Sebastian Meyer

Simona Bennici

Cyril Vaulot

Séverinne Rigolet

& Liva Dzene

INFLUENCE OF THE PRECURSOR AND THE TEMPERATURE OF SYNTHESIS ON THE STRUCTURE OF SAPONITE

Keywords: Synthesis, Minerals, Nanoclay, Saponite, precursor

Several procedures of hydrothermal synthesis of saponite can be found in the literature.

They differ in terms of the preparation conditions of the precursor and of the synthesis temperature. The objective of this study was to investigate how these two parameters influence the structure of the final synthesis product. The precursor was prepared from Mg(NO3)2, Al(NO3)3 and Na4SiO4 in three different ways: as a gel, a dried gel and a calcined gel. The influence of the synthesis temperature on the structure of saponite was investigated in 90 -200 °C range. The results showed that the use of calcined precursor yields a single mineral phase, saponite with up to 90% aluminum in tetrahedral configuration. In comparison, the use of gel precursor, resulted in a product with only 60% of aluminum in the tetrahedral configuration. The synthesis temperature did not have a significant effect on saponite structure. The reported synthesis method showed the possibility to obtain saponite with superior characteristics, in terms of crystallinity, surface acidity and thermal stability compared to natural mineral, even at 90 °C and then with a higher potential for industrial applications.

INTRODUCTION

For solid state catalysts, the surface and pore characteristics play a decisive role and are investigated with many different approaches. Mesophase templating allows to synthesise amorphous silica with defined mesoporosity of up to 10 nm [START_REF] Gleiter | Nanostructured materials: basic concepts and microstructure[END_REF].

Limitations in this approach are difficulties in large-scale production and the possibility to functionalise the surface to create catalytic active sites. Zeolites are easy to synthesise, but the access to the sites positioned inside the pores, which constitutes the biggest part due to the high internal surface area, is restricted due to the presence of micropores with a dimension < 0.8 nm and therefore not reachable by larger molecules.

However, a partial inhibition of crystallization using an excess of grafted organic species can lead to intracrystalline mesoporosity with increased accessibility for active sites [START_REF] Xue | Hierarchical structure and catalytic properties of a microspherical zeolite with intracrystalline mesopores[END_REF]. Saponite clay minerals with an important anysotropy due to their layered structure allow a better adsorption of more complex molecules onto their surface. This makes saponite clay minerals suitable for a wide range of catalytic applications for organic molecules from cracking, isomerisation [START_REF] Vogels | Catalytic activity of synthetic saponite clays: Effects of tetrahedral and octahedral composition[END_REF] and oxydation [START_REF] Trujillano | Preparation, alumina-pillaring and oxidation catalytic performances of synthetic Ni-saponite[END_REF] to the Mannich reaction [START_REF] Gómez-Sanz | Acid clay minerals as eco-friendly and cheap catalysts for the synthesis of β-amino ketones by Mannich reaction[END_REF]. In polymer composite materials saponite nanoparticles are used to give origin to novel properties not shown before by one of the components at the macro or micro scale [START_REF] Nathani | On the reduced susceptibility to stress whitening behavior of melt intercalated polybutene-clay nanocomposites during tensile straining[END_REF]. Improvements in the mechanical properties and hindrance to diffusion were observed. The particular properties of saponites, as the high cation exchange capacity and important specific surface area, make them also attractive for such applications as adsorbents and in wastewater treatment [START_REF] Marchesi | Synthetic saponite clays as promising solids for lanthanide ion recovery[END_REF].

Their high surface area, thermal stability, large availability and resource friendly characteristics make the use of synthetic and natural saponites increasingly attractive in research and industry [START_REF] Zhou | Modification, hybridization and applications of saponite: An overview[END_REF][START_REF] Carniato | An overview of the recent synthesis and functionalization methods of saponite clay[END_REF].

Saponites are part of the smectite group and belong to the 2:1 clay minerals, which consist of layers composed of two tetrahedral silica sheets sandwiching an octahedral sheet. For synthetic saponites, the octahedral sheet can contain various divalent metals, like Mg 2+ , Ni 2+ , Zn 2+ , Co 2+ , and Fe 2+ [START_REF] Decarreau | Cristallogenese à basse température de smectites trioctaèdriques par vieillissement de coprécipités silicométalliques[END_REF][START_REF] Trujillano | Rapid microwave-assisted synthesis of saponites and their use as oxidation catalysts[END_REF]. One of the requirements of the materials intended for the catalytic use, is their thermal stability.

The thermal stability is found to be higher in smectites like saponites (with Mg 2+ as the major cation in the octahedral sheet) that are stable up to 750 °C, when compared to smectites like montmorillonites (with Al 3+ as a major cation in the octahedral sheet) with a maximum stability temperature of 480 °C [START_REF] Ames | Factors effecting maximum hydrothermal stability in montmorillonites[END_REF].

In the tetrahedral silica sheet functional ions like Al 3+ can isomorphously substitute Si 4+ to create active sites. Si-O-Al groups have shown strong acidic properties as a Lewis site [START_REF] Aquino | Modeling catalytic effects of clay mineral surfaces on peptide bond formation[END_REF][START_REF] Zhou | Modification, hybridization and applications of saponite: An overview[END_REF]. In addition, it has been suggested in the literature that the acid activity of clay minerals could be correlated to the tetrahedral surface charge density [START_REF] Davidtz | The acid activity of 2:1 layer silicates[END_REF]). Thus, a product with the most of Lewis sites and with high thermal stability at the same time, could be a structure with a high amount of Al 3+ substitution in tetrahedral sheet, i.e. having Al/Si molar ratio equal to 0.33. The possibility to synthesize saponite clay mineral with such high amount of Al 3+ has been previously demonstrated [START_REF] Suquet | Synthesis and Swelling Properties of Saponites with Increasing Layer Charge[END_REF].

In general, the formation of saponites is possible under vastly different hydrothermal conditions. Moderate conditions with temperatures between 100 and 575 °C and pressures to several kbars are most promising in terms of growth rate, yield and purity [START_REF] Kloprogge | Synthesis of smectite clay minerals: A critical review[END_REF]). In the 125 -280 °C hydrothermal temperature range, a higher temperature results in an increase of the crystalline phases for ammonium saponites, shown by Fourier-transform Infrared (FTIR) spectroscopy [START_REF] Kloprogge | The effect of synthesis temperature on the FT-Raman and FT-IR spectra of saponites[END_REF].

However, the chemical composition, and thus to some extent saponite crystal structure, would not be expected to significantly change by increasing the temperature. To test this hypothesis, saponite synthesis was performed from 90 to 220 °C and a detailed characterization of the chemical composition and structure of final synthesis product was performed.

While in nature the formation of smectites is determined by the availability of elements and their speciation [START_REF] Christidis | Compositional variations in smectites. Part II: alteration of acidic precursors, a case study from Milos Island, Greece[END_REF], in hydrothermal synthesis the most common starting materials are solid phases, such as other aluminosilicates, glasses and gels, whose composition is close to the that of the desired clay mineral [START_REF] Kloprogge | Synthesis of smectite clay minerals: A critical review[END_REF]. The effect of precursor treatments on the structure of the final synthesis products is not well investigated and therefore investigated in this work. EXPERIMENTAL Synthesis 0.2 M solutions of magnesium nitrate (Mg(NO3)2•6H2O) from Sigma Aldrich (India), aluminum nitrate (Al(NO3)3•9H2O) from Carlo Erba (Val de Reuil, France) and sodium orthosilicate (Na4SiO4) from Alfa Aesar (Karlsruhe, Germany) in distilled water (18.2 MΩ•cm) were prepared. The solutions were mixed in a beaker to form a precursor that contained a Mg/Al/Si molar ratio of 1/0.33/1. The precursor was then stirred at room conditions for 3 h to achieve a homogeneous suspension. For the precursor experiments the suspension was put into the autoclave directly at this stage (sample labeled "Gel") followed by the hydrothermal treatment. To reduce the water content, the suspension was dried in an oven at 200 °C (sample labeled "Dried gel"). For the rest of the experiments the dried gel was put into a crucible and calcined at 450 °C (sample labeled "Calcined gel"). At maximum temperature the holding time was 1 h after a heating rate of 5 °C/min and a cooling time of less than 10 °C/min afterwards.

For the hydrothermal synthesis a 0.01 M NaOH solution was prepared using NaOH powder from Sigma Aldrich (St Louis, USA) and then added to the grinded calcined precursor gel. The autoclave with the suspension was securely closed and placed into an oven for 120 hours. The synthesis was conducted at 90, 120, 150, 180 and 200 °C. In the autoclave a pressure corresponded to a water pressure at those temperatures. The pH was measured with Hi2211 pH ORP Meter by Hanna Instruments (Tanneries, France) with two point calibration at pH = 7 and pH = 10 after the precipitation of the precursor and after the synthesis.

After the synthesis, samples were recovered by centrifugation at 8000 rpm for 10 min, which was followed by washing with distilled water two times and ion exchange with Ca 2+ using 1 M CaCl2 solution [START_REF] Carrado | Synthetic Clay Minerals and Purification of Natural Clays[END_REF]. The choice to saturate the samples with Ca 2+ , was driven by a practical conveniency. As Ca 2+saturated smectites have two water layer hydrate structure over a large relative humidity range [START_REF] Suquet | Swelling and structural organization of saponite[END_REF], such material is then less sensitive to relative humidity changes and more practical to handle during the different characterization experiments.

After the ion-exchange and washing, the suspension was dried in an oven at 60 °C and ground in a mortar.

