Marion Tharcis

Thierry Badel

Stéphane Jéol

Françoise Etienne Fleury

Françoise Méchin
email: francoise.mechin@insa-lyon.fr

High elongation thermoplastic Polyester-urethanes based on widely available diacid intermediates

Keywords: polyurethanes, polyester-polyol, 2-methylglutaric acid, elastomers, mechanical properties, crystallinity, elongation at break

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

INTRODUCTION

Thermoplastic polyurethanes and polyurethane-ureas, TPUs, are an important family of highperformance engineering thermoplastics and elastomers. Due to the versatility of their formulation traditionally based on three main components, i.e. a dihydroxy-terminated oligomer or "macrodiol" with a typical molar mass ranging from 600 to 4000 g.mol -1 (often polyether or polyester, and itself based on a wide variety of possible precursors), a diisocyanate and a short diol or diamine known as "chain extender", a huge range of materials with adjustable thermal or mechanical behaviors and processing methods can be obtained. But despite this theoretical adaptability, in practice TPU formulations are often based on a very limited number of precursors. For example, the diisocyanate is predominantly 4,4'-bis(isocyanatophenyl)methane, MDI, while 1,4-butanediol is most often used as the chain extender. As well, a great part of the TPUs commercialized worldwide are indeed based on a few polyester macrodiols, and for the most part derived either from polycaprolactone (e.g. biodegradable materials) or from the condensation of various diols with adipic acid. Yet over the years, lots of academic works were devoted to the preparation of TPUs from more or less modified crude vegetable oils containing ester groups. [1][2][3][4][5] More recently, soft TPUs were even obtained from polyester polyols based on purified dicarboxylic or hydroxylated-carboxylated molecules derived from triglycerides from seed oils; [6,7] but the possibility of industrial transposition of these syntheses often still needs further proof. Attempts were otherwise made to synthesize new TPUs from polyester polyols based on the renewable succinic acid, and to compare them with polyadipatebased counterparts. [8] Both TPU families were found to differ essentially by the degree of crystallinity of their soft segments, by their glass transition temperatures and by the nature and intensity of the physico-chemical interactions between soft and hard phases.

The production of polyurethanes accounts for ~5% of the total adipic acid consumption worldwide. Adipic acid, AA is traditionally produced from petroleum-based feedstocks, and more specifically from a 2-stage oxidation of cyclohexane. [9] However, various mixtures of aliphatic carboxylic acids are now easily available from the conversion of the corresponding nitriles obtained as by-products in the hydrocyanation of butadiene. [START_REF] Buisine | Rhodia Operations[END_REF]11] Among these, the diacid blend used in this work, Rhodiacid™-MGA, is a mixture of 3 chain isomers of adipic acid rich in 2-methylglutaric acid.

The use of this blend as the basis of new polyester polyols would represent a way of giving value to a current product of the polyamide industry. However to the best of our knowledge, in the academic literature no polyurethane materials were ever prepared and fully characterized from this kind of diacid blend that is only mentioned in a patent by BASF. [12] Indeed the effect of the presence of alkyl side groups in the polyester soft segments of polyurethane elastomers on their physical properties (mechanical or thermal behavior and hydrolysis resistance) was mainly studied by modifying the chemical structure of the starting diol, but scarcely by playing on the diacid structure. [13][14][15] Moreover a recent study proved that high molar mass thermoplastic polyesters based on MGA could display equivalent toughness together with higher values of elongation at failure when compared with counterparts based on linear aliphatic diacids and especially adipic acid. [16] Therefore in the present work, after synthesizing and thoroughly characterizing a series of polyester polyols from the novel diacid MGA, together with their AA-based analogs, linear segmented polyurethanes were obtained from their condensation with 4,4'-diphenylmethane diisocyanate and 1,4-butanediol. The thermal and dynamic and static mechanical properties of the resulting materials were then systematically compared.

EXPERIMENTAL Materials

The polyester polyols were synthesized by Solvay at lab-scale and then used as received, though after being thoroughly characterized. For polyurethane elastomer synthesis, they were reacted with pure 4,4'-diisocyanatodiphenylmethane, MDI, and 1,4-butanediol, BDO, both from Sigma Aldrich France. These precursors were used as received.

Thermoplastic Polyurethane Synthesis

Series of TPUs derived from all the above polyester polyols, MDI and BDO, and containing various amounts of hard segments, HS% were successively prepared (for each formulation 30, 40 or 50 wt% with , in which m i is the total mass of component i used for the synthesis). These materials were named as (name of the polyol)-X% with X the HS weight%.

In a typical synthesis (the following data refer to the sample MGA-BDO- conditions. In some cases (high HS%) the samples had to be initially pressed at higher temperature (180-200°C) for about 5 minutes before being cured at 110°C. Cylindrical small but thicker pieces (> 6 mm) were also molded to allow the measurement of Shore hardness with a hand durometer (AFNOR standard NF T46-052).

Size Exclusion Chromatography

All the prepared materials were first characterized by Size Exclusion Chromatography, SEC. A high molar mass is indeed required to measure representative mechanical properties for the different materials and to be able to compare samples with one another. SEC was carried out using either tetrahydrofurane, THF, at 35°C (for oligodiols) or dimethylformamide, DMF, + LiBr (0.05M) at 70°C (for oligodiols and TPUs) as eluent. The THF flow rate was 1.0 mL/min and that of DMF 0.8 mL/min. Analyses were run on a Viscotek chromatograph equipped with 3 Styragel HR 4E columns; detectors consisted of a differential refractometer VE3580 and a Dual T60A detector (combining a right angle light scattering detector and a viscosimeter), allowing the determination of absolute molar masses after evaluating dn/dc for all the samples. This was done in DMF for all the synthesized TPUs by injecting samples with 3 different concentrations. These measurements led to values ranging from 0.052 to 0.077 mL/g, with the highest data systematically associated with the highest HS%.

H Nuclear Magnetic Resonance

The chemical structure of the starting polyols was characterized by 1 H NMR spectroscopy performed on a Bruker DRX 400 spectrometer operating at 250 or 400 MHz. Samples were dissolved in CDCl 3 . In some cases, a few drops of trifluoroacetic acid were added in order to shift the signals of the labile protons.

Chemical Determinations

The precise knowledge of the hydroxyl number of a macrodiol, i OH , is essential for the proper synthesis of polyurethane materials, since an optimal stoichiometry is required to obtain high molar masses and good mechanical properties. In this work the values retrieved from NMR spectra were compared with those calculated from chemical determination. For this purpose, the conventional method using p-toluene sulfonyl isocyanate (TSI) (ASTM E1899-08) had to be somewhat adapted.

Here the polyols were first thoroughly dried under a vacuum at 80°C for at least 2 hours. They were then dissolved in anhydrous THF and reacted with a known excess of TSI (Aldrich) for 2 hours. A solution of N,N-dibutylamine in anhydrous THF (~2M) was then added and the mixture was allowed to react 15 more minutes. The residual amine was finally back titrated using a 1N hydrochloric acid solution. The accuracy of this method was checked with simple aliphatic alcohols such as 1,4butanediol.

The acid numbers, i COOH were determined by the supplier after an acido-basic titration.

Finally, a chemical determination (AFNOR standard NF T52-132) was sometimes used to check MDI purity or to monitor the consumption of NCO groups during the polyol-MDI addition reaction. For this purpose samples were regularly collected, weighed and reacted with a known excess of N,N-di(n-butylamine). The residual amine was finally back titrated with 1N HCl, and the resulting isocyanate contents were expressed by the NCO percentage:

(1)

Differential Scanning Calorimetry

The thermal behavior of the various polyols and polyurethane materials was studied by differential scanning calorimetry, DSC, using a TA Q22 apparatus. Samples (~4-7 mg) were heated from -100 to 200°C (polyols) or -80 to 250°C (polyurethanes) under nitrogen. Two heating ramps (20K/min) were successively performed with intermediate cooling at the same rate.

Thermogravimetric Analysis

Small samples (m < 10 mg) were heated at 10K/min from room temperature up to 580°C

under air or nitrogen. The analyses were run on a Q500 apparatus from TA Instruments.

Viscosity Measurements

The viscosity of the polyols was measured with an Anton Paar MCR301 rheometer, using a cone/plate geometry. Isothermal time sweep experiments were run in the linear domain of the samples (strain 2 %) at 2 or 3 different temperatures: 25°C (only for amorphous liquid samples), 50 and 80°C.

Tensile Testing

Uniaxial tensile measurements were carried out on a MTS 2/M electromechanical testing system at 22 ± 1°C and 50 ± 5 % relative humidity. H3-type dumb-bell specimens (AFNOR T51-034)

were cut from the cast plates with a 10 mm nominal gauge length and an approximately 1.3 x 4 mm cross-sectional area. These samples were tested with a 20 mm/min constant speed, with the help of an extensometer and pneumatic grips, in order to determine Young's modulus, E, the ultimate stress,  R , and ultimate elongation,  R , and finally the secant moduli for 100% or 300% strain. For each material, 3 to 5 specimens were tested.

For hysteresis measurements, the test consisted of 10 tensile loads from 0 to either 100%, 400% or 800% deformation with a crosshead speed of 50 mm/min.

Dynamic Mechanical Thermal Analysis (DMTA)

DMTA experiments were carried out on a Rheometrics Solid Analyzer RSA II at 1Hz with 0.1 % tensile strain. The heating rate was 3°C/min for the temperature range from -100 to +225°C. The size of the samples cut from the cast plates was approximately 40x8x1.3 mm.

RESULTS AND DISCUSSION

Polyester Macrodiols Synthesis

A typical synthesis can be described as follows: both monomers, the diol and the diacid, were poured in a glass reactor equipped with a mechanical stirrer with vacuum/nitrogen inlet and a distillation column followed by a condenser. The diol/diacid ratio was determined considering the final molar mass expected after full conversion, always with an excess of hydroxyl functions. For instance, for the polyol AA-1,6-hexanediol (HDO) targeting 1000g/mol, the molar ratio diol/diacid = 1.25 was fixed. Then the nitrogen inlet was opened and the temperature was increased progressively to 140°C, with continuous stirring, and kept for 1h at this temperature. Condensation started. The mixture was then slowly heated up to 240°C and kept at this temperature for 3h. No catalyst was used. Then the heater was stopped and the adduct was let to cool down to room temperature. The day after the temperature was progressively increased back to 240°C under nitrogen, then nitrogen was stopped and vacuum was set for 4h 30min. The heater was stopped, the vacuum was stopped and replaced by nitrogen flow and the adduct was allowed to cool down to about 100°C and finally poured in a glass bottle.

