-5,0.8) and their coefficient involved in Theorem 3

	j	c j	U j
	1	0.16929006	x
	2	0.04824241	x + 1
	3	0.00180977	x + 2
	4	0.09384986	x 2 + x -1
	5	0.01670291	x 2 + 2x -1
	6	0.02553475	x 2 + 2x -2
	7	0.00074233	x 2 + 3x + 1
	8	0.04999466