X-ray diffraction of powder and oriented preparations

The ground saponite powder was placed into a backloading sample holder and compacted. The prepared sample was measured by X'Pert Pro from PANalytical (Malvern, United Kingdom) equipped with an X'Celerator real-time multiple strip detector operating with an angular aperture of 2.12°2θ in 3° to 70°2θ range using CuKα radiation with the wavelength of 0.15406 nm. Diffractogrammes were recorded at room temperature with step size of 0.017°2θ and scan time of 4 s per step. The divergence slit, the anti-scatter slit and the two Soller slits were 0.0625°, 0.125°and 2.3°, respectively.

The oriented slides were prepared by placing a droplet of the diluted suspension on a glass slide to evaporate with time. Particles settled by gravity and adopted a preferable orientation parallel to the substrate. For optimal measurement results 3 to 20 mg of powder were used. The dried slides were then recorded in 3° to 35°2θ range with the same scanning parameters as before. To saturate the samples with ethylene glycol, the glass slides were put into a desiccator with ethylene glycol for 2 days.

Fourier-transform infrared spectroscopy

The synthezised saponite was mixed with potassium bromide in the weight ratio 1:100 and ground with a mortar by hand. The powder was given into a mold and pressed isostatically at 5 bar for 3 minutes. The resulting pellets were put into an oven at 150 °C overnight to reduce the water content. FTIR spectroscopy measurements were conducted with the device Equinox 55 from Bruker (Karlsruhe, Germany). The obtained spectrum was the average of 32 measurements with a resolution of 4 cm -1 . The software OPUS was used to record the spectra and to subtract the reference with CO2 and H2O contributions.

Nuclear magnetic resonance spectroscopy

The 27 Al MAS NMR spectra were recorded at room temperature with a Bruker Avance II 400 MHz spectrometer (Wissembourg, France) operating at B0 = 9.4 Tesla giving a 27 Al Larmor frequency of 104.23 MHz, equipped with a Bruker 4 mm double channels probe. All experiments were performed at room temperature in ZrO2 rotors, with a /12 radio frequency pulse length of 0.5 s, a recycle delay of 1 s, at a spinning rate of 12 kHz. The chemical shifts were referred to external [Al(H2O)6] 3+ in AlCl3 aqueous solution. The decompositions of the spectra were performed using the DMFit software [START_REF] Massiot | Modelling one-and two-dimensional solid-state NMR spectra[END_REF].

X-ray fluorescence spectroscopy

The sample pellet was prepared by applying an isostatic pressure of 5 bar on 200 mg of sample. The spectrometer Zetium from PANalytical (Malvern, United Kingdom) was used to record the spectra that were analysed towards the composition of oxides in weight percent. The atomic weight percent was then calculated.

Nitrogen adsorption

A Micromerics ASAP 2420 (Verneuil en Halatte, France) was used to adsorb nitrogen at the temperature of liquid nitrogen (-196 °C). Prior to adsorption, 100 to 200 mg of the sample were outgassed in vacuum at 120 °C to remove the water adsorbed on the surface of the particles. The sample mass after pretreatment was measured, then the sample flask was connected to the device to be cooled down to -196 °C during the whole measurement. Specific surface area of the sample was derived with the BET method using the software MicroActive version 1.01. The average mesopore size was derived from the adsorption branch of the isotherm using the Barrett-Joyner-Halenda (BJH) method. The micro and mesopore size distribution was derived using the density function theory (DFT). Slit and cylinder shape for oxide surfaces were tested and evaluated by their RMS (Root mean square) error of the isotherm fit and by the correct fit between the experimental isotherm and the different modelizations (step-like fits were avoided and the fits were tested in log and linear scales).

Transmission electron microscopy

Samples were prepared by dispersing the synthesized saponite powder with ultrasound in chloroform. The resulting suspension was evaporated on a carbon coated copper grid at room temperature leaving homogeneously distributed particles. The device ARM200F from JEOL (Tokyo, Japan) with an acceleration voltage of 200 kV was used to record the images. The software Digital Micrograph allowed to measure the average basal spacing by calculating the mean over 6 to 10 clay mineral layers in a selected area.

RESULTS AND DISCUSSION

Influence of precursor preparation on structure formation

The pH was 12.7 after 3 hours of the gel aging and 12.5 after the hydrothermal synthesis. The X-ray powder diffractograms (Fig. 1) of synthesis products using different precursors revealed that typical 2:1 tri-octahedral clay mineral planes (020), ( 200), ( 150) and ( 060) could be identified in all three samples. The basal reflection (001) at 5.9°2θ was only well-defined and symmetrical for the sample prepared from the calcined gel. It indicated an ordered structure with layer-to-layer distances of around 1.5 nm. For the samples prepared from gel and dried gel precursors, XRD peaks at 12° (0.75 nm) and 24°2θ (0.36 nm) were observed that corresponded to heterogeneous formations of polymorphs of the kaolinite-serpentine group and chlorites. The peak at 35.7°2θ (2.51 nm) and the two peaks at 59.8°2θ (0.154 nm) and 61.1°2θ (0.151 nm) suggested the presence of lizardite-like structure. Other phases like chlorites could contribute to a higher amount of octahedral Al, especially for the sample prepared from dried gel precursor. The XRD patterns showed crystalline and single clay mineral saponite formation using the precursor of calcined gel. The product after hydrothermal synthesis then contained a higher purity crystalline saponite. No characteristic peaks of other aluminum phases such as alumina, gibbsite or boehmite, neither hydrotalcite-like phases were identified in X-ray diffractograms. The small sharp peak at 29.5°2θ (0.303 nm) corresponded to calcite, an impurity arising from an incomplete washing prior to the sample saturation with Ca 2+ . FTIR spectroscopy was used to compare the results from XRD, regarding the sample cristallinity, and to obtain the information on the presence of amorphous aggregates and the type of bonds present in the samples. For all three samples, O-H and Si-O vibration modes, characteristic of clay minerals, could be identified (Fig. 2) [START_REF] Petit | Fourier Transform Infrared Spectroscopy[END_REF]). The large band at ~3400 cm -1 and at 1629 cm -1 corresponded to the vibration modes of H2O molecules. The band at 1427 cm -1 corresponded to carbonates, an impurity identified also by XRD. The O-H bending mode identified at 651 cm -1 was characteristic for trioctahedral clay minerals [START_REF] Russell | Infrared methods[END_REF].

Fig. 2 FTIR spectra of saponite obtained after hydrothermal synthesis at 200 °C using different precursors.

The region of O-H stretching vibration mode was rather broad for samples prepared from the gel and dried gel precursors. Two bands of 3718 and 3683 cm -1 could be clearly identified for the sample prepared from calcined precursor. The bands corresponded to Mg3O-H stretching mode. It was shown by [START_REF] Pelletier | Influence of layer charge on the hydroxyl stretching of trioctahedral clay minerals: A vibrational study of synthetic Na-and K-saponites[END_REF] that the bands corresponding to the hydroxyl groups were influenced by the saponite layer charge. The Mg3O-H peak shifted from 3676 to 3692 cm -1 due to bond stretching with increasing tetrahedral Al 3+ substitutions and the compensating cation. The peak visible at 3683 cm -1 and additionally at 3716 cm -1 in our reorded spectra was then an additional evidence for Al 3+ substitutions in the tetrahedral sheet. For the sample prepared from the gel precursor a band at 3656 cm -1 could be identified corresponding to Mg2AlO-H stretching mode [START_REF] Petit | Crystalchemistry of talc: A near infrared (NIR) spectroscopy study[END_REF]). It indicated an important substitution of Al 3+ for Mg 2+ in the octahedral sheet. Among the broad zone of O-H stretching mode, two bands could be distinguished for the sample prepared from the dried gel precursor: 3613 and 3553 cm -1 . These bands were charateristic of Al2O-H stretching modes. In particular, 3553 cm -1 band was characteristic of chlorites, which was in agreement with phases identified in XRD pattern [START_REF] Russell | Infrared methods[END_REF].

In all three samples, bands at 991 and 457 cm -1 could be clearly identified corresponding to Si-O vibration modes. In addition, a band at 692 cm -1 could be identified for the calcined gel sample corresponding to Si-O vibration mode as well.

The substitution of Al 3+ -for-Si 4+ in the tetrahedral sheet is known to shift Si-O stretching mode band value from 1000 cm -1 towards lower values, which corresponds to the present case (991 cm -1 ). Klopproge and Frost (2000) had attributed 457 cm -1 to the Si-O bending mode of amorphous silica. In the present case, as no other Si-O modes in this region could be clearly identified and considering XRD results, this could then correspond to Si-O vibration mode belonging to clay minerals. However, the presence of amorphous silica cannot be discarded. Indeed, a band ~569 cm -1 could be identified for the samples prepared from the gel and dried gel precursors, reported as belonging to amorphous SiO2. Finally, all samples contain a band at 823 cm -1 attributed to Al-O4 band vibration, affirming the presence of tetrahedral aluminum. The characteristic bands of alumina were not identified.