Physico-Chemical Characterization of the Polyester Macrodiols

The new diacid used for the polyester polyols synthesis was Rhodiacid™-MGA. This proprietary blend is obtained by conversion of a byproduct in the synthesis of polyamide 6.6 through the hydrolysis of 2-methylglutaronitrile, possibly via a cyclic imide intermediate. [START_REF] Buisine | Rhodia Operations[END_REF]11] Its main components are indicated in Scheme 1. based formulation, the corresponding oligomer based on pure adipic acid was also prepared for further comparisons. Hereafter, oligomers will be named from the starting diacid and diol, and from their molar mass as follows: AA-BDO-1 is the polyester oligomer prepared from adipic acid and 1,4butanediol with an expected molar mass around 1000 g.mol -1 ; MGA-MPDO-2 is the polyester polyol obtained from Rhodiacid™-MGA and "MPDO" with an expected molar mass around 2000 g.mol -1 .

All the polyester polyols were characterized by 1 H NMR in CDCl 3 , and their hydroxyl number was also systematically determined by chemical titration.

The NMR spectra allowed at first to verify the assumed chemical structure. A typical spectrum obtained for MGA-BDO-1 is displayed in Figure 1. M n = (DP n -1)*M constitutive unit + M diol . In the case of MGA-and MPDO-based polyesters, the attributions are more complicated and a matrix calculation has to be used but the data can still be resolved. All the results are listed in Table 1.

In fact, all the chain ends are not 100% hydroxylated; some of them bear COOH groups as shown by the measured i COOH values (see Table 1). Therefore, the above calculated values for M n are a little overestimated.

Another way of calculating M n relies on the chemical determination of end groups. Assuming there are only OH and COOH terminal groups, M n is given by Equation ( 2):

(2)

The values found by titration for the hydroxyl numbers of all the polyols are listed in Table 1, together with the resulting molar masses. For some samples, the amount of water was also controlled but the amounts found were always quite low (Table 1).

Under the ordinary conditions used for polyurethane synthesis, in particular those used in this work (110°C, see below) and in the absence of any suitable catalyst, carboxylic acids are very scarcely reactive towards isocyanate groups. [17,18] However the maximal "degree of polymerization", P, that can be calculated for every final segmented polyurethane from the proportion between OH and COOH groups is always high enough to ensure appropriate molar masses for the proper comparison of thermal and mechanical properties. P values (i.e. the number of [macrodioldiisocyanate-extender-diisocyanate] units in the chain) can be calculated from Equation ( 3):

(3)

where N 0 stands for the number of monomer molecules in the initial mixture, and N RB is the number of monofunctional molecules. The results displayed in Table 1 show that the lowest calculated value for P is 46.

It also appears that all the M n values of the prepared polyols are a little higher than expected, especially for the BDO-based samples with the higher molar mass (closer to 3000 than 2000 g.mol -1 ). Therefore these 2 samples were finally named AA-BDO-3 and MGA-BDO-3.

By 1 H NMR and size exclusion chromatography, the presence of a small amount of residual starting diol (e.g. 1,4-BDO) was evidenced. But for every polyester polyol this amount was always less than 1%, and as this diol also takes part in the reaction with the diisocyanate no attempt was made to get rid of it.

These polyols were then characterized with respect to their thermal behavior using differential scanning calorimetry. Two different behaviors were observed. Liquid samples were indeed entirely amorphous and displayed only one glass transition at low temperature. In contrast, some polyols were waxy at room temperature and turned out to be semi-crystalline, with a low T g (varying from -71 to -56°C) and a melting and/or crystallization temperature around 25-50°C. More precisely, all the samples derived from linear monomers with no side group (AA-BDO, AA-HDO) showed this kind of behavior, whereas quite logically those bearing methyl and ethyl side groups (AA-MPDO, MGA-BDO, MGA-HDO, MGA-MPDO) were perfectly amorphous. This can be viewed as a first advantage of MGA-based polyester polyols since they do not necessarily require to be heated when processed for polyurethane synthesis (particularly by casting). All the DSC results are summed up in Table 2.

Figure 2. Thermogravimetric analysis of the prepared polyols (10°C/min, air)

By thermogravimetric analysis (TGA), the temperature associated with a 5 % weight loss, T d5% could be measured and was then plotted vs M n as a function of the nature of the diol or diacids. The results are shown in Figure 2. It appears that the thermal stability of the polyesters increases with their molar mass, suggesting that the first degradation reactions are related to the chain ends; and that it is quite independent from the nature of the starting materials. The entire data lie on the same curve, except for two points that were plotted against overestimated M n values determined from NMR, and would probably fall on the same line if their real M n could have been determined from titration. Finally, the complex viscosities were measured at 3 different temperatures, whenever possible. As seen with DSC experiments, the samples based on precursors without side groups were partially crystallized at room temperature and their viscosity could not be measured at 25°C. But all the polyesters could be characterized at 50 and 80°C, and the results are given in Figure 3 as a function of the molar mass evaluated from titration. For all the liquid samples, Newtonian behaviors were always observed. While no special variation was observed with respect to the nature of the starting diol, it appears that for a given molar mass and at both studied temperatures, the viscosity is always slightly lower for MGA-based samples than for their AA-based counterpart. This can once again be interesting as far as processing is concerned, since converters could benefit from more limited heating levels using MGA derivatives.

Thermoplastic Polyurethane Synthesis

To ensure that the first part of the reaction was complete (no remaining hydroxyl groups and a constant NCO concentration), a chemical titration of the residual NCO groups was sometimes used to monitor isocyanate conversion. As shown in Figure 4, it appeared that in any case 30 minutes was the optimal time for a complete conversion of all the hydroxyls, without the loss of isocyanate functions through side reactions. In this particular example the expected final conversion for NCO groups was 24 and was reached a er 0 min, while the side reac ons apparently began only a er 80 min. Therefore 30 minutes was finally the time chosen for the addition of the chain extender BDO for all the syntheses.

Figure 4. Evolution of the NCO % () and of the corresponding NCO conversion () during the addition reaction between MDI and MGA-BDO-2 at 80°C under nitrogen ([NCO] 0 /[OH] 0 = 4.15, i.e. expected final conversion = 24 %).

In order to compensate for the possible loss of some NCO functions by side reactions, the overall NCO/OH target ratio was always 1.03. The precise compositions of all the synthesized materials are available in the Supporting Information (Table S1).

Thermoplastic Polyurethane Characterization

Molar Masses

The results obtained for the molar masses are shown in Table 3. Almost all the M n values are higher than 30,000 g.mol -1 (the lowest 2 still higher than 20,000 g.mol -1 ), which is sufficient to allow the correct measurement of static and dynamic mechanical properties and more generally ensures a proper interpretation of all the physical data. These high molar masses also confirm that an appropriate stoichiometry was respected for all the syntheses and therefore that the results of the chemical titrations were quite reliable.

Thermal Transitions

The thermal behaviors of the segmented polyurethanes were then examined by DSC and TGA. All the results are displayed in Table 4, and typical thermograms are shown in Figure 5.

Figure 5. Typical DSC thermograms obtained for a) MGA-BDO-1 series; b) AA-BDO-3 series; 1 st scans in the left column, 2 nd scans in the right column

In DSC, 2 successive heating scans were performed with intermediate cooling at the same rate (20°C/min). The behavior is that classically observed for segmented polyurethanes with successive phenomena associated respectively with the soft and hard domains. In such materials, a soft phase consisting mainly of the soft segments with sometimes dissolved hard segments, separated amorphous hard segments, separated more or less highly ordered hard segments and even crystalline hard segments can coexist at room temperature. [19,[START_REF] Macosko | Fundamentals of reaction injection molding[END_REF] Here in both heating runs, all the materials show a clear T g at low temperature that can be associated with the polyester soft domains. In the first scan, this T g lies between -50 and -24°C, depending on the sample. In the second scan, the value systematically increases with respect to the first scan, revealing that the hard segments that partially dissolved in the soft phase above the order-disorder transition temperature (or microphase mixing temperature MMT, see below) [START_REF] Saiani | [END_REF]22] during the first heating run did not have enough time to phase separate during the cooling stage; therefore the soft phase is less pure at the beginning of the second heating ramp and the observed T g is then higher. In agreement with this, the slight phenomenon detected in the first scan at 40-50°C (see Fig. 5a), that could be associated with the T g of separated amorphous MDI-BDO hard segments (or separated amorphous domains very rich in hard segments) also disappears in the second scan; this was then confirmed by DMA (see below).

Knowing that the value of T g globally reflects the degree of purity of the soft domains and/or the amount of dissolved HS, [23] several comparisons can be made. First when comparing 3 samples based on the same polyester diol but with increasing HS %, for almost every series T g regularly increases. This can be attributed to the presence of increasing amounts of dissolved MDI-BDO hard segments in the soft domains (for a pure MDI-BDO polyurethane with a high molar mass, T g = 107°C [24] ). The values of T g recorded during the second scans are usually in the same order as in the corresponding first scans.

For the TPUs derived from AA-diol-1 type polyester diols, the observed T g is about the same for BDO and MPDO-based macrodiols whereas it is slightly higher for the samples with a polyester based on HDO. The same holds for the T g recorded during the second scans, and suggests that the AA-HDO-1 polyester chain is slightly more miscible with the MDI-BDO hard segments than those based on the other 2 diols.

For the series of TPUs derived from MGA-diol-1 polyesters, almost the same conclusion can be drawn, except that the case of the MPDO-based sample is somewhat intermediate between BDO and HDO.

When comparing series based on an AA-diol-1 type macrodiol with the materials derived from the corresponding MGA-diol-1, slightly higher T g s seem to be found in the case of MGA, suggesting that the latter would slightly increase the miscibility with MDI-BDO hard segments.

However this trend is rather subtle, and is not observed in the case of BDO-based polyesters.

Finally the comparison between samples based on AA-BDO-1 and -3, or between MGA-BDO-1 and -3, clearly shows that the longer starting macrodiols lead to better phase-separated polyurethanes with a much lower T g of the soft domains, even in the second scan for which T g increases by about only 5°C, i.e. much less than for the materials derived from shorter macrodiols.

The immiscibility between soft and hard segments increases with their length. This was already observed in the past for other segmented polyurethanes. [25] Moreover, in the case of AA-BDO-3, the soft chains of the macrodiol are sufficiently long to be able to partially crystallize within the TPU. For the samples derived from this particular diol a melting peak is clearly visible during the first heating scan in the range 36-43°C (see Figure 5). In addition the material with the highest content in soft domains, AA-BDO-3-30%, even displays a crystallization followed by a melting peak during the second heating scan, denoting the strong tendency of the soft phase to crystallize for this particular formulation. For the other AA-BDO-3 samples though, the cooling rate is probably too high to allow this re-crystallization, especially as part of the hard segments remain dissolved in the soft phase as confirmed by the increase in T g , and no melting of the soft phase can be observed in the 2 nd scan.

Apart from these transitions related to the soft phase, endothermic peaks are also detected at high temperature for all the materials and can be associated with the polyurethane hard domains.

In the literature, segmented polyurethanes based on MDI-BDO hard segments have been widely studied, starting with the works of Cooper, [26][27][28] or Koberstein [19,29,30] and still very recently by Li. [31] DSC annealing experiments were combined with WAXD, AFM, FTIR or more recently with movingwindow two-dimensional correlation spectroscopy to identify the observed transitions. Multiple endotherms detected at high temperatures, up to 250°C, can be associated successively with the disruption of ordered microdomain structure at the micro-phase mixing transition (MMT), and for those at still higher temperatures to the melting of microcrystalline HS. From Koberstein's experiments, for all annealing conditions the disruption of the pseudo-ordered hard MDI-BDO microdomains precedes melting of the microcrystalline HS.

In this work, melting peaks were observed up to 224°C (for samples with 50wt% HS) in the first scans (polymer annealed at 110°C). Note that in the past, the presence of MDI-BDO spherulites melting at 224-226°C has been experimentally clearly evidenced in HS-rich segmented polyetherurethanes. [32] However other authors associated this high melting peak with the solvent processing used for this particular sample preparation, [33] but they nevertheless pointed out a melting endotherm around 215°C for analogous bulk-processed polyurethanes. [30] Here within a series, samples with the highest HS% also display endotherms at the highest temperature, denoting increasing amounts of better ordered hard domains. Moreover the comparison between corresponding polyurethanes derived either from diacid-BDO-1 or -3 macrodiols shows that those with the longest soft segments display the most highly ordered hard phase, with melting peaks at higher temperature and higher melting enthalpies, and consistently with a purer soft phase (see above). Quite logically, using a longer soft segment while keeping the same weight proportion of hard segments must induce higher average lengths for these HS as well; this can explain their higher degree of ordering. [25] These samples definitely have a higher degree of microphase separation than those based on the shorter macrodiols.

In the second scans, the increase in the T g of the soft phase is logically accompanied by a decrease in the number of melting peaks of the hard domains and in their temperature.

Thermal Stability

The thermal stability of the prepared TPUs was assessed through TGA measurements performed in nitrogen and in air. Typical curves are displayed in the Supporting Information (Figure S1) and systematically show 2 main degradations at ca. 350°C and 420°C. The values of T d5% (listed in Table 4) are not influenced by the length of the polyol, yet slightly decrease when increasing the HS proportion; this seems rather logical since the thermally reversible urethane bonds should be the first to degrade when heating the material. [34][35][36] The esters and other functions related to the soft segment are expected to degrade at higher temperature (2 nd peak on the derivative curves at ≈ 420°C that indeed increases with the relative amount of soft segment). Finally if T d5% does not seem to depend on the nature of the diol used to synthesize the starting polyester macrodiol (BDO, HDO or MPDO), it is systematically slightly lower (by about 5-10°C) for a sample based on MGA than for its counterpart derived from AA, at least in nitrogen. In air though, this discrepancy seems to fade.

Dynamic Mechanical Behavior

Samples were systematically analyzed from -100°C to 150°C in tensile mode. Typical curves are depicted in Figure 6a-c. First, Figure 6a displays the thermomechanical behavior of sample MGA-HDO-1 50% that is representative of all the studied TPUs, i.e. elastomeric materials with a single principal mechanical relaxation that tends to broaden towards high temperatures. This main relaxation is associated with the glass transition of the soft domains, while the small shoulder around 50°C could be related to the glass transition of some separated amorphous hard domains, although this phenomenon is not always clearly visible in the DSC thermograms. As these hard segments begin to disorder and melt, the conservation modulus E' finally initiates a sharp decrease above 150°C.

In Figure 6b, E' curves recorded for all the BDO-1 samples with varying HS % and based on both diacids were superimposed; the same was done for tan curves in Figure 6c. Figure 6b shows that very close thermomechanical behaviors are observed when comparing each AA-based sample with its MGA-based counterpart. For both diacids, the rubbery modulus is strongly dependent on the amount of hard segments [37] , in agreement with the fact that the latter act as physical crosslinks and reinforcing, covalently linked fillers. From the superimposition of tan curves, other observations can be made (Figure 6c): first, low amounts of HS are associated with a high and narrow mechanical transition, whereas it becomes broader (especially towards high temperatures) and displays lower and lower amplitude as HS% increases. This phenomenon can be related to the increase in the elastic modulus with increasing amounts of hard segments, and reflects the associated decrease in mobility of the soft segments as their moves are hampered by more and more physical crosslinks. For a given HS%, this amplitude is almost equal for the AA-and the MGA-based sample. However the temperature at the relaxation maximum is slightly higher for the MGA-based polyurethane, which might indicate a slightly higher miscibility between soft and hard segments in this case.

Figure 6. Dynamic mechanical analysis of segmented PU samples: a) E', E" and tan curves recorded for MGA-HDO-1 50%; b) E' curves for BDO-1 series with various HS % : AA (---); MGA(-); c) tan curves for BDO

Hardness

Table 4 shows that the values of Shore A hardness measured at room temperature for AA-or MGA-based samples are systematically remarkably close for a given composition, with one exception. Very generally the observed Shore A hardness increases when increasing the weight proportion of hard segments HS%, consistently with the elastic moduli of these materials (Figure 6b).

Samples of the BDO-3 series, i.e. those based on the longer soft segments, indeed display different tendencies. More precisely in this case, whereas the MGA-based samples still show a regularly increasing hardness from 30 to 50% HS, the 3 AA-based polyurethanes show the same, high Shore hardness (~95 ShA) for all the HS % values. This simply reflects the fact that in this case the soft segment is long enough to be able to crystallize, as revealed by the additional melting observed around 35-40°C by DSC (see Table 4 and Figure 5b), and contrary to the MGA-based polyester. This is thus once again a real advantage of MGA, which allows the preparation of elastomers with adjustable hardness over a wider range of composition, i.e. over a wider range of mechanical characteristics at high temperature.

Tensile Testing

The mechanical behavior of the TPUs was finally evaluated by tensile tests on specimens cut from plates. Representative stress-strain curves are shown in Figure 7a to c, for all the samples based on macrodiols with M n ≈ 1000 g.mol -1 . It appears that in the first 150-200 % elongation, the curves associated with each AA-based sample and with its MGA-based counterpart are more or less superimposed. Beyond this elongation, 2 situations are encountered: for samples derived from BDO or HDO, MGA-based samples keep on elongating whereas AA-samples display a strong strain hardening attributed to the partial crystallization of the soft segments. This results in much higher ultimate elongations for the TPUs prepared from MGA-based macrodiols, compared to AA-based samples. In contrast, samples based on "MPDO" (that did not display any crystallinity in their DSC curves) are not prone to strain hardening, and even those derived from AA show a purely elastomeric behavior. In this case it is very difficult to differentiate AA-and MGA-based materials.

For all the studied series, the values of Young moduli E evaluated for AA-based samples and for their MGA-based counterpart are identical (see Fig. S7 in Supplementary Material). Other parameters have to be considered in order to be able to differentiate both TPU families. In this respect, the ultimate stress  B and elongation at break  B can be plotted vs the amount of hard segments for all series. This was done in Figure 8 & 9. It appears first that for both starting diacids,  B increases regularly with HS%. The observed values are higher for AA-based samples when the starting diol is BDO, whereas they become more comparable for both acids in the case of HDO and almost identical in the case of "MPDO"; this is consistent with the strain hardening phenomenon reported above for the crystallizable materials. In contrast, the elongation at break  B is systematically higher for TPUs derived from MGA, as opposed to AA (see Figure 9). This was also observed on high-molar mass thermoplastic MGA-based vs AA-based polyesters. [16] Quite logically here, this parameter regularly decreases when increasing HS%, since more and more physical crosslinks due to hydrogen bonding between hard segments are present in the materials, and both curves even tend to merge as the composition gets closer to that associated with phase inversion between soft and hard phases. [38] But the phenomenon is particularly marked for softer samples with lower amounts of hard segments, and constitutes a real advantage for MGA-derived samples over the classical materials obtained from AA-based polyester polyols, since all the other mechanical characteristics are comparable.

Finally samples with 40 wt% HS were subjected to 10 consecutive tensile cycles (no delay between cycles) with variable maximal strain (100, 400 or 800%) at 50 mm/min. Typical curves recorded for MGA-BDO-1 40% are displayed in Figure 10a. During the first loading stage, all curves superimpose on a single "master curve". After unloading and once the stress value reaches zero, a residual strain is observed that increases with the maximum strain or with the number of cycles. This residual strain reflects the partially viscoplastic character of the studied materials. Some stress softening is also observed, with the stress at the maximum strain regularly decreasing as the number of cycles increases ("Mullins effect, see also below). [39] In the literature, and following the pioneering work from Kilian et al, [40] such phenomena were attributed to a change in the polyurethane microstructure occurring during the first loading cycle (often named "conditioning cycle" [41] ): some hard segments are assumed to be expelled from the hard domains, reducing their volume fraction and increasing that of the soft phase. [39] The new soft phase coupled with the remaining hard domains can be seen as a more compliant material dominating the following deformations. [42] Note that the strain hardening phenomenon observed for AA-based samples is clearly highlighted when comparing the first load-unload cycles for the various elastomers (see Fig. S8 in Supplementary Material), above ca. 300% strain.

Figure 11. Energy of the elastic hysteresis obtained for various AA-BDO-1 and MGA-BDO-1 based polyurethanes undergoing tensile load-unload cycles with maximal nominal strain 400%.

Figure 10b shows the values of residual strain measured after the 1 st cycle (exactly the same tendency is observed after 10 cycles) for several formulations based either on AA or MGA. Two main features can be deduced from this figure: first, samples based on MGA show a slightly higher residual strain compared to their AA-based counterpart; this is apparently not due to the molar masses that are all comparable; all 4 samples also have about the same hardness (from 79 to 82 ShA). -Then, samples based on HDO-polyester polyols clearly display lower residual strains. The main difference deduced from previous experiments was that these samples had a lower degree of microphase separation compared to those based on BDO-polyester polyols. Referring to the above explanations for stress softening and residual strain, in the case of this less-separated material more HS are already present in the soft phase before the 1 st loading cycle, and consequently maybe a lower proportion can be expelled from the hard phase to the soft phase during this 1 st cycle. Therefore the structure of the material would not change as much between the 1 st and 2 nd loading cycle. This is consistent with the fact that while the first loading curves are quite different between BDO-polyester polyol and HDO-polyester polyol based polyurethanes (HDO curve lower), there is almost no difference between them for the following loading cycles. Both types of material would thus be brought to a rather similar microstructure by their first tensile loading.