The NMR results (Fig. 3) revealed the aluminum resonances for the tetrahedral ( IV Al) and octahedral ( VI Al) configuration of all precursor types after synthesis. Signals observed around 60 and 8 ppm were attributed to IV Al and VI Al respectively. Using the gel precursor, the Al content was nearly evenly distributed between the octahedral and the two tetrahedral sheets in the final product with IV Al = 55%. The NMR spectroscopy results were in agreement with the FTIR spectroscopy results, where Mg2AlO-H stretching mode was identified suggesting an important substitution of Al 3+ for Mg 2+ in the octahedral sheet. Using the dried and calcinated gel precursors the content of tetrahedral aluminum increased with IV Al = 80% and IV Al = 85%, respectively. NMR results showed a higher amount of octahedral Al in the samples prepared from the gel and dried gel precursors. In agreement with XRD results, the structure with maximum Al in the tetrahedral position was obtained using the calcined gel as precursor. In summary, three different characterization techniques: XRD, FTIR and NMR spectroscopies, indicated that the sample prepared from calcined gel precursor gave better defined structure of saponite with a maximum substitution in the tetrahedral sheet compared to samples prepared with the gel and dried gel precursors. The initial precipitation of precursor was expected to create a homogenous gel from its constituents [START_REF] Hamilton | The preparation of silicate compositions by a gelling method[END_REF]. The calcination of precursor removed water, and it was then expected that more amorphous regions with a large number of defects were created. Moreover, amorphous and crystalline Al(OH)3 containing compounds could undergo phase transitions at temperatures between 130 and 450 °C and were more likely to disappear during the calcination of the precursor [START_REF] Walker | Chlorite polytype geothermometry[END_REF][START_REF] Földvári | Handbook of Thermogravimetric System of Minerals and Its Use in Geological Practice[END_REF]. Also, during drying, water was gradually evaporated and a very basic environment could be locally created thus favoring the formation of tetrahedral Al 3+ . The obtained powder from calcined precursor was amorphous and very reactive upon dissolution and subsequent forming of crystalline phase in hydrothermal synthesis experiments.

Influence of synthesis temperature on structure formation

The increase of temperature could be expected to enhance the sample crystallinity, as shown by [START_REF] Kloprogge | The effect of synthesis temperature on the FT-Raman and FT-IR spectra of saponites[END_REF]. The calcined gel precursor was used to synthesize saponites at five different temperatures from 90 to 200 °C. The powder XRD pattern (Fig. 4) showed the regular saponite structure nearly independent from the synthesis temperature, without crystalline secondary phases. The layer-to-layer distances were between 1.43 and 1.48 nm with a slight trend to increase with synthesis temperature. Such layer-to-layer distance corresponds to Ca 2+ -saturated saponite with two layers of the water in the interlayer [START_REF] Suquet | Synthesis and Swelling Properties of Saponites with Increasing Layer Charge[END_REF]. The swelling property (typical of smectite clay minerals) was assessed using the saturation of oriented preparations with ethylene glycol (Fig. 5). For such preparations, the 001 reflection was pronounced because of the alignment of particles, and shifted its position when saturated with ethylene glycol due to the increase of the layer-to-layer distance. Only a slight shift of around 0.05 to 0.1 nm was observable and indicated a high charge of the layers as expected. Substitution of Al 3+ -for-Si 4+ in the silicate sheets resulted in a more localized charge that had to be compensated by interlayer cations compared to the aluminum substitution in the octahedral sheet, which was compensated over a larger volume by slightly more negatively charged oxygen atoms [START_REF] Kloprogge | Synthesis of smectite clay minerals: A critical review[END_REF]. The analysis of oriented slides indicated smaller layer-to-layer distances for the lower synthesis temperatures (90 and 120 °C) suggesting a stronger interaction between the layers and more pronounced localization of the charge in the tetrahedral sheets. Moreover, the diffraction peak was broadened with the decrease of the temperature, suggesting a smaller particle size and/or more heterogeneous structure. For a better insight into the amorphous phases and their bond structure, FTIR spectra were acquired for sample prepared at each synthesis temperature. The identified bands corresponded to bonds in saponite structure, as previously discussed in section 3.1 (Fig. 6). The peak shapes at 991 and 457 cm -1 corresponding to Si-O vibration modes expressed a higher crystallinity for the synthesis at 200 °C. At lower synthesis temperatures the reduced and broader peaks indicated smaller particle size and/or more heterogenous structures. Apart from the Si-O vibration bands, the rest of the spectrum seemed nearly unaffected by the synthesis temperature. To obtain structural information about the location of aluminum in the structure, NMR spectroscopy measurements were performed (Fig. 7). With the synthesis procedure used, the aluminum content in the tetrahedral sheet was about 80 to 90%, while the octahedral sheet only contained the small remaining amount. There was a slight trend towards a higher amount of octahedral Al at high synthesis temperatures (Table 1). This result was in agreement with the trend observed by the XRD experiment performed on oriented slides, which suggested the localization of the charge in the tetrahedral sheet with the decrease of the synthesis temperature. But because the quantification of the NMR data is rather approximate and a deviation of 5 to 10% should be assumed for interpretation, this trend had to be handled with care.

Nevertheless, it can be stated that even at low synthesis temperatures the vast majority of the substitution by Al happened in the tetrahedral sheet. The chemical composition of synthesized products was analyzed by XRF spectroscopy. The highest Al/Si and Ca/Si molar ratios were obtained for samples synthesized at 90 and 120 °C (Table 1). The higher amount of interlayer Ca 2+ correlated with the increased amount of Al substitutions in the structure. Assuming that no other phases were present in significant amounts, and that probably a slightly higher amount of Al was located in the tetrahedral sheets compared to octahedral sheets for samples prepared at a lower synthesis temperature, these results could then also be correlated to the XRD experiments on oriented slides. The analysis of oriented slides lead to the hypothesis that for the lowest synthesis temperatures (90 and 120 °C) the charge was mainly localized in the tetrahedral sheets. These results were then in good agreement with the chemical composition and NMR results. The creation of functional groups, as the Lewis acid groups, requires a maximum tetrahedral substitution. Here, the sample synthesized at 90 °C, using calcined precursors, showed that a high amount of functional groups could be created even at low temperature. In comparison with the study of [START_REF] Kloprogge | The effect of synthesis temperature on the FT-Raman and FT-IR spectra of saponites[END_REF], where a shorter synthesis time (3 days instead of 5 days) and a temperature of 280 °C were used, well defined saponite structure could be obtained even at lower temperature.

Nanostructure and porosity

Although the synthesis at 90 °C seemed to be the most promising, the FTIR spectroscopy results suggested a higher homogeneity and/or better crystallinity for the sample synthesized at 200 °C. Thus, with XRD and FTIR spectroscopy results alone, it remained unclear how amorphous constituents might be distributed. The observations of samples synthesized at three different temperatures (90, 150 and 200 °C) under TEM were undertaken to obtain an overview of particle size and shape (Fig. 8). The particle size was visually estimated to be between 50 and 100 nm. The TEM images showed a majority of crystalline regions, which was in agreement with the XRD and FTIR spectroscopy results. No particular difference, in terms of particle size and presence of amorphous aggregates, was observed among the samples synthesized at different temperatures. Thus, the slight differences observed by the XRD and FTIR spectroscopy analysis of oriented slides among the various samples could be due to the inhomogeneity of structure in terms of Al distribution in the structure, which increased with decreasing the temperature. In the TEM pictures the layer fringe contrast was used to approximate the basal spacing. In all locations chosen, the values were found between 0.9 to 1.1 nm. This was considerably smaller than the 1.4 to 1.5 nm obtained by powder XRD. This was due to the removal of interlayer water in the ultra-high vacuum. It has to be noted that the estimation of basal spacing was only approximate due to the unknown exact orientation angle of the layers. N2 adsorption measurements (Fig. 9) provided additional insight about the microstructure of the samples giving information about the accessible surface area, pore volume, and pore size distribution and shape. Adsorption at low relative pressure corresponded to the filling of the microporosity and at higher pressure to the mesoporosity. The specific surface area for each synthesis temperature, determined by the BET method, was up to 179 m 2 /g (Table 1). The samples prepared at 120 and 180 °C presented the lowest specific surface area. This did not correlate to the microstructural features investigated earlier and might therefore be connected to the sample preparation previous to the isotherm acquisition. The analysis of the shape of the isotherms (and of the hysteresis) gave information about the pore shape. The adsorption curve was a type IV and hysteresis was of H2 type, according to IUPAC classification.

The hysteresis shape corresponded to ink bottle shape cavities, which needed to reach a critical pressure (quite lower than during adsorption) in order to desorb nitrogen.

Applying the BJH method, average pore sizes of 6 to 8.7 nm was obtained (Table 1). With the t-plot technique, a small amount of micropores could be identified. [START_REF] Occelli | Surface area, pore volume distribution, and acidity in mesoporous expanded clay catalysts from hybrid density functional theory (DFT) and adsorption microcalorimetry methods[END_REF] found micropores with the diameter of 1.5 nm using the DFT model. The derivative of the pore volume over the pore width calculated by the DFT model was then also applied in this study. Clay minerals are known for their layered structure, which means that a slit-like pore shape with constant height but larger and varying width and length could be expected. However, the overall RMS error of the isotherm fit was significantly lower assuming a cylindrical pore shape than a slit one. The distribution showed two pore domains (Fig. 10).

Fig. 10

Pore size distribution derived from the adsorption isotherm with the DFT method assuming cylindrical pores on oxide surface with regularization of 0.1 .

The micropores were narrowly distributed around 1.1 nm. For the samples obtained at 120 and 180 °C nearly no micropores could be found by DFT. No clear trend in the pore size was observable for increasing synthesis temperature. Instead, the pore distribution seemed to remain the same. The mesoporous domain above 2 nm was at the origin of the most part of the pore volume. On the other hand, micropores contributed to the increase of surface area and their absence in these two samples explained the lower BET specific surface area measured. If existing, pillars in between the layers could result in more cylindrical shapes, but such structures were not revealed by XRD, FTIR spectroscopy or TEM, and they could be then excluded. However, applying a cylindrical pore shape a better fit with the presence of mesopores could be obtained, which contributed largely to the total volume. These observations showed the discrepancy on the application of the various models, since it is not easy to differentiate among different pore shapes for a certain pore dimension. Cylindrical mesopores could be more likely created in the intercrystallite spaces formed during aggregation.

Therefore, the pore size distribution and shape did not correlate (or only partially) to the materials intrinsic properties, as the charge, but to the preparation unitary steps, as grinding, sieving and outgassing.