Work imputs and hystereses were also evaluated for all the tested samples (see Table 5). For example Figure 11 shows the comparison for a series of samples based on MGA-BDO and AA-BDO macrodiols tested to a maximal nominal strain 400%. Several key features can be pointed out.

First, the classical Mullins effect [42,43] is clearly evidenced for all the samples, with an important decrease in the cycle hysteresis due to the stress softening phenomenon between the first and second loading cycles. This is attributed to some restructuring with irreversible deformation of the hard segments accompanying the stretching of the soft segments [41] . For all the studied samples, the dissipated work keeps decreasing slightly during the following cycles and finally stabilizes (see Fig. S9 in Supplementary Material).

Then, the dissipated work logically increases with the amount of hard segments, in agreement with the literature [41] . The phenomenon is quite clear for the first cycle, whereas it is less pronounced for the following cycles that do not induce significant additional restructuring of the hard segments.

Finally, the energy hysteresis during the first cycle is systematically higher for the MGA-based samples than for their AA-based counterpart. However the order is often reversed for all the following cycles, starting from the second cycle. As both series are made of similar hard segments, this could be related to the interactions between soft and hard domains which could be initially higher for the MGA series; in this respect the stretching of the soft segments would induce more deformation and breakup of the hard domains during the "conditioning cycle" [41] .

CONCLUSIONS

MGA, a new diacid and abundant indirect product of the polyamide industry, was reacted with several diols to prepare series of new polyester polyols with adjustable molar masses. These polyols were then considered as innovative precursors for segmented polyurethanes. From the comparison with analogous hydroxylated oligomers or polyurethane materials based on adipic acid instead of MGA, several benefits can be attributed to the MGA derivatives. First concerning the starting polyols, MGA-based compounds turned out to be systematically amorphous, unlike polyadipates that are often semi-crystalline and therefore need to be heated to be processed; MGA derivatives also show a lower viscosity than their polyadipate counterpart with the same molar mass and thus can be considered more easily processable in the manufacture of segmented polyurethanes, especially by casting.

Segmented polyurethanes with varying hard segment content were then obtained from analogous series of AA-and MGA-based polyester polyols with different compositions or lengths, after bulk polyaddition with MDI and 1,4-butanediol. For all these formulations, elastomers with high molar masses were always obtained and could be rather easily processed in the form of homogeneous plates. Even after reaction, once again AA-based soft segments retained their tendency to crystallize within the polyurethane material, either spontaneously (longer macrodiols) or when submitted to a mechanical stress (strain hardening), while segmented polyurethanes derived from MGA-polyester polyols always showed perfectly amorphous soft domains. Therefore although most of the prepared materials were finally equivalent between MGA and adipic acid derivatives, the MGA-based TPUs generally displayed a more adjustable hardness, together with a much higher elongation at break.

FIGURE CAPTIONS

  molten and heated up to 80°C; afterwards, the proper amount of polyol ([NCO]/[OH] > 1) (MGA-BDO-1, 32.81 g) was added and the mixture allowed to react. It usually became homogeneous and transparent within 5-10 min at 80°C. After 30 minutes stirring (see § 3.3), the flask was removed from the oil bath 3-5 min before the previously weighed proper amount of BDO (4.15 g) could be added with the help of a syringe and the increasingly viscous mixture was homogenized by vigorous stirring for 1-2 min. It was then degassed under vacuum and then rapidly cast in a previously warmed flat metal mold. Homogeneous plates with a regular thickness about 1.3 mm were obtained by curing under a hot press (4 tons, 110°C) for 18 h and cooling back to room temperature under ambient

Scheme 1 .

 1 Scheme 1. Constituents of Rhodiacid™-MGA

Figure 1 : 1 H

 11 Figure 1: 1 H NMR spectrum of sample MGA-BDO-1 (23°C, CDCl 3 , 250 MHz)

Figure 3 .

 3 Figure 3. Viscosities of the prepared polyester polyols at a) 50°C ; b) 80°C

-

  -1 series : : AA, 30%; : AA, 40%; : AA, 50%; : MGA, 30%; : MGA, 40%; : MGA, 50%

Figure 7 .Figure 8 .Figure 9 .

 789 Figure 7. Stress-strain curves for TPU series based on various macrodiols with M ≈ 1000 g.mol -1 ; (a) starting diol: BDO; (b) starting diol: HDO; (c) starting diol: "MPDO"

Figure 10 .

 10 Figure 10. (a) Example of stress-strain curves and hysteresis observed for the sample MGA-BDO-1 40% subjected to various maximal strains (100, 200, 400 or 800%) at 50 mm/min; (-) : first cycle, (---): 10 th cycle; (b) residual strain (arrows in the 1 st graph) observed after the 1 st cycle for various formulations (40% HS)

FIGURE 1 : 1 HFIGURE 2 :FIGURE 3 :FIGURE 4 :

 11234 FIGURE 1 : 1 H NMR spectrum of sample MGA-BDO-1 (23°C, CDCl 3 , 250 MHz)

FIGURE 5 :FIGURE 6 :FIGURE 7 :FIGURE 8 :FIGURE 9 :FIGURE 11 :

 5678911 FIGURE 5 : Typical DSC thermograms obtained for a) the MGA-BDO-1 series; b) the AA-BDO-3 series

  Fischer determination b) conventional calibration according to polystyrene standards c) in THF d) In DMF/LiBr

Table 3 . Absolute molar masses of the TPUs determined from SEC/triple detection

 3

	polyester starting diol	polyester polyol	M n polyol (titration, g.mol -1 )	HS %	M n (kg.mol -1 )	M w (kg.mol -1 )
				30	50.0	118.0
		AA-BDO-1	1250	40	45.5	101.5
				50	45.4	116.0
				30	36.2	101.0
		AA-BDO-3	3550	40	31.1	84.0
	BDO			50 30	45.7 33.5	114.5 71.0
		MGA-BDO-1	1400	40	37.7	88.5
				50	45.8	122.0
				30	21.1	47.2
		MGA-BDO-3	2640	40	28.4	58.8
				50	36.2	68.3
				30	58.3	114.3
		AA-HDO-1	1160	40	43.5	80.9
	HDO			50 30	66.5 47.4	123.0 90.0
		MGA-HDO-1	1140	40	43.0	87.2
				50	61.4	173.5
				30	50.0	181
		AA-MPDO-1	1750	40	40.6	130.6
	MPDO			50 30	40.8 54.0	121.1 155.0
		MGA-MPDO-1	1690	40	36.1	84.3
				50	43.6	118.0

Table 4 . Thermal transitions and degradation temperatures of the segmented polyurethanes

 4

	TPU	Hardness (Shore A)	T g 1 st scan (°C)

Table 5 : Hysteresis during tensile testing for the first, second and tenth loading/unloading cycles to nominal strains 100, 400 or 800%.

 5 From DSC measurements (1 st and 2 nd heating scans at 20°C/min) b) From dynamic mechanical analysis (heating scan at 3°C/min) c) From thermogravimetric analysis (10°C/min); T d5% corresponds to a 5% weight loss d) Not determined

		64 ± 1	-38	-22	161	134	-30	325
		82 ± 1	-39	-26	153/174	163	-26	324
		94 ± 1	-32	-14	139/216	175	-11	320
	MGA-BDO-1-30%	62 ± 2	-38	-24	156	143	-25	312
	MGA-BDO-1-40%	80 ± 2	-33	-25	156/203	169	-22	314
	MGA-BDO-1-50%	94 ± 1	-30	-21	156/190/221	180	-14	313
	AA-BDO-3-30% AA-BDO-3-40%	94 ± 1 96 ± 1	-50 -45	-45 -40	38/177 43/205	40/174 179	ND d)	327 322
	AA-BDO-3-50%	96 ± 1	-47	-32	36/190/224	184/210		320
	MGA-BDO-3-30% MGA-BDO-3-40%	66 ± 1 86 ± 2	-43 -46	-38 -40	167 215	169 189/205	ND d)	318 311
	MGA-BDO-3-50%	95 ± 1	-44	-38	216/220	185/207/220		312
	AA-HDO-1-30%	59 ± 2	-33	-19	-	114 (?)	-21	318
	AA-HDO-1-40%	79 ± 1	-28	-8	164/183	145	-16	319
	AA-HDO-1-50%	93 ± 1	-24	-6	158/200/207	171	-7	314
	MGA-HDO-1-30%	59 ± 2	-30	-17	-	-	-19	313
	MGA-HDO-1-40%	79 ± 1	-28	-14	162/185	151	-17	313
	MGA-HDO-1-50%	91 ± 1	-24	-7	158/207	171	-8	304
	AA-MPDO-1-30%	57 ± 2	-39	-24	173 (?)	151	-30	319
	AA-MPDO-1-40%	79 ± 2	-38	-29	165/206	172	-29	315
	AA-MPDO-1-50%	92 ± 1	-36	-18	170/203/212	182	-23	317
	MGA-MPDO-1-30%	58 ± 2	-33	-20	-	-	-23	312
	MGA-MPDO-1-40%	79 ± 2	-30	-20	160/180/200	169	-18	310
	MGA-MPDO-1-50%	92 ± 1	-27	-12	208/215	183	-19	312
	a)

a) Integration of the whole load-unload sequence

ACKNOWLEDGEMENTS

The authors thank the NMR Polymer Center of the "Institut de Chimie de Lyon" (FR 02 ), for assistance and access to the NMR facilities.

Table 2. Thermal characteristics of the synthesized polyester polyols


Stéphane Ballet
email: stephane.ballet@univ-amu.fr

Julia Pieltant
email: pieltant@lix.polytechnique.fr

Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of 2 and 3

Keywords: Algebraic function field, tower of function fields, tensor rank, algorithm, finite field

Tower of algebraic function fields with maximal

Hasse-Witt invariant and tensor rank of multiplication in any extension of F _2 and F _3

Introduction

General context

The determination problem of the tensor rank of multiplication in finite fields has been widely studied over the past 20 years, as shown by the growing number of publications on this topic including amoung others [START_REF] Chudnovsky | Algebraic complexities and algebraic curves over finite fields[END_REF], [START_REF] Shparlinski | Curves with many points and multiplication in finite fields[END_REF], [START_REF] Arnaud | Évaluations dérivées, multiplication dans les corps finis et codes correcteurs[END_REF], [START_REF] Cenk | On multiplication in finite fields[END_REF], [START_REF] Ballet | On the tensor rank of multiplication in any extension of 2[END_REF], [START_REF] Randriambololona | Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method[END_REF]. This problem is worthwhile both because of its theoretical interest and because it has several applications in the area of information theory such as cryptography and coding theory.

Tensor rank of multiplication

Let q be a finite field with q elements where q is a prime power and let q n be a q -extension of degree n. The multiplication in the q -vector space q n is a bilinear map from q n × q n into q n , thus it corresponds to a tensor t ∈ q n q n q n

where q n denotes the dual of q n over q . Hence the product of two elements x and y of q n is the convolution of this tensor with x ⊗ y ∈ q n q n . If

t = λ =1 a ⊗ b ⊗ c
where a ∈ q n , b ∈ q n , c ∈ q n , then

x • y = t(x ⊗ y) = λ =1 a (x)b ( y)c . (1)
Every expression ( 1) is called a bilinear multiplication algorithm (resp. a symmetric bilinear multiplication algorithm if a = b for all ∈ {1, . . . , λ}). The integer λ is called the tensor rank (resp. symmetric tensor rank) of the algorithm , or the bilinear complexity (resp. symmetric bilinear complexity) of the algorithm . Let us set µ q (n) := min µ( ),

respectively µ sym q (n) := min sym µ( sym ),
where is running over all bilinear multiplication algorithms (resp. sym is runnig over all symmetric such algorithms) in q n over q .

It is interesting to study the minimal length of a symmetric multiplication algorithm since it turns out that it plays an important role in several other areas such as Riemann-Roch system of equations, arithmetic secret sharing, multiplicationfriendly codes, etc, as mentioned in [START_REF] Cascudo | Torsion limits and Riemann-Roch systems for function fields and applications[END_REF].

Basic notions related to function fields and notation

Let F/ q be an algebraic function field of one variable of genus g, with constant field q . For any integer k ≥ 1, we denote by P k (F/ q ) the set of places of degree k, by B k (F/ q ) the cardinality of this set and by P(F/ q ) = ∪ k P k (F/ q ) the set of all places in F/ q . For any place P, we define F P to be the residue class field of P and P its valuation ring. Every element t ∈ P such that P = t P is called a local parameter for P. The support of = P a P P is the set of the places P such that a P = 0.

For any divisor of F/ q , the Riemann-Roch q -vector-space associated to is the set

L ( ) = { f ∈ F/ q | + ( f ) ≥ 0} ∪ {0}.
The Riemann-Roch Theorem states that the dimension dim of the vector space L ( ) is related to the degree of the divisor and to the genus of F/ q by: dim

= deg -g + 1 + dim(κ -), (2)
where κ denotes a canonical divisor of F/ q . In this relation, the complementary term i( ) := dim(κ -) is called the index of speciality of . Note that in any case, we have i( ) ≥ 0. In particular, a divisor is called a non-special divisor when the index of speciality i( ) is zero and is called a special divisor if i( ) > 0.

Known results

General results

The bilinear complexity µ q (n) of the multiplication in the n-degree extension of a finite field q is known for certain values of n. In particular, Winograd [START_REF] Winograd | On multiplication in algebraic extension fields[END_REF] and de Groote [START_REF] De Groote | Characterization of division algebras of minimal rank and the structure of their algorithm varieties[END_REF] have shown that this complexity is ≥ 2n -1, with equality holding if and only if n ≤ 1 2 q + 1. Using the principle of the Chudnovsky-Chudnovsky algorithm [START_REF] Chudnovsky | Algebraic complexities and algebraic curves over finite fields[END_REF] applied to elliptic curves, Shokrollahi has shown in [START_REF] Shokrollahi | Optimal algorithms for multiplication in certain finite fields using algebraic curves[END_REF] that the bilinear complexity of multiplication is equal to 2n for 1 2 q + 1 < n < 1 2 (q + 1 + ε(q)) where ε is the function defined by: ε(q) = greatest integer ≤ 2 q prime to q, if q is not a perfect square 2 q, if q is a perfect square.

Eventually, the original algorithm of D.V. and G.V. Chudnovsky introduced in [START_REF] Chudnovsky | Algebraic complexities and algebraic curves over finite fields[END_REF] leads to the following theorem by [START_REF] Ballet | Curves with many points and multiplication complexity in any extension of q[END_REF]: Theorem 1. Let q be a prime power. The tensor rank µ q (n) of multiplication in any finite extension q n of q is linear with respect to the extension degree; more precisely, there exists a constant C q such that for any n, it holds that:

µ sym q (n) ≤ C q n.
Moreover, one can give explicit values for C q ; in particular for q = 2 or q = 3:

Proposition 2.
The best known values for the constant C q defined in the previous theorem are:

C q =    19.6 if q = 2 [11], [5] 27 if q = 3 [3]
Remark. Let us present the best finalized version of this algorithm in its symmetrical version, which is a generalization of the algorithm of Chudnovsky-Chudnovsky type introduced by Arnaud in [START_REF] Arnaud | Évaluations dérivées, multiplication dans les corps finis et codes correcteurs[END_REF] and developed later by Cenk and Özbudak in [START_REF] Cenk | On multiplication in finite fields[END_REF]. This generalization uses several coefficients in the local expansion at each place P i instead of just the first one. Due to the way to obtain the local expansion of a product from the local expansion of each term, the bound for the symmetric bilinear complexity involves the complexity notion M q (u) introduced by Cenk and Özbudak in [START_REF] Cenk | On multiplication in finite fields[END_REF] and defined as follows: Definition 3. We denote by M q (u) the minimum number of multiplications needed in q in order to obtain coefficients of the product of two arbitrary u-term polynomials modulo x u in q [x].

Note that in [START_REF] Randriambololona | Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method[END_REF], Randriambololona gives an even more general version of the Chudnovsky-Chudnovsky algorithm, which encompass the case of non-necessarily symmetric algorithms. This generalization is not relevant here, since we focus on the symmetric tensor rank; thus we introduce the generalized symmetric algorithm Chudnovsky-Chudnovsky type described in [START_REF] Cenk | On multiplication in finite fields[END_REF].

Theorem 4. Let

• q be a prime power,

• F/ q be an algebraic function field,

• Q be a degree n place of F/ q ,

• be a divisor of F/ q ,

• P = {P 1 , . . . , P N } be a set of N places of arbitrary degree,

• t 1 , . . . , t N be local parameters for P 1 , . . . , P N respectively,

• u 1 , . . . , u N be positive integers.

We suppose that Q and all the places in P are not in the support of and that: a) the map

Ev Q : L ( ) → q n F Q f -→ f (Q) is onto, b) the map Ev P : L (2 ) -→ q deg P 1 u 1 × q deg P 2 u 2 × • • • × q deg P N u N f -→ ϕ 1 ( f ), ϕ 2 ( f ), . . . , ϕ N ( f )
is injective, where each application ϕ i is defined by

ϕ i : L (2 ) -→ q deg P i u i f -→ f (P i ), f (P i ), . . . , f (u i -1) (P i ) with f = f (P i ) + f (P i )t i + f (P i )t 2 i + . . . + f (k) (P i )t k i + .
. ., the local expansion at P i of f in L (2 ), with respect to the local parameter t i . Note that we set f (0) := f . Then

µ sym q (n) ≤ N i=1 µ sym q (deg P i ) M q deg P i (u i ).
In particular, we will consider in this paper a specialization of this algorithm which is described in Section 4 and requires the additional hypothesis that there exists a non-special divisor of degree g -1; this will motivate the study of ordinary towers.

Asymptotic bounds for the extensions of 2 and 3

The following asymptotic bound for the bilinear complexity was introduced in [18]:

M sym q := lim sup k→∞ µ sym q (k) k .
Recently, with the help of the torsion-limit technique and Riemann-Roch systems, Cascudo, Cramer and Xing improved in [START_REF] Cascudo | Torsion limits and Riemann-Roch systems for function fields and applications[END_REF] the upper bounds for M sym q in the case where q is small (q ∈ {2, 3, 4, 5}). In particular, from optimal towers with asymptotically few 2-torsion points relatively to the genus they obtained: ≤ 5.45.

Motivations -New results established in this paper

As mentioned in [START_REF] Cascudo | Torsion limits and Riemann-Roch systems for function fields and applications[END_REF] by Cascudo, Cramer and Xing, and in [START_REF] Bassa | The Hasse-Witt invariant in some towers of function fields over finite fields[END_REF] by Bassa and Beelen, it is widely accepted that towers of algebraic function fields having a large p-rank are less efficient for certain applications to information theory. Indeed, for example, in [START_REF] Bassa | The Hasse-Witt invariant in some towers of function fields over finite fields[END_REF], it reads: "A detailed study of the p-rank in towers is relevant for their applications, see [START_REF] Cascudo | Torsion-limits for towers and asymptotically good special codes in secure computation and complexity[END_REF]. For example, although both of the towers introduced in [START_REF] Garcia | On the asymptotic behaviour of some towers of function fields over finite fields[END_REF] and [START_REF] Garcia | A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vlȃduţ bound[END_REF] have the same limit and hence are equally influential for applications in coding theory, a detailed study of their p-rank reveals that in fact the latter 2 is more appropriate for other kinds of applications, e.g. secure multiparty computation and fast bilinear multiplication." In particular, it appears that having a large p-rank may be a barrier for a tower to lead to competitive bounds for the symmetric tensor rank of multiplication in every extension of the finite field q , with q a power of p. In this paper, we show that there are two exceptional cases, namely the extensions of 2 and 3 . More precisely, we will show that an ordinary tower may lead to better uniform results for the tensor rank of multiplication in any extension of 2 and 3 than a non-ordinary one because of the link between maximal p-rank and existence of a non-special divisor of degree g -1. Indeed, we know that the existence of a non-special divisor of degree g -1 in the function field F/ q is of crucial importance in the performance of Chudnovsky-Chudnovsky type algorithms over q designed from F/ q [START_REF] Ballet | On the existence of non-special divisors of degree g and g -1 in algebraic function fields over q[END_REF], [START_REF] Randriambololona | Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method[END_REF]. When the definition field q is such that q ≥ 4, then according to [START_REF] Ballet | On the existence of non-special divisors of degree g and g -1 in algebraic function fields over q[END_REF] there always exists a non-special divisor of degree g -1. Nevertheless, the problem persists in the case where the definition field is 2 or 3 . In [START_REF] Ballet | On the tensor rank of multiplication in any extension of 2[END_REF], to avoid this obstacle, we substituted non-special divisors of degree g -1 for zero-dimensional divisors whose degree is as close as possible to g -1, in the descent over 2 of the original Garcia-Stichtenoth tower presented in [START_REF] Garcia | A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vlȃduţ bound[END_REF] and defined over 16 ; nonspecial divisors of degree g -1 being the borderline case of zero-dimensional divisors. However, Bassa and Beelen established in [START_REF] Bassa | The Hasse-Witt invariant in some towers of function fields over finite fields[END_REF] that the second optimal Garcia-Stichtenoth tower introduced in [START_REF] Garcia | On the asymptotic behaviour of some towers of function fields over finite fields[END_REF] and defined over q 2 is ordinary. On the other hand, it was shown in [START_REF] Ballet | On the existence of dimension zero divisors in algebraic function fields defined over q[END_REF] that there always exists a non-special divisor of degree g -1 in any ordinary function field. In this article, we combine these two results to a modified version of the optimal tower studied by Bassa and Beelen which improves the results obtained in [START_REF] Ballet | On the tensor rank of multiplication in any extension of 2[END_REF]. Namely we define intermediate steps for the tower and descend the definition field from 16 to 2 , and from 9 to 3 respectively, which lead us to the two following bounds:

µ sym 2 (n) ≤ 15.23n + 9 2
and µ sym 3 (n) ≤ 7.732n.