CONCLUSIONS

In this study, saponites were synthesized and the influence of the heat treatment of the precursor was shown. The characterization of the synthesized products by XRD and FTIR spectroscopy indicated that a homogenous powder of saponite (in terms of structure and chemical composition) could be obtained using a precursor calcined at 450 °C. Additionally, a higher quantity of aluminum in tetrahedral configuration was measured for this sample, which might bring to a higher thermal stability and a higher amount of acidic groups useful in catalysis. By using the calcined precursor, hydrothermal synthesis performed at low temperature (90 °C) allowed to obtain a high purity saponite. A synthesis temperature of 200 °C resulted in a very similar product, probably presenting a slightly improved homogeneity, but with lower layer charge and consequently less tetrahedral aluminum.

The distribution of mesopores was found to be in the 2 to 11 nm range, promising for the accessibility of large organic molecules. Additionally, it was shown that pillaring or acidic activation could further increase the accessible surface area [START_REF] Kloprogge | Synthesis of smectites and porous pillared clay catalysts: a review[END_REF][START_REF] Michot | Surface heterogeneity in micropores of pillared clays: The limits of classical porefilling mechanisms[END_REF][START_REF] Trombetta | Solid acid catalysts from clays Evaluation of surface acidity of mono-and bi-pillared smectites by FT-IR spectroscopy measurements, NH3-TPD and catalytic tests[END_REF]. In contrast to montmorillonite, saponite can be more easily modified by acidic leaching due to of the presence of octahedral magnesium. Such acidic treatments could greatly enhance the microporosity and specific surface area (up to 510 m 2 /g) [START_REF] Vicente | Characterization, surface area, and porosity analyses of the solids obtained by acid leaching of a saponite[END_REF][START_REF] Bisio | On the acidity of saponite materials: A combined HRTEM, FTIR, and solid-state NMR study[END_REF]).

Fig. 1

 1 Fig. 1 Powder XRD patterns after hydrothermal synthesis at 200 °C using different

Fig. 3

 3 Fig. 3 27 Al MAS NMR spectra of samples synthesized at 200 °C with different

Fig. 4

 4 Fig. 4 Powder XRD pattern of synthetic saponite with indexed diffraction planes at

Fig. 5

 5 Fig. 5 XRD patterns of 001 reflection of oriented slides at ambient conditions (solid

Fig. 6

 6 Fig. 6 FTIR spectra of saponite synthesized at different temperatures.

Fig. 7

 7 Fig. 7 27 Al MAS NMR spectra of saponite synthesized at different temperatures indicating aluminum in octahedral ( VI Al) and tetrahedral sheet ( IV Al).

Fig. 8

 8 Fig. 8 TEM images of saponite synthesized at different temperatures. Basal spacings

Fig. 9

 9 Fig. 9 Isotherms of nitrogen adsorption and desorption for saponites synthesized at

Table 1 .

 1 Element molar ratios obtained by XRF and calculated ratio of tetrahedral aluminum obtained by NMR spectroscopy. Surface area and average pore size calculated from N2 adsorption measurement.

	Synthesis	Al/Si molar	Mg/Si	Ca/Si molar	IV Al by	SBET (m 2 /g)	Average
	temperature	ratio	molar ratio	ratio	NMR (%)		pore size
							(nm)
	90 °C	0.42	1.13	0.21	90	172	6.0
	120 °C	0.41	0.96	0.20	85	131	8.7
	150 °C	0.35	0.99	0.17	90	173	6.2
	180 °C	0.36	1.01	0.16	80	113	7.0
	200 °C	0.35	0.98	0.17	80	179	6.7

ACKNOWLEDGEMENTS

N2 adsorption experiments, XRD, XRF, NMR and TEM analysis were performed on the technical plateforms of IS2M. The authors are grateful to Laure Michelin, Loïc Vidal and Habiba Nouali for their contribution. The internship of Sebastian Meyer was funded by IS2M/UHA in the frame of an "Emerging" project call.

On behalf of all authors, the corresponding author states that there is no conflict of interest.


published or not. The documents may come

Given a field of scalars K and an indeterminate x, a natural way to describe a function field K(C) K(x) over K(x) with degree n and genus g is to choose a primitive element in it and to give its minimal polynomial. This is a polynomial in K(x) [y] and even in K[x, y] once we have cleared denominators. It has degree n in y and its degree in x cannot be expected to be significantly smaller than the genus g of the curve C in general. So this plane model involves about ng coefficients while the dimension of the underlying modular problem is rather linear in n+g. Indeed a plane model of a general curve of genus g tends to have singularities and tells more about these singularities than about the function field itself. Another more natural possibility to represent a function field K(C) K(x) is to consider the canonical model of the curve C and express x as a rational fraction of the holomorphic differentials. Even Petri's description of the ideal of canonical curves [3, Chapter III, §3] however involves a number of parameters that is quadratic in the genus g.

Similarly, a natural way to describe a number field K Q of degree n and root discriminant δ K is to choose a primitive element in it and give its minimal polynomial. This is a polynomial in Q[x] and even in Z[x] once we have cleared denominators. It has degree n in x and its height cannot be expected to be significantly smaller than the absolute value of the discriminant d K = δ n K in general. So this model has bit size of order n×log d K while the logarithm of the number of number fields of degree n and discriminant bounded by D in absolute value is conjectured [START_REF] Cohen | Counting discriminants of number fields[END_REF][START_REF] Malle | On the distribution of Galois groups[END_REF], and even proven for n ⩽ 5 [START_REF] Davenport | On the density of discriminants of cubic fields[END_REF][START_REF] Bhargava | The density of discriminants of quartic rings and fields[END_REF][START_REF] Bhargava | The density of discriminants of quintic rings and fields[END_REF], to be bounded by log D plus a function of n alone. The reason for this discrepancy is that the quotient ring of Z[x] by a unitary irreducible polynomial tends to be highly singular and provides more information than the mere isomorphism class of the corresponding number field.

In this text we construct two families of natural models for a function field of genus g and degree n over K(x) where K is a field of cardinality q. These models involve a number of coefficients in K of order n(log n) 3 (g n+1+log q n) and n(log n) 2 (g n+1+log q n) respectively. In case q = +∞ we set log q n = 0 in these formulae. The first model is obtained by picking few functions (κ j ) 1⩽j⩽r of small degree in K(C) and finding enough relations with small degree between them to produce local equations for an affine model of C. The second model is obtained from the first one by providing long enough formal series expansions of the κ j to be able to recover the equations. This second model is unambiguous: we can recover the full ideal of relations from these formal expansions, not just local equations. It is also a bit smaller. It consists of a finite 0-dimentional affine K-scheme (a finite set in a finite K-algebra) that pins down a model of C.

Using (not so) parallel constructions in number theory we propose two families of models for a number field, having bit size of order n(log n) 3 (log δ K + log n) and n(log n) 2 (log δ K + log n) respectively. The first model, which was already presented in [START_REF] Couveignes | Enumerating number fields[END_REF], is obtained by picking few small integers (κ j ) 1⩽j⩽r in K and finding enough relations with small degree and small height between them to produce local equations for an affine model of Spec K. The second model is obtained from the first one by picking a small unramified prime p an providing long enough p-adic approximations of the κ j to characterize the number field. Again this second model is unambiguous and a bit smaller. It consists of a finite 0-dimentional affine scheme (a finite set in a finite algebra) that pins down a model of Spec K.

In this text, the notation Q stands for a positive absolute constant. Any sentence containing this symbol becomes true if the symbol is replaced in every occurrence by some large enough real number.

DEFINITIONS AND MAIN STATEMENTS

We recall notation about function fields and number fields. We then define two ways of specifying a function field K(C). We can provide local equations for a 0-dimentional variety over the rational field K(x) such that the function field K(C) is the residue field of this variety.

We can also provide formal expansions for the coordinates of some formal point on the former model. We state the existence of small models of either kind for any function field K(C) K(x). We similarly define two ways of specifying a number field K. One option is to provide local equations for a 0-dimentional variety over Q such that the number field K is the residue field of this variety. Another option is to provide p-adic expansions for the coordinates of some pinfinitesimal point on the former model, with enough accuracy to recover the equations. We state the existence of small models of either kind for any number field K.

1.1. Function fields. Let K be a field. Let q be the cardinality of K. If q is finite we write log q for the logarithm function with base q. If q = +∞ then log q is the constant zero function. Let

P 1 K = Proj K[x 0 , x 1 ] be the projective line over K. Let x = x 1 x 0 and let A 1 K = Spec K[x
] be the affine line over K. We call ∞ the point (0, 1) on P 1 K with projective coordinate x 0 = 0. Let C be a smooth absolutely integral projective curve over K. Let f ∶ C → P 1 be a finite separable map and let K(C) K(x) be the corresponding regular field extension. In this paper we shall call such an extension a function field. We shall denote n the degree of f and g the genus of C.

1.2. Function fields as incomplete intersections. Let K(C) K(x) be a function field as in Section 1.1. Let r ⩾ 1, d x ⩾ 1 and d y ⩾ 1 be integers. Let y 1 , . . . , y r be indeterminates. Let E 1 , E 2 , . . . , E r be polynomials in K[x][y 1 , . . . , y r ] with total degree ⩽ d y in the y j and with degree

⩽ d x in x. Let I → A 1 = Spec K[x] be the K[x]-scheme with equations E 1 = E 2 = ⋅ ⋅ ⋅ = E r = 0 and det (∂E i ∂y j ) 1⩽i, j⩽r = 0.
We assume that the generic fiber I ⊗ K(x) → Spec K(x) has an irreducible component isomorphic to Spec K(C) → Spec K(x). We call C → A 1 = Spec K[x] the Zariski closure of this component in I. We say that C → A 1 is an incomplete intersection model of the function field in dimension r and degrees (d x , d y ).

1.3.

Interpolant of an infinitesimal point. Let r ⩾ 1, m ⩾ 1, d x ⩾ 1 and d y ⩾ 1 be integers. Let K be a field. Let L ⊃ K be a finite separable field extension and let l be its degree. Let λ be a primitive element in L. Let S ⊃ L be a finite separable field extension and let s be its degree. Let t be an indeterminate.

Let b m 1 , . . . , b m r be r elements in S[[t]] t m . Let a ∞ = λ + t ∈ S[[t]] and a m = a ∞ mod t m ∈ S[[t]] t m .
Let x, y 1 , . . . , y r be r + 1 indeterminates. Let

β m ∶ K[x][y 1 , . . . , y r ] → S[[t]] t m
be the morphism of K-algebras that maps x onto a m and y j onto b m j for 1 ⩽ j ⩽ r. We assume that β m is surjective. We denote by b m the corresponding S[[t]] t m point on A r+1 .

Let J dx,dy be the ideal of K[x][y 1 , . . . , y r ] generated by all polynomials in Ker β m having total degree ⩽ d y in y 1 , . . . , y r and degree ⩽ d x in x. We say that a point

b ∞ = (a ∞ , b ∞ 1 , . . . , b ∞ r ) ∈ (S[[t]]) r+1 is the interpolant of b m in degrees (d x , d y ) if (1) b ∞ j mod t m = b m j for every 1 ⩽ j ⩽ r, ( 2
) all polynomials in J dx,dy vanish at b ∞ , (3) and b ∞ is unique with the two above properties.

1.4. Short models of function fields. We shall prove that both incomplete intersections and interpolants provide short descriptions of function fields.

Theorem 1 (Function fields as small incomplete intersections). The exists an absolute constant Q such that the following is true. Let K be a field and let K(C) K(x) be a function field of genus g ⩾ 2 and degree n as in Section 1.1. Let q be the, possibly infinite, cardinality of K. Call r the smallest positive integer such that 2r r ⩾ n(r + 1).

Let h be smallest non-negative integer such that q h+1 > nr(r + 1).

Let ν = 2 + ⌈ 2(g -1) n ⌉ and d x = r(h + ν) and d y = r.
There exists an incomplete intersection model of K(C) K(x) in dimension r and degrees (d x , d y ).

The total number of K-coefficients in this model is

⩽ Q(log n) 3 (g + n(1 + log q n)).
The meaning of this theorem is that we have a short description of K(C) K(x) as a quotient of a finite algebra : the smooth zero-dimensional part of a complete intersection of small degree in a projective space of small dimension. In particular the number of affine parameters required to decribe the incomplete intersection is almost linear in g + n. There remains a little uncertainty concerning which irreducible component of the complete intersection is of interest to us. One way of removing this uncertainty if to specify a geometric point on the targeted component. We then realize that giving enough such geometric points (counting multiplicities) enables us to reconstruct the equations. This leads us to the next theorem.

Theorem 2 (Function fields from interpolants). The exists an absolute constant Q such that the following is true. Let K be a field and let K(C) K(x) be a function field of genus g ⩾ 2 and degree n as in Section 1.1. Let q be the, possibly infinite, cardinality of K. Let r, h, ν, d x and d y be as in Theorem 1. Let ρ = d x + r(ν + h). There exist integers l, s, m and finite separable field extensions L K and S L of respective degrees l and s, and an element λ in K such that L = K(λ), and a point

b m = (λ + t mod t m , b m 1 , . . . , b m r ) ∈ (S[[t]] t m ) r+1 such that b m has an interpolant b ∞ = (λ + t, b ∞ 1 , . . . , b ∞ r ) ∈ (S[[t]]) r+1 in degrees (d x , d y ).
And there is an isomorphism of K-algebras from

K(λ + t, b ∞ 1 , . . . , b ∞ r ) ⊂ S[[t]] onto K(C) ⊃ K(x) that sends λ + t onto x. Further 1 ⩽ l ⩽ O(1 + log q (1 + g n) + log q n) and 1 ⩽ s ⩽ n,
and m is the smallest positive integer such that mls > nρ, and

rmls ⩽ Q(log n) 2 n(1 + log q n) + g .
The meaning of this theorem is that a model of the function field can be recovered from short formal expansions of a few functions in it. The function field is recovered from r formal expansions of length m having coefficients in an extension of degree ls of K. This involves rmls coefficients in K. So the number of parameters in this model is bounded by

Q(log n) 2 n(1 + log q n) + g . 1.
5. Number fields. By a number field we mean a finite field extension K Q of the rational field. We denote by n the degree of this extension and by O the ring of integers of K. We denote by (ρ i ) 1⩽i⩽r the r real embeddings of K and by (σ j , σj ) 1⩽j⩽s the 2s complex embeddings of K. We also denote by (τ k ) 1⩽k⩽n the n = r + 2s embeddings of K. We let

K R = K ⊗ Q R = R r × C s
be the Minkowski space. We follow the presentation in [14, Chapitre 1, §5]. An element x of K R can be given by r real components (x ρ ) ρ and s complex components (x σ ) σ . So we write x = ((x ρ ) ρ , (x σ ) σ ). For such an x in K R we denote by x the maximum of the absolute values of its r + s components. The canonical metric on K R is defined by

< x, y >= 1⩽i⩽r x i y i + 1⩽j⩽s x j ȳj + xj y j .
The corresponding Haar measure is said to be canonical also. The canonical measure of the convex body {x,

x ⩽ 1} is 2 r (2π) s ⩾ 2 n .
The map a ↦ a ⊗ 1 injects K and O into K R . For every non-zero x in O we have

x ⩾ 1. Let (α i ) 1⩽i⩽n be any Z-basis of O. Set A = (τ j (α i )) 1⩽i,j⩽n . The product A Āt is the Gram matrix B = (< α i , α j >) 1⩽i
,j⩽n of the canonical form in the basis (α i ) i . This is a real symmetric positive matrix. The volume of O according to the canonical Haar measure is

v O = det(B) = det(A) .
We let

d K = det(AA t )
be the discriminant of K and we denote by

δ K = d K 1 n
the root discriminant. The square of the volume of O is d K .

1.6. Number fields as incomplete intersections. Let K Q be a number field. Let r ⩾ 1, d ⩾ 1, and H ⩾ 1 be integers. Let E 1 , E 2 , . . . , E r be r polynomials of total degree ⩽ d in Z[x 1 , . . . , x r ] all having coefficients bounded in absolute value by H. Let I be the Z-scheme with equations

E 1 = E 2 = ⋅ ⋅ ⋅ = E r = 0 and det (∂E i ∂x j ) 1⩽i, j⩽r = 0.
We assume that I ⊗ Q → Spec Q contains Spec K as one of its irreducible components. We call C → Spec Z the Zariski closure of this component in I. This is the spectrum of some order in K.

We say that C → Spec Z is an incomplete intersection model of K Q in dimension r, degree d and height H.

1.7. Interpolant of a p-infinitesimal point.
Let p be a prime integer. We fix some algebraic closure of Q p and call Q nr p the maximal unramified extension in it. For every power q = p f of p we call Q q the unique subextension of Q nr p Q p of degree f and Z q its ring of integers. Let r ⩾ 1,

s ⩾ 1, m ⩾ 1, d ⩾ 1 and H ⩾ 1 be integers. Let b m 1 , . . . , b m r be r elements in Z p s p m . Let x 1 , . . . , x r be r indeterminates. Let β m ∶ Z[x 1 , . . . , x r ] → Z p s p m be the ring homomorphism sending x j onto b m j for 1 ⩽ j ⩽ r. We assume that β m is surjec- tive. We call b m = (b m 1 , . . . , b m r ) the corresponding point in A r (Z p s p m ). Let J H,d be the ideal of Z[x 1 , . . . , x r ]
generated by all polynomials in Ker β m having total degree ⩽ d and all their coefficients bounded by H in absolute value. We say that a point

b ∞ = (b ∞ 1 , . . . , b ∞ r ) ∈ (Z p s ) r is the interpolant of b m in degree d and height H if (1) b ∞ j mod p m = b m j for every 1 ⩽ j ⩽ r, ( 2
) all polynomials in J H,d vanish at b ∞ , (3) and b ∞ is unique with the two above properties.

1.8. Short models of number fields. We shall prove that both incomplete intersections and interpolants provide short descriptions of number fields.

Theorem 3 (Number fields have small incomplete intersection models).

There exists a positive constant Q such that the following is true. Let K be a number field of degree n ⩾ Q and root discriminant δ K . Let r be the smallest positive integer such that 2r r ⩾ n(r + 1).

Set (1) = 2r r -n and H = 2n × 2r r 1 2 n 2 r(r + 1)δ 2 K r .
Then r ⩽ Q log n and log H ⩽ Qlog n(log n + log δ K ) and K has an incomplete intersection model in dimension r, degree d = r and height H. Further d + r r = 2r r ⩽ Qn log n so that the total number of coefficients in the r equations is ⩽ Qn(log n) 2 .

The meaning of Theorem 3 is that we have a short description of K Q as a quotient of a finite algebra associated with the smooth zero-dimensional part of a complete intersection of small degree and small height in a projective space of small dimension. The bit size of the model is ⩽ Qn(log n) 3 (log δ K + log n). Every factor in this estimate is awaited but the (log n) 3 . Another annoyance is the remaining little uncertainty about which irreducible component of the complete intersection is of interest to us. One way of removing this uncertainty if to specify a geometric point on the targeted component. We then realize that giving enough such geometric points (counting multiplicities) enables us to reconstruct the equations. This leads us to the next theorem.