Note that the difficulty to obtain non-asymptotic estimations of the 2-torsion points in all steps of the tower used in [START_REF] Cascudo | Torsion limits and Riemann-Roch systems for function fields and applications[END_REF] is an obstruction to obtain uniform bounds as we get in this paper.

Definitions and related properties of the p-rank

Definition 5. The p-rank γ(F), also called Hasse-Witt invariant, of a function field F with constant field p , the algebraic closure of the finite field p , is defined as the dimension over p of the group of divisor classes of degree zero of order p.

If the function field is defined over a finite field q , we define its p-rank as the p-rank of the function field F q , obtained by extending the constant field to the algebraic closure of q .

It can be shown that :

Proposition 6. If F/ q be a function field of genus g(F), then 0 ≤ γ(F) ≤ g(F). Definition 7. A function field F/ q is called ordinary if γ(F) = g(F). A tower of function fields = F n / q n∈ is said ordinary if for any n ≥ 0, F n is such that γ(F n ) = g(F n ), i.e.

if any step of the tower is an ordinary function field.

Let us recall the following result from [START_REF] Ballet | On the existence of dimension zero divisors in algebraic function fields defined over q[END_REF]:

Corollary 8.
If F is an ordinary function field of genus g > 0 defined over 2 or over 3 , then there is always a degree g -1 zero-dimensional divisor in F.

Moreover, directly from Definition 5, we can deduce the following lemma:

Lemma 9. Let r ≥ 0 be an integer. If we set F/ q r := H/ q ⊗ q r , then H/ q is ordinary if and only if F/ q r is ordinary.

PROOF. Note that the genus does not change under constant field extension or descent. It follows from Definition 5 that p-rank does not change under constant field extension or descent since the p-rank of a function field F/ q defined over a finite field q is equal to the p-rank of F q .

To conclude this section, we recall the following result which is proven in [7, Lemma 6, 2.]:

Lemma 10. If H/F is a finite extension of function fields with same constant field q , then g(H) -γ(H) ≥ g(F) -γ(F).
In particular, if H is ordinary then so is F.

Good ordinary sequences of function fields defined over 2 or 3

In this section, we present sequences of algebraic function fields defined over 2 or 3 , constructed from the well-known Garcia-Stichtenoth tower defined in [START_REF] Garcia | On the asymptotic behaviour of some towers of function fields over finite fields[END_REF], which will be used to obtain new bounds for the tensor rank of multiplication.

Definition of the Garcia-Stichtenoth's tower

Let us consider a finite field q 2 with q = p r , for p a prime number and r an integer. We consider the Garcia-Stichtenoth's elementary abelian tower T 0 over q 2 constructed in [START_REF] Garcia | On the asymptotic behaviour of some towers of function fields over finite fields[END_REF] and defined by the sequence (F 0 , F 1 , F 2 , . . .) where

F 0 := q 2 (x 0 )
is the rational function field over q 2 , and for any i ≥ 0, F i+1 := F i (x i+1 ) with x i+1 satisfying the following equation:

x q i+1 + x i+1 = x q i x q-1 i + 1 .
Let us denote by g i the genus of F i in T 0 / q 2 and recall the following formulae:

g i = (q i+1 2 -1) 2 for odd i, (q i 2 -1)(q i+2 2 -1) for even i. (3)
Thus, according to these formulae, it is straightforward that the genus of any step of the tower satisfies:

(q i 2 -1)(q i+1 2 -1) < g(F i ) < (q i+2 2 -1)(q i+1 2 -1). (4)
Moreover, a tighter upper bound will be useful and can be obtained by expanding expressions in (3):

g(F i ) ≤ q i+1 -2q i+1 2 + 1. ( 5
)
If the characteristic p = 2 and r = 2, i.e. q = 4, then we can densify the Garcia-Stichtenoth's tower with steps defined over the finite field q 2 by considering the following completed tower:

T 1 / 16 : F 0,0 ⊆ F 0,1 ⊆ F 0,2 = F 1,0 ⊆ F 1,1 ⊆ F 1,2 = F 2,0 ⊆ • • • such that F i ⊆ F i,s ⊆ F i+1 for any integer s ∈ {0, 1, 2}, with F i,0 := F i and F i,2 := F i+1 . Indeed:
Proposition 11. There exists a tower T 1 defined over 16 whose recursive equation is defined over 2 . More precisely, the tower T 1 is the densified Garcia-Stichtenoth's tower over [START_REF] Randriambololona | Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method[END_REF] and is defined by

T 1 = F i,s i≥0 s∈{0,1}
where for any i ≥ 0:

F i,0 := F i and F i,1 := F i (t i+1 )
with t i+1 satisfying the equation:

t 2 i+1 + t i+1 = x 4 i x 3 i + 1 for i = 0, . . . , n -1. (6)
PROOF. Let x 0 be a transcendental element over 2 and let us set

F 0 := 16 (x 0 ).
We define recursively for i ≥ 0

(i) x i+1 such that x 4 i+1 + x i+1 = x 4 i x 3 i +1 for i = 0, . . . , n -1, (ii) t i+1 such that t 2 i+1 + t i+1 = x 4 i x 3 i +1 for i = 0, . . . , n -1, (or alternatively t i+1 = x 2 i+1 + x i+1 ).
Thus, we can define recursively the tower T 1 by setting:

F i,1 = F i,0 (t i+1 ) = F i (t i+1 ) and F i+1,0 = F i+1 = F i (x i+1 ).
Let us remark that it is possible to densify the general Garcia-Stichtenoth's tower over q 2 for any characteristic p and for any integer r since each extension F i+1 /F i is Galois of degree q = p r with full constant field q 2 . However, in the general case the equation ( 6) for the intermediate steps is not defined over p but over q . For example, for p = 3 and r = 2, we obtain an equation which is defined over 9 .

Notation. In the sequel, we will denote by B k (F/K) the number of places of degree k of an algebraic function field F/K defined over a finite field K; we will also denote by g i,s the genus of F i,s / 16 in T 1 / 16 .

Descent of the definition field of the Garcia-Stichtenoth's tower on the fields 2 and 3

First we state that when q = 3, one can descend the definition field of the tower T 0 / q 2 from q 2 to q since the recursive equation defining the tower has coefficients lying in q . Thus, we have the following result: Proposition 12. If q = p = 3, there exists a tower E/ q defined over q given by a sequence:

G 0 ⊆ G 1 ⊆ G 2 ⊆ G 3 ⊆ • • •
defined over the constant field q and related to the tower T 0 / q 2 by F i = q G i for all i, namely F i / q 2 is the constant field extension of G i / q . Now, we are interested in the descent of the definition field of the tower T 1 / q 2 from q 2 to p if it is possible. In fact, for the tower T 1 / q 2 , one can not establish a general result but one can prove that it is possible in the case where the characteristic is 2 and r = 2, i.e. q = 4. Note that in order to simplify the presentation, we are going to set the results by using the variable p. Proposition 13. If p = 2 and q = p 2 , the descent of the definition field of the tower T 1 / q 2 from q 2 to p is possible. More precisely, there exists a tower T 2 / p given by a sequence:

H 0,0 ⊆ H 0,1 ⊆ H 0,2 = H 1,0 ⊆ H 1,1 ⊆ H 1,2 = H 2,0 ⊆ • • •
defined over the constant field p and related to the tower T 1 / q 2 by F i,s = q 2 H i,s for all i ≥ 0 and s ∈ {0, 1, 2}, namely F i,s / q 2 is the constant field extension of H i,s / p .

PROOF. It is a straightforward consequence of Proposition 11.

In order to draw consequences for the previously descended towers, let us recall the known results concerning the number of places of degree one of the tower T 0 / q 2 , established in [START_REF] Garcia | On the asymptotic behaviour of some towers of function fields over finite fields[END_REF] and [START_REF] Aleshnikov | On the splitting of places in a tower of function fields meeting the Drinfeld-Vlȃdu ¸t bound[END_REF]. Proposition 14. If q = p r ≥ 2, then for any n > 2:

B 1 (F n / q 2 ) = q n (q 2 -q) + 2q 2 if p = 2, q n (q 2 -q) + 2q if p > 2.
Now, we deduce some straightforward properties concerning the towers T 2 / 2 and E/ 3 . Proposition 15. Let q = p 2 = 4. For any integers i ≥ 0 and s ∈ {0, 1, 2}, the algebraic function field H i,s / p in the tower T 2 / p has B 1 (H i,s / p ) places of degree one, B 2 (H i,s / p ) places of degree two and B 4 (H i,s / p ) places of degree four and satisfies:

(i) H i / p ⊆ H i,s / p ⊆ H i+1 / p with H i,0 = H i and H i,2 = H i+1 ,
(ii) the genus g i,s of H i,s / p satisfies:

(ii.a) g i,s ≤ g i+1 p 2-s (ii.b) g i,s ≤ p s-2 (q i+2 -2q i 2 +1 ) + p s-2 (iii) B 1 (H i,s / p ) + 2B 2 (H i,s / p ) + 4B 4 (H i,s / p ) ≥ q i (q 2 -q)p s .
Moreover, p is algebraically closed in each algebraic function field H i,s of the tower T 2 / p .

Remark.

Bound (ii.a) is tighter than Bound (ii.b), but when we will need an estimate for g i,s which does not depend on the parity of the step i of the tower, Bound (ii.b) will be useful.

PROOF. Property (i) follows directly from Proposition 13.

Each extension H i+1 /H i,s is a Galois extension of degree [H i+1 : H i,s ] = 2 2-s . Moreover, according to [19, Prop. 3.7.8] the full constant field of H i,s is p since at least one place of H 0 is totally ramified in H i,s . Indeed, the place at infinity of F 0 is totally ramified in the tower T 0 / q 2 . Hence, the same holds for the place at infinity of H 0 in T 2 / p . Since the algebraic function field F i,s is a constant field extension of H i,s , for any two integers i ≥ 0 and s ∈ {0, 1, 2}, F i,s and H i,s have the same genus, so by the Hurwitz Genus Formula [START_REF] Stichtenoth | Algebraic Function Fields and Codes[END_REF], one has:

g i,s ≤ g i+1 p 2-s (7)
with g(H i+1 / p ) = g(F i+1 / q 2 ) = g i+1 given by (3). Finally, applying Bound (5) on g i+1 , we get (ii.b). Moreover, for α ∈ q 2 \{ω ∈ q 2 | ω q + ω = 0}, let P α denote the place of degree one in the rational function field F 0 which is the zero of x 0 -α, then P α splits completely in

F i+1 /F 0 by [15, Lemma 3.9]. Let us set d := [F i+1 : F 0 ] and d := [F i+1 : F i,s ].
If denotes the number of places of F i,s lying over the place P α of F 0 , it is well known that ≤ d d with equality holding if and only if P α splits completely in F i+1 /F 0 ; so = d d = [F i,s : F 0 ] which proves that the place P α splits completely also in F i,s /F 0 . Thus, there are exactly q i p s places of degree one above P α in F i,s , so there are at least q i (q 2 -q)p s places of degree one in F i,s , since

q 2 \{ω ∈ q 2 | ω q + ω = 0} = q 2 -q.
To conclude, we deduce from [START_REF] Garcia | On the asymptotic behaviour of some towers of function fields over finite fields[END_REF] (where the same formula with s = 0 is proven for F i,0 ) that the number of places of degree one of F i,s / q2 is such that B 1 (F i,s / q 2 ) ≥ (q 2 -q)q i p s . Eventually, F i,s being a degree four constant field extension of H i,s , it is clear that for any two integers i ≥ 0 and s ∈ {0, 1, 2}, it holds that B 1 (H i,s / p ) + 2B 2 (H i,s / p ) + 4B 4 (H i,s / p ) ≥ (q 2 -q)q i p s .

Similar results than those of Proposition 15 can be obtained for the tower E/ 3 , namely: Proposition 16. Let q = p = 3. For any integer i ≥ 0, the algebraic function field G i / q in the tower E/ q has the same genus g i than the corresponding step F i / q 2 of the tower T 0 / q 2 . Moreover, the number of places of degree one and two of each function field G i / q is related to the number of rational places of F i / q 2 by:

B 1 (F i / q 2 ) = B 1 (G i / q ) + 2B 2 (G i / q )
thus, the following bound holds:

B 1 (G i / q ) + 2B 2 (G i / q ) ≥ q i (q 2 -q). ( 8
)
To conclude this section, let us recall that in [START_REF] Bassa | The Hasse-Witt invariant in some towers of function fields over finite fields[END_REF], the authors established the ordinarity of the classical tower over q 2 : Theorem 17. For any prime power q, the tower T 0 / q 2 is ordinary.

Thus, we can deduce that the ordinarity of T 0 / q 2 provides the same property to the towers T 2 / 2 and E/ 3 : Proposition 18. The towers T 2 / 2 and E/ 3 are ordinary.

PROOF. Since constant field descent preserves ordinarity from Lemma 9, the tower E/ 3 is ordinary and so are the steps F i,0 of the tower T 2 / 2 . Moreover Lemma 10 implies that the intermediate steps F i,1 are also ordinary since each one belongs to a finite extension F i+1,0 /F i,1 with same constant field, where F i+1,0 is ordinary.

A straightforward consequence of this last proposition and Corollary 8 is the following result: Corollary 19. For any function field F in the towers T 2 / 2 and E/ 3 , there exists a non-special divisor of degree g(F) -1.

New bounds for the tensor rank

Preliminary results

To obtain our new estimates for µ sym Theorem 20. Let n and d be two fixed integers. Let F/ q be an algebraic function field of genus g with at least B k places of degree k for any k|d. If the three following conditions are satisfied: (a) B n (F/ q ) > 0, (b) there exists a non-special divisor of degree g -1,

(c) k|d k(B k + b k ) ≥ 2n + 2g -1, where the integers b k are chosen such that 0 ≤ b k ≤ B k , then µ sym q (n) ≤ k|d µ sym q (k)(B k + b k ) + k|d µ sym q (k)b k , so µ sym q (n) ≤ η   k|d k(B k + b k ) + k|d kb k   with η := max k|d µ sym q (k) k .
PROOF. The algorithm recalled in Theorem 4 is applied on a set P = ∪ k|d P k with P k a subset of P k (F/ q ) with cardinality B k . Among each P ∈ P k , b k are used with multiplicity u = 2; all such places form a subset R of P. The others B k -b k places of P k are used with multiplicity u = 1. From the existence of a non-special divisor of degree g -1 provided by Hypothesis (b) and the existence of a place Q of degree n, one constructs an effective divisor such that deg = n + g -1 and dim = n. Precisely, one can choose any divisor which is equivalent to Q + , but whose support is disjoint from the support of Q + . For such a divisor , the map Ev Q is bijective: indeed its kernel ( -Q) is trivial since -Q is equivalent to which is non-special of degree g -1 and so is zero-dimensional;

thus Ev Q is injective and actually bijective by dimension reasons. Moreover, from

Hypothesis (c) it holds that 2

- P∈P P + R∈R R = {0} since deg 2 -( P∈P P + R∈R R) = 2 deg - k|d k(B k + b k ) < 0
thus Ev P is injective. From [START_REF] Cenk | On multiplication in finite fields[END_REF], it holds that M q (2) ≤ 3, so Theorem 4 then gives the following bound:

µ sym q (n) ≤ P∈P µ sym q (deg P) + 2 R∈R µ sym q (deg R).
Rearranging summation to group places with the same degree, we get the result.

Here we state two special cases of Theorem 20 which are adapted to the study of the tensor rank on 2 and 3 respectively. The first one is adapted to the case where places of degree one, two and four are taking into account: Proposition 21. Let p = 2. If F/ 2 is an algebraic function field of genus g with at least B k places of degree k for k =1, 2 and 4, such that the three following conditions are satisfied: (a) B n (F/ 2 ) > 0, (b) there exists a non-special divisor of degree g -1,

(c) k|4 k(B k + b k ) ≥ 2n + 2g -1, where the integers b k are chosen such that 0 ≤ b k ≤ B k , then µ sym 2 (n) ≤ 9 2 (n + g + 1) + 9 4 k|4 kb k .
PROOF. It is a straightforward consequence of Theorem 20 with q = p = 2 and d = 4. We recall that µ

k 's such that k|4 k(B k + b k ) = 2n + 2g -1 + ε, with ε ∈ {0, 1, 2,
3}: we must consider the less favorable case where there only exists places of degree four and so we have to choose ε = 3. This second specialization corresponds to the case where only places of degree one and two are considered: Proposition 22. Let q = 3. If F/ 3 is an algebraic function field of genus g with at least B k places of degree k for k =1, 2 such that the three following conditions are satisfied:

(a) B n (F/ 3 ) > 0, (b) there exists a non-special divisor of degree g -1, For i ≤ n- 13 4 , it holds that p 2i+6 ≤ p n-1 2 .

(c) k|2 k(B k + b k ) ≥ 2n + 2g -1,
Then we get that 13 4 . On the other hand, for i such that i > log q (n) - 1 2 , one has q i+1-1 2 ≥ n + 1, so q i+1 p s-1 (q -3) ≥ n + 1 since p s-1 ≥ q -1 2 = p -1 , which gives that q i+1 p s (q -3) ≥ 2n + 2 and so q i+1 p s (q -1 -2) + q i-1 2 p s ≥ 2n + 2. Thus it holds that: q i+1 p s (q -1) ≥ 2n + 2 + 2q i+1 p sq i-1

p 2i+6 1 -1 p i+2 + 1 p 2i+3 ≤ p n-1 2 p 2 ( p -1), since 1 -1 p i+2 + 1 p 2i+3 ≤ 1 ≤ p 2 (
2 p s = 2n + 2p s-2 (q i+2 -q i 2 +1
) + 2. Eventually, one gets that q i+1 p s (q -1) ≥ 2n + 2p s-2 (q i+2 -q i 2 +1 ) + 2p s-2 -1 since 2p s-2 -1 ≤ 2 for s ∈ {0, 1}, and Condition (c) is satisfied according to the inequalities (ii.b) and (iii) established in Proposition 15. Thus, for n ≥ 21 one can find at least one integer i in the interval log q (n) -1 2 ; n- 13 4 , and so a corresponding step of the tower H i,0 for which Proposition 21 holds. Note that in any case, Condition (a) is satisfied for lower steps than Condition (c), so it may happened that the first suitable step that satisfy both conditions is not H i,0 itself but one of the previous step. For i ≤ n- 5 2 , Condition (a) is satisfied since one has:

q i ≤ q n-4 2 q ≤ q n-4 2 q+1 = q n-4 2 q-1 2
and so (q

i+2 2 -1)(q i+1 2 -1) ≤ q n-1 2 q-1 2
which gives that the inequality

2g i + 1 ≤ q n-1
2 ( q -1) of [19, Corollary 5.2.10] holds according to (4). On the other hand, when i ≥ 2 log q n 2 -1 , Condition (c) is satisfied. Indeed, for such i one has: q i 2 ≥ n 2 -1, so 4q i 2 ≥ 2n -5, which gives that 4q i 2 + 2q ≥ 2n + 1. Adding 2q i+1 , which equals (q 2 -q)q i , to both sides it follows that: (q 2 -q)q i + 2q ≥ 2n + 2q i+1 -4q i 2 + 1 = 2n + 2(q i+1 -2q i 2 + 1) -1. Thus from (8) and ( 5) we get that inequality of Condition (c) holds with b 1 = b 2 = 0. To conclude, one can see that for n ≥ 13, the interval 2 log q n 2 -1 ; n-5 2 contains at least an integer i and so G i / 3 is a suitable step of the tower; moreover the smallest such integer is the smallest i ≥ 2 log q n 2 -1 , i.e. the smallest one for which Condition (c) is satisfied.