Theorem 4 (Number fields from interpolants). The exists an absolute constant Q such that the following is true. Let K be a number field of degree n ⩾ Q and root discriminant δ K over Q. Let r and H be as in Theorem 3. There exist positive integers s ⩽ n, and m and a prime integer

p ⩽ Qn(log n) 2 (log n + log δ K ) and a point b m = (b m 1 , . . . , b m r ) ∈ (Z p s p m ) r such that b m has an interpolant b ∞ = (b ∞ 1 , . . . , b ∞ r ) ∈ (Z p s ) r
in degree r and height H. And the field

Q(b ∞ 1 , . . . , b ∞ r ) ⊂ Q q is isomorphic to K. And rms log p ⩽ Qn(log n) 2 (log n + log δ K ) .
The meaning of this theorem is that a model of the number field can be recovered from a few short p-adic expansions of a few algebraic numbers in it. The number field is recovered from s, m, p and by r elements in Z q p m Z q where q = p s . Each of these r ⩽ Q log n elements is a q-adic expansion of bit size ms log p. So the bit size of this model is bounded by Qn(log n) 2 (log n + log δ K ).

CONSTRUCTING MODELS OF FUNCTION FIELDS

In this section we prove Theorems 1 and 2. In Section 2.1 we recall the definition of the Maroni invariants and we bound them. In Section 2.2 we use a generic well-posedness theorem in multivariate Hermite interpolation due to Alexander and Hirschowitz in order to prove the existence of small affine models of function fields. In Section 2.3 we prove that formal expansions of coordinates at a smooth point in these models characterize the function field.

2.1. Small degree functions. Let K be a field. Let K(C) K(x) be a function field as in Section 1.1. We call C aff the affine part

C aff = f -1 (A 1 ) ⊂ C of C. The sheaf f ⋆ O C is
a rank n locally free P 1 -module. Every such sheaf decomposes as a direct sum of invertible sheaves. So (2)

f ⋆ O C = ⊕ 0⩽i⩽n-1 O P 1 (-a i )
where a 0 = 0 and the remaining a i form a non decreasing sequence of strictly positive integers, called the scrollar invariants, or the Maroni invariants of the cover. We call O i the i-th term on the right hand side of Equation ( 2). Let ωi be a generator of H 0 (P 1 , O i ⊗ O P 1 (a i )) and set

ω i = x -a i 0 ωi
. This is a function on C, regular on C aff , and the maximum order of its poles above ∞ is a i . The family

(ω i ) 0⩽i⩽d-1 is a K[x] basis of H 0 (A 1 , f ⋆ (O C )) = K[C aff ] the integral closure of K[x] in K(C). The determinant sheaf det(f ⋆ O C ) = ∧ d f ⋆ O C is isomorphic to O P 1 (-a) where a = 0⩽i⩽n-1 a i = n + g -1.
Because f ⋆ O C is a sheaf of P 1 -algebras, there exists a multiplication tensor

m ∈ H 0 (P 1 , f ⋆ O C ⊗ f ⋆ O C ⊗ f ⋆ O C ).
For every triple (i, j, k) of integers with 0 ⩽ i, j, k ⩽ n -1, we call

m k i,j ∈ H 0 (P 1 , Ôi ⊗ Ôj ⊗ O k ) the corresponding component of m. Since m k i,j is a global section of a sheaf isomorphic to O P 1 (a i + a j -a k ) we have (3) a i + a j ⩾ a k whenever m k i,j is non-zero. We call µ k i,j ∈ K[x] the coefficients of the multiplication table of K[C aff ]
in the basis (ω 0 , . . . , ω n-1 ). We have

ω i ω j = k µ k i,j ω k
and µ k i,j is a polynomial of degree ⩽ a i + a ja k . In particular µ k i,j is zero unless inequality (3) holds true.

There exists a permutation σ of the set of integers in the interval [1, n -2] such that, for every integer i in this interval, the coefficient µ n-1 i,σ(i) is non zero. Otherwise the determinant det(µ n-1 i,j ) 1⩽i,j⩽n-2 would be zero and there would exist a function η in the K(x)-vector space generated by ω 1 , ω 2 , . . . , ω n-2 such that ηW ⊂ W where W is the K(x)-vector space generated by ω 0 = 1, ω 1 , . . . , ω n-2 . This would turn W into a K(x, η)-vector space. But this is imposible because the extension K(x, η) K(x) has degree at least two and the codimension of W in K(C) is 1. We deduce that a i + a σ(i) ⩾ a n-1 for 1 ⩽ i ⩽ n -2 and summing out we find

(n -2)a n-1 ⩽ 2 1⩽i⩽n-2 a i = 2(a -a n-1 ) that is (4) a n-1 ⩽ 2a n = 2 1 + g -1 n .
This inequality and its proof are the function field analogue to [START_REF] Bhargava | Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves[END_REF][Theorem 3.1].

Lemma 1. The Maroni invariants of a function field of degree n and genus g are bounded form above by

ν = 2 + ⌈ 2(g -1) n ⌉
the smallest integer bigger than or equal to 2 + 2(g -1) n.

So the field extension K(C) K(x) has a basis made of functions in K[C aff

] with small degree.

2.2.

Small equations for function fields. Having found small degree functions (ω i ) 0⩽i⩽n-1 we now look for small degree relations between them. Let r ⩾ 2 and h ⩾ 0 be integers. For every pair of integers (i, j) with 0 ⩽ i ⩽ n -1 and 1 ⩽ j ⩽ r we let

u i,j (x) ∈ K[x] h
be a polynomial of degree ⩽ h in x. For 1 ⩽ j ⩽ r we set

κ j = 0⩽i⩽n-1 u i,j ω i ∈ K[C aff ].
The functions κ j have all their poles above ∞ and the order of these poles is bounded from above by h + ν. We define a morphism of K[x]-algebras

aff ∶ K[x][y 1 , . . . , y r ] → K[C aff
] by sending y j to κ j for 1 ⩽ j ⩽ r. This results in a morphism

e aff ∶ C aff / / f aff ) ) A r+1 K = Spec K[x][y 1 , . . . , y r ] s s A 1 K = Spec K[x] The image by e aff of the cycle [C aff ⊗ f aff K(x)] is a 0-cycle P of degree n on A r K(x)
. We let I be the corresponding ideal of K(x)[y 1 , . . . , y r ] and denote by 2P ⊂ A r K(x) the subscheme associated with I 2 . We denote by C → A 1 the Zariski closure of P K(x) in A r+1 . Let d ⩾ 1 be an integer such that ( 5)

n(r + 1) ⩽ d + r d . Let V = K(x)[y 1 , . . . , y r ] d
be the K(x)-vector space of polynomials having total degree ⩽ d in the variables y 1 , . . . , y r and let M be the basis of V consisting of monomials. We say that 2P is well poised in degree d if the restriction to V of the quotient map

K(x)[y 1 , . . . , y r ] → K(x)[y 1 , . . . , y r ] I 2
is surjective. We let Ω be a separable closure of K(x) and we call T the set of K(x)-embeddings τ ∶ K(C) → Ω.

There are n such embeddings. For every τ in T we call

P τ = (τ (κ j )) 1⩽j⩽r ∈ A r (Ω)
the corresponding geometric point of P . The scheme 2P is well poised if and only if the matrix The maximal minors of M 1 P are polynomials of degree ⩽ dn(r + 1) in each of the u i,j and one of them is not identically zero. The latter determinant cannot vanish on the cartesian product (K[x] h ) nr as soon as the cardinality of K[x] h is bigger than dn(r + 1). If K is a finite field with cardinality q this condition is granted as soon as q h+1 > dn(r + 1).

M 1 P = [(m(P τ )) τ ∈T,
If K is infinite we can afford h = 0. We will assume that 2P is well poised in degree d. So P is well poised also and the map e aff ⊗ K(x) is a closed immersion.

We look for small degree equations between the (κ j ) 1⩽j⩽r . More precisely we look for polynomials in Ker aff ⊂ K[x][y 1 , . . . , y r ] having total degree ⩽ d in the y j and degree in x as small as possible. We denote K[x][y 1 , . . . , y r ] d the K[x]-module of polynomials with total degree ⩽ d in the y j . This is a free K[x]-module of rank d+r d . We define a morphism of

K[x]-modules K[x][y 1 , . . . , y r ] → K[C aff ] ⊗ x d(h+ν) 0 = H 0 (A 1 , f ⋆ O C ⊗ P 1 O P 1 (d(h + ν)))
by sending ∏ j y

γ j j to x d(h+ν) 0 ∏ j κ γ j j for all exponents (γ 1 , . . . , γ r ) with ∑ j γ j ⩽ d. This morphism of K[x]-modules extends to a morphism of P 1 -modules ⩽d ∶ Sym ⩽d O ⊕r P 1 → f ⋆ O C ⊗ P 1 O P 1 (d(h + ν))
that is generically surjective. The kernel of ⩽d is a locally free sheaf of rank

= d + r d -n.
The image of ⩽d is a locally free sheaf of rank n and its determinant has degree

deg det Im ⩽d ⩽ deg det f ⋆ O C ⊗ P 1 O P 1 (d(h + ν)) = -a + nd(h + ν).
Ker ⩽a ≃ ⊕ 1⩽i⩽ O P 1 (-e i ).
More explicitly there exist polynomials

(E i ) 1⩽i⩽ in K[x][y 1 , . . . , y r ] d such that the coefficients of E i are polynomials in K[x] of degree ⩽ e i and E i (κ 1 , . . . , κ r ) = 0. Further 1⩽i⩽ e i ⩽ n(h + ν)d -a.

On the one hand

e i ⩽ ⌊ n(h + ν)d -a n ⌋ ⩽ (h + ν)d
for every 1 ⩽ i ⩽ + 1n. On the other hand the scheme 2P is well poised and the Ω-vector space generated by the E i for 1 ⩽ i ⩽ + 1n has codimension n -1 < n in Ker ⩽d ⊗ P 1 Ω. So there exists at least one embedding τ ∈ T such that the ( + 1n) × r matrix

((∂E i ∂y j )(P τ )) 1⩽i⩽ +1-n, 1⩽j⩽r
has maximal rank r. This means that there exist r integers

1 ⩽ i 1 < i 2 < ⋅ ⋅ ⋅ < i r ⩽ + 1 -n such that the minor determinant (6) Φ = det ((∂E i k ∂y j )) 1⩽k, j⩽r
does not vanish at P τ for some τ and thus for all τ by Galois action. We now choose d, r, and h depending on g, n, and q. We take r = d and in order to grant the condition in Equation ( 5) we choose r to be the smallest integer such that

1 r + 1 2r r ⩾ n. Since 2k k ⩾ 2 k for every integer k ⩾ 1 we have 1 k + 1 2k k ⩾ 2 k 2
for k large enough. Further

1 k + 2 2k + 2 k + 1 ⩽ 1 k + 1 2k k × 4
for k ⩾ 1. We deduce that

(7) 1 r + 1 2r r ⩽ 4n and r ⩽ Q log n.
If K is infinite we set h = 0. If K has q elements we take h to be the smallest integer such that q h+1 > nr(r + 1).

We check that ( 8)

h ⩽ Q(1 + log q n).
We have

n + = 2r r ⩽ 4n(r + 1)
and ( 9)

e i ⩽ d(h + ν) ⩽ Q(log n)(1 + log q n + g n )
for every

1 ⩽ i ⩽ + 1 -n. The number of K-coefficients in E i for each 1 ⩽ i ⩽ + 1 -n is thus bounded from above by 2r r × (1 + max 1⩽i⩽ +1-n e i ) ⩽ Q(log n) 2 (g + n(1 + log q n)).
This finishes the proof of Theorem 1.

2.3.

Interpolants for function fields. We set

d x = (h + ν)r
an upper bound for the degree in x of the equations E i k . In order to prove Theorem 2 we look for a fiber of f aff ∶ C aff → A 1 where the determinant Φ of Equation ( 6) does not vanish. Let Ψ(x) ∈ K[x] be the product of all Φ(P τ ) for τ ∈ T . We bound the degree of Ψ(x). First Φ has degree ⩽ rd x in x and total degree ⩽ r(r -1) in the y j . Since the α j have poles of order at most ν + h we deduce that the evaluation of Φ at (α j ) 1⩽j⩽r is a function in K[C aff ] with poles of order ⩽ (ν + h)r(r -1) + rd x . The norm of this function is Ψ(x) and its degree in x is bounded from above by n times (ν + h)r(r -1) + rd x . So

deg Ψ ⩽ Qn(log n) 2 (1 + log q n + g n )
according to Equations ( 7), ( 9) and ( 8).

If K is infinite there exist infinitely many unitary polynomials of degree one in K[x] that are prime to Ψ(x). If K is finite with cardinality q then we use the following lemma. Lemma 2. Let K be a finite field with q elements. Let Ψ(x) ∈ K[x] be a polynomial with degree deg Ψ ⩾ 1. There exists an irreducible polynomial in K[x] that is prime to Ψ and has degree ⩽ log q (deg Ψ) + 1.

For every integer k ⩾ 1 we set Π k (x) = x q kx. This is a separable polynomial in K[x] and all its irreducible factors have degree dividing k. If q k > deg Ψ then at least one of these irreducible factors is prime to Ψ. We take k to be the smallest integer bigger than log q deg Ψ. ◻ We apply Lemma 2 to our Ψ(x) and find a unitary irreducible polynomial F (x) ∈ K[x] that is prime to Ψ(x) and has degree

l ⩽ log q (deg Ψ) + 1 ⩽ O(1 + log q (1 + g n) + log q n).
Let K be an algebraic closure of K. Let λ ∈ K be a root of F . Let σ be a point in C( K) such that f (σ) = λ. We set L = K(λ) and S = K(σ) the residual field at σ. Let s be the degree of S L. We set

ρ = d x + r(ν + h) ⩽ Q(log n)(1 + log q n + g n )
and let m be the smallest positive integer such that mls > nρ. We denote b ∞ 1 , . . . , b ∞ r the expansions of the functions of κ 1 , . . . , κ r at the point σ in the local parameter t = xλ. We set

b m j = b ∞ j mod t m ∈ S[[t]] t m for every 1 ⩽ j ⩽ r. The point b m = (λ + t mod t m , b m 1 , . . . , b m r ) in A r+1 (S[[t]] t m ) has an interpolant in degrees (d x , d y ) and this interpolant is b ∞ = (λ + t, b ∞ 1 , . . . , b ∞ r ).
Indeed let E(x, y 1 , . . . , y r ) be a polynomial in K[x][y 1 , . . . , y r ] with degree ⩽ d x in x and with total degree ⩽ d y = r in the y j . The polynomial E induces a function in K[C aff ] having at most n poles with order ⩽ d x + r(ν + h) = ρ each. If this function vanishes at b m then the divisor of its zeros has degree mls > nρ. So E induces the zero function on C aff . Therefore it vanishes at b ∞ . Unicity follows from the existence of r equations with degree ⩽ d x in x and total degree ⩽ d y in the y j having a non-zero Jacobian determinant at σ. Since nρ ls < m ⩽ nρ ls + 1

we have mls ⩽ nρ + ls ⩽ Q(log n)(n(1 + log q n) + g) and r ⩽ Q log n. This finishes the proof of Theorem 2.

CONSTRUCTING MODELS OF NUMBER FIELDS

We prove Theorems 3 and 4. A natural counterpart to sheaves of modules over P 1 in the context of number fields are euclidean lattices. Sections 3.1 and 3.2 summarize [START_REF] Couveignes | Enumerating number fields[END_REF]. These sections realize the spectrum of some order of K as an irreducible component of a complete intersection in some affine space over Z. Section 3.3 finds a smooth F p -point on this irreducible component and thickens it just enough to characterize the number field.

3.1. Small elements in number fields. The ring of integers of a number field can be seen as a euclidean lattice. The existence of an integral domain structure which is compatible with the L ∞ -norm restricts the possibilities for the successive minima: they must all have the same order of magnitude. Lemma 3 (Number fields have small integers). The ring of integers O of a number field K with degree n and root discriminant δ K contains n linearly independant elements (ω i ) 1⩽i⩽n over Z such that all the absolute values of all the ω i are ⩽ δ 2 K . This is Proposition 1 in [START_REF] Couveignes | Enumerating number fields[END_REF]. Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and ZhaoSee prove in [START_REF] Bhargava | Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves[END_REF][Theorem 3.1] a similar statement.

3.2.

Small equations for number fields. Having found small integers (ω i ) 0⩽i⩽n-1 we now look for relations between them with small degree and small height. Let d ⩾ 5 and r ⩾ 1 be two integers. We assume that

n(r + 1) ⩽ d + r d .
Let M be the set of monomials of total degree ⩽ d in the r variables x 1 , . . . , x r . We have

A r C = Spec C[x 1 , . . . , x r ] ⊂ Proj C[x 0 , x 1 , . . . , x r ] = P r C . Let V C
be the C-linear space generated by M . We may associate to every element in M the corresponding degree d monomial in the r + 1 variables x 0 , x 1 , . . . , x r . We thus identify V C with H 0 (O P r C (d)), the space of homogeneous polynomials of degree d. We call T the set of all n embeddings of K into C. We let (ω i ) 1⩽i⩽n be n linearly independant short elements in O as in Lemma 3. We pick rn rational integers (u i,j ) 1⩽i⩽n, 1⩽j⩽r and we set κ j = 1⩽i⩽n u i,j ω i for 1 ⩽ j ⩽ r. For every τ ∈ T we consider the points

P τ = ( 1⩽i⩽n u i,j τ (ω i )) 1⩽j⩽r ∈ C r = A r (C) ⊂ P r (C)
and call P the union of these n points. The maximal minors of the matrix

M 1 P = [(m(P τ )
) τ, m∈M , (∂m ∂x 1 (P τ )) τ, m∈M , . . . , (∂m ∂x r (P τ )) τ, m∈M ] are polynomials of degree ⩽ dn(r + 1) in each of the u i,j and one of them is not identically zero according to the theorem of Alexander and Hirschowitz [START_REF] Alexander | Singularités imposables en position générale à une hypersurface projective[END_REF][START_REF] Alexander | Polynomial interpolation in several variables[END_REF]. The latter determinant cannot vanish on the cartesian product [0, dn(r + 1)] nr . Thus there exist nr rational integers u i,j in the range [0, dn(r + 1)] such that the corresponding scheme 2P is well poised: it imposes n(r+1) independent conditions on degree d homogeneous polynomials. We assume that the u i,j meet these conditions. We look for polynomials with degree ⩽ d and small integer coefficients vanishing at P . We denote by

V R = R[x 1 , . . . , x r ] d the R-vector space of polynomials in R[x 1 , . . . , x r ] of degree ⩽ d.
There is a unique R-bilinear form on V R that turns the set M of monomials into an orthonormal basis. Let V Z = Z[x 1 , . . . , x r ] d . The lattice of relations with integer coefficients and degree ⩽ d is a free

Z-module L ⊂ V Z of rank = d + r d -n.
We set L = L ⊗ Q R the underlying R-vector space and L ⊥ its orthogonal complement in V R . We denote by L ⊥ the intersection

L ⊥ = L ⊥ ∩ V Z . Since V Z is
(M 0 P ) t T t ) = det T M 0 P M 0 P t T t = det M 0 P M 0 P t d K .
So the square of the volume of the lattice of relations is bounded by the determinant of the hermitian positive definite matrix M 0 P M 0 P t . We will use several times the following lemma.

Lemma 4. If E(x 1 , . . . , x r ) ∈ Z[x 1 , . . . , x r ] is a polynomial with degree ⩽ d E and height H E then for every embedding τ the evaluation of E at P τ is bounded in absolute value by

H E n 2 d(r + 1)δ 2 K d E d E + r r .
The coefficients u i,j are bounded by dn(r + 1). All the absolute values of the ω i are bounded by δ 2 K . So the terms in E are bounded from above by

H E n 2 d(r + 1)δ 2 K d E
and there are at most d E +r r of them. ◻ Recall that the coefficients in M 0 P are degree ⩽ d monomials in the κ j . We deduce from Lemma 4 that they are bounded from above by

n 2 d(r + 1)δ 2 K d .
The coefficients in M 0 P M 0 P t are bounded from above by

D = d + r d n 2 d(r + 1)δ 2 K 2d .
The matrix M 0 P M 0 P t being hermitian positive definite, its determinant is bounded from above by the product of the diagonal terms. We deduce that the volume of the lattice L of relations is bounded from above by D n 2 . Recall that the dimension of L is

= d + r d -n.
For any x in V R we denote by x the 2 -norm in the monomial basis.

E i ⩽ 2 D n 2 .
We assume that the sequence i ↦ E i is non-decreasing and deduce that

E i ⩽ 2( +1-i) D n 2( +1-i)
for every 1 ⩽ i ⩽ . We forget the last n -1 equations. On the one hand

E i ⩽ H(K, d, r) for every 1 ⩽ i ⩽ + 1 -n, where (10) H(K, d, r) = 2n D 1 2 = 2n × d + r d 1 2 n 2 d(r + 1)δ 2 K d .
On the other hand the scheme 2P is well poised and the C-vector space generated by the E i for 1

⩽ i ⩽ + 1 -n has codimension n -1 < n in L ⊗ R C. So there exists at least one embedding τ such that the ( + 1 -n) × r matrix ((∂E i ∂x j )(P τ )) 1⩽i⩽ +1-n, 1⩽j⩽r
has maximal rank r. There exist r integers 1 ⩽ i 1 < i 2 < ⋅ ⋅ ⋅ < i r ⩽ + 1n such that the value at P τ of the minor determinant [START_REF] Lee | Upper bound on the number of extensions of a given number field[END_REF] Φ = det ((∂E i k ∂x j )) 1⩽k, j⩽r is non-zero for some τ and thus for all τ by Galois action.

In order to prove Theorem 3 we specialize the values of the parameters r and d. We take d = r.

It is evident that 2r r ⩾ 2 r so 1 r + 1 2r r ⩾ 2 r 2
for r large enough. Further

1 r + 2 2r + 2 r + 1 ⩽ 1 r + 1 2r r × 4.
We choose r to be the smallest positive integer such that n(r + 1) ⩽ 2r r . We have ( 12) n(r + 1) ⩽ 2r r ⩽ 4n(r + 1) and r ⩽ 3 log n for n large enough. We deduce that = 2r rn ⩽ 4n(r + 1) ⩽ Qn log n. So 2n ⩽ n Q log n .

From Equation [START_REF] Malle | On the distribution of Galois groups[END_REF] we deduce that 2r r ⩽ Qn log n. Also n 2 d(r + 1) ⩽ Qn 2 log 2 n and ( 13)

n 2 d(r + 1) d ⩽ n Q log n .
So the coefficients of equations E i are bounded in absolute value by ( 14) H(K, r, r) ⩽ (nδ K ) Q log n .

This proves Theorem 3.

3.3.

Interpolants for number fields. In order to prove Theorem 4 we first look for a prime integer p such that the value at P τ of the determinant Φ of Equation ( 11) is prime to p. Let Ψ ∈ Z be the product of all Φ(P τ ) for all embeddings τ . We first bound Ψ in absolute value. For every 1 ⩽ k, j ⩽ r the polynomial ∂E i k ∂x j ∈ Z[x 1 , . . . , x r ] has total degree ⩽ r and height ⩽ rH(K, r, r). Using Lemma 4, Equation [START_REF] Neukirch | Algebraic number theory[END_REF] and Equation [START_REF] Martinet | Perfect lattices in Euclidean spaces[END_REF] we deduce that the evaluation of ∂E i k ∂x j at P τ is bounded in absolute value by [START_REF] Robert | Upper bounds on number fields of given degree and bounded discriminant[END_REF] G = n 2 r(r + 1)δ 2 K r rH(K, r, r)

2r r ⩽ (nδ K ) Q log n
We deduce that

Φ(P τ ) ⩽ G r r! ⩽ (nδ K ) Q(log n) 2 and Ψ ⩽ (n n d K ) Q(log n) 2 .
Using an estimate [18, Chapter I §2.6, Corollary 10.1] for the Tchebychev function θ we deduce that there exists a prime p ⩽ Qn(log n + log δ K )(log n) 2 that does not divide Ψ.

The ring Z[κ 1 , . . . , κ r ] is an order in K which is unramified at p. Let q be a place above p. Let s be the degree of inertia at q. We set q = p s and we fix an isomorphism between Z q and the completion of Z[κ 1 , . . . , κ r ] at q. We call b ∞ 1 , . . . , b ∞ r the images of κ 1 , . . . , κ r by this bijection. We let m be the smallest positive integer such that q m = p ms is bigger than G n where G is given in Equation [START_REF] Robert | Upper bounds on number fields of given degree and bounded discriminant[END_REF]. We check that ms log p ⩽ n log G + s log p ⩽ n(log G + log p) ⩽ Qn log n(log n + log δ K ).

We set b m j = b ∞ j mod p m ∈ Z q p m for every 1 ⩽ j ⩽ r. The point b m = (b m 1 , . . . , b m r ) has an interpolant in degree d = r and height H(K, r, r) and this interpolant is b ∞ . Indeed let E(x 1 , . . . , x r ) be a polynomial vanishing at b m and with total degree ⩽ r and height ⩽ H(K, r, r). The evaluation of E at P τ is bounded from above by G according to Lemma 4. Its norm is thus an integer bounded in absolute value by G n and divisible by p ms > G n . So E vanishes at P τ and at b ∞ just as well. Unicity follows from the existence of r equations with degree ⩽ r and heigth ⩽ H(K, r, r) having a non-zero Jacobian determinant modulo q. This finishes the proof of Theorem 4.

CONCLUSION

We have constructed short models of two kinds for a global field: as an irreducible component of some complete intersection or as an interpolant of a finite affine scheme. Theorems 1 and 2 on the one hand and Theorems 3 and 4 on the other hand present evident similitudes. The awaited factor g + n(1 + log q n) in the geometric situation has counterpart n(log n + log δ K ) in the arithmetic situation. The not so welcome factor (log n) 3 or (log n) 2 is the same in either cases. The methods are parallel to some extent. Interpolation plays a crucial role on either sides,

  unimodular, L and L ⊥ have the same volume. See[START_REF] Martinet | Perfect lattices in Euclidean spaces[END_REF] Corollary 1.3.5.]. We denote by Ô = Hom(O, Z) the dual of O, the ring of integers of K, as a Z-module. The evaluation map at (x 1 , . . . , x r ) = (κ 1 , . . . , κ r ) is denotedZ,d ∶ Z[x 1 , . . . , x r ] d → O.We observe that L ⊥ contains the image of Ô by the transpose mapˆ Z,d ∶ Ô → Z[x 1 , . . . , x r ] dwhere we have identified Z[x 1 , . . . , x r ] d with its dual thanks to the canonical bilinear form. So the volume of L is bounded from above by the volume of ˆ Z,d ( Ô). We consider the matrixM 0 P = [(m(P τ )) τ,m∈M ] of the map C,d = Z,d ⊗ Z C in the canonical bases. If we prefer to use an integral basis of O on the right we should multiply M 0 P on the left by the inverse T of the matrix of a basis of O in the canonical basis. We deduce that the square of the volume of ˆ Z,d ( Ô) is the determinant of T M 0 P (M 0 P ) t T t . Since T M 0 P has real coefficients we have det(T M 0 P

  m∈M , (∂m ∂y 1 (P τ )) τ ∈T, m∈M , . . . , (∂m ∂y r (P τ )) τ ∈T, m∈M ] with n(r + 1) rows and d+r d columns has maximal rank n(r + 1). We note that M 1 P consists of r + 1 blocks of size n × d+r d piled vertically. It has maximal rank for a generic P when d ⩾ 5, according to the theorem of Alexander and Hirschowitz[START_REF] Alexander | Singularités imposables en position générale à une hypersurface projective[END_REF][START_REF] Alexander | Polynomial interpolation in several variables[END_REF].

  Since det Sym ⩽d O ⊕r P 1 ≃ det Ker ⩽d ⊗ P 1 det Im ⩽d we deduce that deg det Ker ⩽d ⩾ -n(h + ν)d + a. There exists non negative integers (e i ) 1⩽i⩽ that form a non decreasing sequence and such that

  Lecture III, §4, Theorem 16] to the gauge function x ↦ x we find that L contains linearly independant elements E 1 , E 2 , . . . , E such that

	1⩽i⩽

The volume of the sphere {x ∈ L, x ⩽ 1} is ⩾ 2 -2 . Applying Minkowski's second theorem

[17,

JEAN-MARC COUVEIGNES

especially the theorem of Alexander and Hirschowitz. We do not use the geometric equivalent of Minkowski space however although it exists, see [16]. The reason is that basic arguments about vector bundles over P 1 suffice for our purpose. Another difference is that in the arithmetic case our constructions are consistent with the best known bounds for the number of number fields of given degree and bounded discriminant [8,15,11]. In the geometric situation however, cohomological methods produce much better estimates [10], but no effective construction.