Till the end of this section, we will deal with the following notations:

n 2,i,s def := max m 2m + 2g(H i,s ) -1 ≤ k|4 kB k (H i,s / 2 )
and

n 3,i def := max m 2m + 2g(G i ) -1 ≤ k|2 kB k (G i / 3 ) .
Let us explain the relevance of these definitions, focusing on the case of the role of n 3,i in the tower E/ 3 (the same holds for the tower T 2 / 2 when one replaces n 3,i by n 2,i,s ). The integer n 3,i is the biggest one for which it holds that:

k|2 kB k (G i / 3 ) ≥ 2n 3,i + 2g i -1
i.e. q n 3,i is the biggest extension of 3 for which G i / 3 could be a suitable step of the tower to apply Proposition 22 with b

1 = b 2 = 0. If n > n 3,i , then k|2 kB k (G i / 3 ) < 2n + 2g i -1 but one has k|2 kB k (G i / 3 ) + 2(n -n 3,i ) ≥ 2n + 2g i -1
which means that G i is still a suitable step of tower to apply Theorem 22 if we can choose the b k 's such that k|2 kb k ≥ 2(nn 3,i ).

Thus, we are interested in the determination of a lower bound for n 3,i and n 2,i,s . It is the purpose of the two following lemmas: Lemma 25. If p = 2 and q = p 2 = 4, then n 2,i,s ≥ q i+1 p s + q i 2 +1 p s -1.

PROOF. According to Proposition 15 (iii) and (ii.a), and Formula (5), we get:

k|4 kB k (H i,s / 2 ) -2g(H i,s ) + 1 ≥ (q 2 -q)q i p s -2p s-2 (q i+2 -2q i 2 +1 + 1) + 1 = q i+2 p s -q i+1 p s -p s-1 (q i+2 -2q i 2 +1 + 1) + 1 = q i+2 p s-1 (p -1) -q i+1 p s + q i 2 +1 p s -p s-1 + 1 ≥ q i+1 p s-1 (q -p) + q i 2 +1 p s -1 since s ∈ {0, 1, 2} and p -1 = 1 = q i+1 p s + q i 2 +1 p s -1. Lemma 26. If q = 3, then n 3,i ≥ 4q i+1 2 -1.
PROOF. Proposition 16 and Formula (5) give:

k|2 kB k (G i / 3 ) -2g(G i ) + 1 ≥ q i (q 2 -q) -2(q i+1 -2q i+1 2 + 1) + 1 ≥ q i+1 (q -1) -2q i+1 + 4q i+1 2 -1 = q i+1 (q -3) + 4q i+1 2 -1 = 4q i+1 2 -1.
Now, we establish a lower bound for the gap between the genus of two successive steps of each tower T 2 / 2 and E/ 3 : Lemma 27. (i) If q = p 2 = 4 then ∆g i,s def := g(H i,s+1 ) -g(H i,s ) ≥ p s (2q i -3q i 2 ).

(ii) If p = q = 3 then ∆g i def := g(G i+1 ) -g(G i ) ≥ (q -1)(q i+1 -q i/2 ).

PROOF. (i) For any s ∈ {0, 1}, since [H i,s+1 : H i,s ] = p the Hurwitz Genus Formula gives that g i,s+1 -1 ≥ p(g i,s -1) and it follows that g i,s+1 -g i,s ≥ (p -1)(g i,s -1). If s = 0, then g i,s -1 = g i -1 and according to (4), it holds that g i -1 ≥ (q i 2 -1)(q i+1 2 -1). Thus, we get g i ≥ qq i -(1 + q)q i 2 = 2q i -3q i 2 , which gives that g i,s+1 -g i,s ≥ 2q i -3q i 2 = p s (2q i -3q i 2 ). If s = 1, then g i,s+1 -g i,s ≥ (p -1)(g i,s -1) holds, with g i,s -1 = g i,1 -1 ≥ p(g i -1) from Hurwitz Genus Formula. Thus we get g i,s+1 -g i,s ≥ (p -1)p(g i -1) ≥ (p -1)p(2q i -3q i 2 ) = p s (2q i -3q i 2 ).

(ii) From Formulae (3), we get g i =    (q -1) q i+1 -q i 2 for even i, (q -1) q i+1 -q i+1 2 for odd i, which gives the result. (n) ≤ 6(nn 3,i ) + 3(n 3,i + g i ) we see that the second one is better than the other when nn 3,i < ∆g i , under the assumption that 2(nn 3,i ) ≤ k|2 kB k (G i / q ). So if D 3,i is such that D 3,i ≤ ∆g i and 2D 3,i ≤ k|2 kB k (G i / q ), then when 2(nn 3,i ) < D 3,i , the second bound is better and can be reached since we can choose the b k 's such that then µ sym 3 (n) is bounded above by any linear function whose graph lies above all the points n 3,i + D 3,i , Φ 3 (n 3,i + D 3,i ) i . We fix X := n 3,i + D 3,i where D 3,i := min (q -1)(q i+1 -q i/2 ) ; 1 2 q i (q 2 -q) . Thus, for any i ≥ 2, D 3,i = (q -1)(q i+1 -q i/2 ); and it holds that

Φ 3 (X ) ≤ 3 1 + g i+1 X X .
One has:

g i+1 X ≤ (q i+3
2 -1)(q i+2 2 -1) 4q i+1 2 -1 + (q -1)(q i+1 -q i/2 ) = q i+ 5 2 -q i+2 2 (1 + q) + 1 q i+2 + 4q i+1 2 -q i+1 -(q -1)q i/2 -1 ≤ q i+2 qq -i+2 2 (1 + q) + q -i-2

q i+2 1 + 4q -i+3 2 -q -1 -(q -1)q -i+3 2 -q -i-2

which gives that: g i+1 X ≤ qq -i+2 2 (1 + q) + q -i-2

1 -q -1 -(q -1)q -i+3 2 -q -i-2 so since i ≥ 2: g i+1 X ≤ qq -2 (1 + q) + q -4 1 -q -1 -(q -1)q -5 2 -q -4 .

sym 2 ( 2 ) = 3 and µ sym 2 ( 4 )

 2224 ≤ 9; so we obtain η = max k|4 result follows from a choice of the B k 's and the b

2 in

 2 where the integers b k are chosen such that 0 ≤ b k ≤ B k , same proof than the previous one with q = 3 and d = 2, and so η = 3 Theorem 20 gives the result. Lemma 23. Let q = 4 = p 2 and n ≥ 19. There exists a step H i,s / 2 of the tower T 2 / 2 such that the three conditions of Proposition 21 are satisfied with b 1 = b 2 = b 4 = 0. Moreover, if Condition (c) is satisfied then the two others also are. PROOF. Thanks to Corollary 19, Condition (b) is satisfied for any step of the tower.

  Moreover one can check that for n = 19, H 1 is the first suitable step of the tower to apply Proposition 21 with b 1 = b 2 = b 4 = 0. Indeed, it holds that g(H 1 / 2 ) = 9 so Condition (a) is satisfied and since B 1 (H 1 / 2 ) = 4, B 2 (H 1 / 2 ) = 2 and B 4 (H 1 / 2 ) = 12, Condition (c) is also satisfied for H 1 but it is not the case for H 0,1 . Similarly for n = 20, H 1 does not satisfy Condition (c), but H 1,1 does satisfy both Conditions (a) and (c) since g(H 1,1 / 2 ) = 21, B 1 (H 1,1 / 2 ) = 4, B 2 (H 1,1 / 2 ) = 2 and B 4 (H 1,1 / 2 ) = 25. Lemma 24. Let q = 3 and n ≥ 13. There exists a step G i / 3 of the tower E/ 3 such that the three conditions of Proposition 22 are satisfied with b 1 = b 2 = 0. Moreover, if Condition (c) is satisfied then the two others also are. PROOF. Thanks to Corollary 19, Condition (b) is satisfied for any step of the tower.

  k|2 k b k := 2(nn 3,i ). The particular case where n = n 3,i + D 3,i will give us an upper bound for µ sym 3 (n) as follows: define the function Φ 3 (x) := min i Φ 3,i (x), withΦ 3,i (x) =    6(xn 3,i ) + 3(n 3,i + g i ) if xn 3,i < D 3,i 3 x + g i+1 else,

  The estimate C 2 = 19.6 is obtained by combining the general uniform bound µ Table 1] for n ≤ 18 (see Appendix).

	µ	sym 2 (n) ≤ 477 26 n + 45 2 from [5] for n greater than 19, and the values of 2 (n) given in [11, sym

  ( p -1) according to Proposition 15 (ii.b) with s ∈ {0, 1} (one can always assume that s = 2 since H i,2 = H i+1,0 ). Hence, from [19, Corollary 5.2.10] Condition (a) is satisfied for any step H i,s such that i ≤ n-

							p -1).
	It	follows	that	p 2i+4 -p i+2 + p ≤ p	n-1 2 ( p -1),	which	leads	to
	2g i,s + 1 ≤ p	n-1 2

which turns out not to be ordinary, according to[START_REF] Cascudo | Torsion limits and Riemann-Roch systems for function fields and applications[END_REF]

(n) and µ sym

(n) from the towers described in the previous section, we will need some technical results which are proven below.

Acknowledgments

The first author wishes to thank Inria for funding his venue at Saclay where this work was partly done while he was visiting the GRACE team at LIX.

The second author was partially supported by the Labex Mathematics Hadamard (LMH) through its International PostDoctoral Scholarschip program.

PROOF. We first set p = 2 and q = p 2 . Note that for n ≤ 18, the result already holds from Section 1.4.1 and [11, Table 1] (see Appendix). So, fix n ≥ 19 and choose i ≥ 0 and s ∈ {0, 1} such that k|4 kB k (H i,s+1 / p ) ≥ 2n + 2g i,s+1 -1 but k|4 kB k (H i,s / p ) < 2n + 2g i,s -1.

We can apply Proposition 21 in the two following ways:

Rewriting those two bounds respectively as:

n) is bounded above by any linear function whose graph lies above all the points n 2,i,s + D 2,i,s , Φ p (n 2,i,s + D 2,i,s ) i,s . We fix X := n 2,i,s + D 2,i,s where

17 so that one has D 2,i,s ≤ ∆g i,s from Lemma 27, and

according to Theorem 15. Thus, for any i, s, Φ 2 (X ) ≤ 9

One has

which gives that g i,s+1

X ≤ 81 34 , so

2 . Now we consider the case q = p = 3. Since the result already holds for n < 13 from [11, Table 1] (see Appendix), fix n ≥ 13, and choose i ≥ 0 such that

We can apply Proposition 22 in the two following ways:

(a) on G i+1 / q with b 1 = b 2 = 0, which gives:

which leads to:

Rewriting those two bounds respectively as:

Finally, with q = 3, one gets:

Remark. In the case of 2 , the descent of the tower T 0 defined over q 2 with q = 2 from q 2 to q = 2 is not sufficient to obtain a competitive bound for the tensor rank. Indeed, in this case, we get: