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Géraldine Dantelle
email: geraldine.dantelle@neel.cnrs.fr

Valérie Reita
email: valerie.reita@neel.cnrs.fr

Cécile Delacour
email: cecile.delacour@neel.cnrs.fr

Luminescent Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals for neuronal network bio-imaging and nanothermometry

Keywords: luminescence, harmonic iodate nanocrystals, neurons, biophotonics, nanothermometry

Dual-light emitting Yb 3+ ,Er 3+ -codoped α-La(IO3)3 nanocrystals, known to exhibit both second harmonic signal and photoluminescence (PL), are evaluated as optical nanoprobes and thermal sensors using both conventional microscope and more sophisticated micro-PL setup. When loaded in cortical and hippocampal neurons for a few hours at a concentration of 0.01 mg/ml, visible PL signal arising from the nanocrystals can be clearly detected using an epifluorescent conventional microscope, enabling to localize the nanocrystals along the stained neurons and to record PL variation with temperature of 0.5 % K -1 . No signal of cytotoxicity, associated with the presence of nanocrystals, is observed during the few hours of the experiment. Alternatively, a micro-PL setup can be used to discriminate the different PL lines. From ratiometric PL measurements, a relative thermal sensitivity of 1.2 % K -1 was measured.

Introduction

Temperature is a robust parameter that describes and predicts the state of many systems [1]. In neuronal networks, it is strongly linked with the metabolism and state of cells, triggering biochemical reactions, activation rates and thermal sensitive ionic channels. Slight changes of temperature can affect the dynamics and the reactivity of molecular processes, or reveal disfunction and abnormal developments leading to pathological states. At sub-cellular scales, the expected changes of temperature associated to cellular functions or disfunction is expected to be very weak (100 mW, 10 -5 K) or/and localized (100 -1 nm) which makes the measure highly challenging [START_REF] Baffou | A critique of methods for temperature imaging in single cells[END_REF]. Such small variations could remain hidden and screened by the surrounding extracellular solution which exhibits high thermal conductivity, close to water around 0.6 Wm K -1 [START_REF] Ramires | Standard Reference Data for the Thermal Conductivity of Water[END_REF].

In order to detect subtle temperature differences in neurons, one needs non-invasive, local, stable measurements and highly sensitive thermosensors around metabolism temperature (37 ± 3 °C). The use of luminescent nanoprobes, whose emission properties are dependent on temperature, can be the solution for intracellular thermometry as it is still less invasive and can provide high spatiotemporal resolution [START_REF] Bai | Micro/nanoscale thermometry for cellular thermal sensing[END_REF][START_REF] Brites | Thermometry at the nanoscale[END_REF]. Luminescent nanoprobes can be internalized by neurons or adsorbed on their membranes and be in closed contact with the reactive sites, providing a way to sense local and weak variations of temperature along many individual cells at the same time [START_REF] Simpson | Non-neurotoxic nanodiamond probes for intraneuronal temperature mapping[END_REF][START_REF] Tanimoto | Detection of temperature difference in neuronal cells[END_REF].

There are many ways to measure temperature through photoluminescence (PL), monitoring either the excited state lifetime, the luminescence intensity, the variation of emission peak position, etc. [8]. Several thermal nanoprobes have been investigated, such fluorescent proteins [START_REF] Donner | Mapping intracellular temperature using green fluorescent protein[END_REF], gold metal nanoclusters [START_REF] Zhang | Metal nanoclusters: new fluorescent probes for sensors and bioimaging[END_REF], nanodiamonds [START_REF] Simpson | Non-neurotoxic nanodiamond probes for intraneuronal temperature mapping[END_REF], polymer [START_REF] Okabe | Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy[END_REF] or quantum dots [START_REF] Yang | Quantum dot nano thermometers reveal heterogeneous local thermogenesis in living cells[END_REF], carbon nanodots [START_REF] Gan | Temperature-Dependent Dual Emission from Sucrose-Derived Carbon Nanodots: A Ratiometric Fluorescent Thermometer[END_REF], perovskite nanocrystals [START_REF] Yuan | Fluorescence enhancement of perovskite nanocrystals using photonic crystals[END_REF]. Among them, lanthanide-doped inorganic nanocrystals are of particular interest for their narrow emission lines, related to the f-f orbitals of lanthanide ions involved in the optical transitions and allowing spectral discrimination, for their relatively long excited state lifetime allowing autofluorescence removal by time-gated experiments and for their numerous emission lines covering the visible and near-infrared range [START_REF] Brites | Lanthanide-Based Thermometers: At the Cutting-Edge of Luminescence Thermometry[END_REF]. With these nanoprobes, typical luminescence-based nanothermometry consists in a ratiometric approach, consisting in measuring the luminescence intensity ratio (LIR) of two emission peaks that are in thermal equilibrium. To compare the thermal performances of different types of nanoprobes, a relative thermal sensitivity, Sr, has been defined as [START_REF] Zhou | Advances and challenges for fluorescence nanothermometry[END_REF]: (1) Many Nd 3+ -doped nanocrystals (SrF2 [START_REF] Villa | 1.3 μm Emitting SrF2:Nd 3+ Nanoparticles for High Contrast In Vivo Imaging in the Second Biological Window[END_REF], LaF3 [START_REF] Rocha | Subtissue Thermal Sensing Based on Neodymium-Doped LaF3 Nanoparticles[END_REF], NaGdF4 [START_REF] Chen | Core/Shell NaGdF4:Nd 3+ /NaGdF4 Nanocrystals with Efficient Near-Infrared to Near-Infrared Downconversion Photoluminescence for Bioimaging Applications[END_REF], CaF2 [START_REF] Yu | Luminescence enhancement of CaF2:Nd 3+ nanoparticles in the second nearinfrared window for in vivo imaging through Y 3+ doping[END_REF], NaHoF4 [START_REF] Feng | Neodymium-doped NaHoF4 nanoparticles as nearinfrared luminescent/T2-weighted MR dual-modal imaging agents in vivo[END_REF], YAG [START_REF] Dantelle | Nd 3+ -doped Gd3Sc2Al3O12 nanocrystals: towards efficient nanoprobes for temperature sensing[END_REF]) have been developed for in-vivo imaging as Nd 3+ present emission lines in in the biological transparency windows situating in the near-infrared range (680 to 1700 nm), limiting autofluorescence from the tissues [START_REF] Del Rosal | Neodymium-doped nanoparticles for infrared fluorescence bioimaging: The role of the host[END_REF][START_REF] Hemmer | Exploiting the Biological Windows: Current Perspectives on Fluorescent Bioprobes Emitting above 1000 nm[END_REF]25]. However, these Nd 3+doped single-band nanothermometers present a moderate Sr (~ 0.1 to 0.5 % K -1 , depending on the host matrix [26]), leading to a relatively high temperature uncertainty δT, of the order of 2°C at 37 °C [27,28,29] with: [START_REF] Baffou | A critique of methods for temperature imaging in single cells[END_REF] with δLIR, the standard deviation of the LIR.

For in-vitro experiments, for which visible-emitting nanoprobes can be used, Er 3+ ,Yb 3+ -doped nanocrystals present a good alternative as they offer higher Sr values (> 0.2 % K -1 ), reducing consequently the uncertainty to less than 1 °C and even 0.1 °C in some matrices and in some more complex core-shell architectures [START_REF] Brites | Lanthanide-Based Thermometers: At the Cutting-Edge of Luminescence Thermometry[END_REF]30,31]. The high performances of Er 3+ -based nanothermometers have been very recently confirmed theoretically by M. Suta and A.

Meijerink, who demonstrated that Er 3+ ions fulfil all requirements for optimum performances (high Sr in the physiological temperature range) [32].

Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals were recently developed with the goal of proposing multi-functional nanoprobes for in-vitro for imaging. The key advantage of these nanocrystals lies in their dual emission properties [33,34]: (1) as non-centrosymetric nanostructures, non-linear optical signal such as Second Harmonic can be generated upon near-infrared femto-second excitation [35] and (2) the facile, thus efficient, substitution of La 3+ by Er 3+ and Yb 3+ ions leads to PL properties in the visible and near-infrared range [34]. Such combination paves the way to bio-imaging using multiple and complementary imaging techniques: fluorescence imaging using a conventional epifluorescent microscope and in-depth imaging using a two-photon microscope equipped with a short-pulse excitation laser. We report here an additional feature of these Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals and demonstrate, as a proof of principle, their capacity to be used as nanothermometers in neuronal cells. After the measurement of the thermal sensitivity in various environments (powder, liquids), the nanocrystals are loaded in cortical and hippocampal neurons for in-vitro bioimaging and temperature measurements.

Materials and Methods

Nanocrystal synthesis. Yb 3+ ,Er 3+ doped α-La(IO3)3 nanocrystals (La0.895Yb0.1Er0.005(IO3)3) synthesis. LaCl3•6H2O, ErCl3•6H2O and YbCl3•6H2O (Strem Chemicals, purity > 99.9%) were mixed, in stoichiometric proportions, in water (10 mL) with a total concentration of 0.05 mol/L. Separately, HIO3 (Sigma Aldrich, purity > 99.5 %) was also dissolved in water, with a molar concentration of 0.15 mol/L. Both solutions were mixed together and poured into a quartz vessel, then placed into a Anton-Paar Multiwave PRO microwave oven. The heating was conducted at 250 °C for 10 minutes to crystallize the α-La(IO3)3 phase [36]. After cooling down, the white precipitate is retrieved by filtering and washed several times with distilled water before drying at room temperature for 24 h. Nanocrystal structural characterizations. Powder X-ray diffraction (XRD) pattern was recorded on a Siemens D8 Advance diffractometer (λCu = 1.54056 Å) at room temperature. Scans were collected in the 2Ɵ range 16-90°, with a step of 0.005° and 1 s of acquisition time. Transmission electron microscopy (TEM) images were recorded using a Philips CM300 microscope, operating at 300 kV and equipped with a TemCam F416 TVIPS camera. Because of the lack of stability of the iodate nanocrystals under the electron beam, the experiment was performed using a liquidnitrogen cooled sample holder. The experiment at lower temperature (estimated between 77 and 170 K) allows preserving the nanocrystals from being modified under the beam.

Photoluminescence measurements. PL and PL excitation spectra of powdered Yb 3+ ,Er 3+ doped α-La(IO3)3 nanocrystals were measured using a commercial Safas fluorimeter (Xenius) at room temperature. Using a tunable vertically polarized femtosecond laser (Insight X3 Spectra-Physics, 80 MHz, 100-120 fs) as the excitation source, the non-linear properties of these nanocrystals dispersed in solution were recorded, as detailed in [34]. Temperaturedependent PL measurements were performed using a micro-PL setup coupled with a temperature stage (Linkam THMS600). The excitation was performed using a CW Argon laser (488 nm, typical power: 100 kW.cm -2 ) and focused on the sample through an Olympus objective (x 60). The emission was collected through the same microscope objective and detected, after a razor edge filter at 488 nm, through a T6400 spectrometer equipped with a Symphony Camera detector cooled at 77 K. During the cell culture, the photoluminescence is collected in liquids using an upright-style epifluorescent Olympus BX51 microscope. The incident illumination (white light, Xenon lamp) is filtered at λexc= 488 ± 20 nm and focused on the sample with a × 60 water-immersion objective. The emitted photons are collected through the same objective and filtered at λem = 535 ± 25 nm. A High pass dichroic mirror at λdic = 510 nm suppresses the contribution of the incidents photons on the signals that is recorded with sCMOS camera cooled between 80-70K.

Cell culture. E16.5 pregnant mice were killed by cervical elongation. Hippocampi were dissected from the mouse embryos in HBSS-HEPES (Gibco), incubated 15 min in Trypsin 2,5% (Gibco), rinsed in HBSS-HEPES twice, and mechanically dissociated by repeatedly pipetting in 1 mL attachment medium (MEM supplemented with 10% foetal bovine serum, 1% glutamine and 0.05% peniciline/streptomycine, Gibco). Cells were seeded with a density of 150 000 cells/mL on the glass coverslip previously sterilized and coated with poly-L-lysine (PLL, 1mg/mL) to ensure neurons adhesion. Neurons were then incubated at 37°C and 5% CO2 in the attachment medium, and replaced a few hours later by serum-free glial conditioned medium supplemented with 1 mM AraC.

Nanocrystals loading. Nanocrystals were dispersed in 37°C culture medium with ultrasonic bath to prevent large aggregates, and add to the cells medium at a final concentration ranging from 0.1 to 0.001 mg/mL. Constant agitation was maintained during few minutes. Then cells with nanocrystals were incubated at 37 °C and 5% CO2 until observations (around 3 h).

Immunostaining. After PL acquisition neurons were fixed in 3,9% paraformaldehyde (10 min), permeabilized in PBS-0.1% Triton-X100 (3 min) and blocked in PBS-2% bovine serum albumin (10 min). After each step, the solution (PFA, PBS-Triton or PBS-BSA) is removed and cells are rinsed at least three times in PBS. Primary antibody anti-YL1/2

(1:1000, BioRad) is diluted in PBS-2% BSA (1:1000), incubated with cell 1 hour and incubated with cell for 1 hour. Then the solution is replaced by FITC-conjugated secondary antibodies (Alexa Fluor, dilution 1/500) for 1 hour. Lastly, cells are then incubated with DAPI (1:1000, Sigma-Aldrich) to visualize cells body. The two stainings (YL1/2 and DAPI) enable to label the microtubule cytoskeleton and the soma respectively to locate the neurites and the cells body.

Finally, cells were washed again in PBS, mounted in Mowiol (Sigma-Aldrich) mounting medium between two coverslips. Optical and immunofluorescence images were collected using either an Olympus BX51 and Zeiss AxioImager M2 microscope.

Results

Temperature dependent PL of nanocrystals

Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals (with doping concentrations of 0.5 mol.% Er 3+ and 10 mol.% Yb 3+ ) were prepared by a microwave-assisted hydrothermal method as reported in [36]. Obtained nanocrystals, imaged by lowtemperature transmission electron microscopy (TEM, Figure 1a), present an average size of 40 nm and a good polydispersity (± 10 nm), obtained by the size analysis on more than one hundred nanocrystals. The PL and PL excitation properties of powdered nanocrystals were recorded in the visible range and are presented on Figure 1b.

The main excitation peak is situated at 380 nm, corresponding to the 4 I15/2 → 4 G11/2 transition of Er 3+ . There are two intense emission peaks upon a 380 nm excitation, situated at 522 nm and 544 nm, corresponding to the 2 H11/2 → 4 I15/2 and 4 S3/2 → 4 I15/2 transitions of Er 3+ , respectively. A small PL peak is observed around 660 nm, corresponding to the 4 F9/2 → 4 I15/2 transition of Er 3+ . This weak intensity attests from a low population rate of the emitting level ( 4 F9/2). As this level is populated either from non-radiative de-excitations from the 4 S3/2 level or through Er 3+ -Er 3+ cross-relaxation mechanisms, the latter being very unlikely at this low Er 3+ concentration (0.5 mol%) [37,38], it shows that nonradiative mechanisms are not dominating in the considered nanocrystals. Under a 1064-nm femto-second excitation, these nanocrystals exhibit Second Harmonic signal (Figure 1c). Using a micro-PL setup, the emission spectrum of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals dried on a silicon substrate was recorded in the 500 to 560 nm range. Because of the much better spectral resolution of this setup, compared to the one of the fluorimeter, the sub-structure of the emission peaks, whose maximal intensity lies at 522 nm and 544 nm, can be visualized (Figure 2a), corresponding to the transitions from the multiple Stark sublevels of the 2 H11/2 and 4 S3/2 levels to the Stark sublevels of the 4 I15/2 ground state of Er 3+ .

Thanks to a temperature stage (Linkam THMS600) coupled to the micro-PL setup, PL spectra were recorded at different temperatures (varying between 25 and 70 °C) after a 10 min thermalisation at each temperature. When temperature increases, the intensity of the two emission peaks at 522 nm and at 544 nm, respectively labelled I522 and I544, tends to equalize, as the two emitting levels, 2 H11/2 and 4 S3/2, are in close proximity in energy and their population is ruled by Boltzmann equation [39,40]:

(3

)
with A is a constant depending on the host matrix, ∆E the energy difference between the 4 S3/2 and 2 H11/2 levels, kB the Boltzmann constant, and T the absolute temperature. The evolution of LIR, plotted in a logarithmic scale, as a function of 1/T, follows a linear evolution in agreement with Boltzmann theory (Figure 2b). The slope allows for the calculation of ∆E ~ 730 cm -1 . This value can be compared to the value calculated from the PL spectrum, taking into account the barycentre of both peaks at 522 and 544 nm, situated respectively at 19160 cm -1 and 18410 cm -1 , which gives a gap between the 2 H11/2 and 4 S3/2 levels of 750 cm -1 , in good agreement with the ∆E value obtained from the nanothermometry measurements.

In order to compare the thermal sensitivity of these Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals with other luminescent nanoprobes, their relative thermal sensitivity Sr was calculated from Eq. 1. As the ∂LIR/∂T term is the derivative of the function described in Eq. 3, Sr can be further evaluated by [41,42]:

At 25 °C, the Sr value for our system is 1.2 % K -1 , which is comparable the values obtained in other Yb 3+ ,Er 3+ doped oxide or fluoride nanocrystals [START_REF] Brites | Lanthanide-Based Thermometers: At the Cutting-Edge of Luminescence Thermometry[END_REF]43]. Under those experimental conditions, the temperature uncertainty, δT, is evaluated at ~ 0.5 °C.

The same experiment was repeated on nanocrystals dispersed in two different liquid environments: in ethylene glycol, which is known to stabilize these nanocrystals [33] and in water at pH 7 for further prospects in biology. Once the nanocrystals have sedimented onto the silicon substrate, their emission was recorded as a function of temperature and the LIR value can be studied as a function of temperature (Figure 2b). Like in powder, the linear evolution of LIR

Impact of the nanocrystals on cultured neurons.

As previously described [46], cortical and hippocampal neurons are extracted from mouse embryos (E16.5) (detailed in Material and Methods). Cells are plated on glass coverslip previously coated with cell adhesive polymers (poly-Llysine at 1mg/ml), and then incubated at 37°C and 5% CO2. Then nanocrystals are loaded several days of culture (DIV4 or 5) and few hours before the recordings. Depending on the incubation time, nanocrystals will penetrate the plasmid membrane or remain adsorbed on neurons membranes. For each condition, Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals are dispersed in water, at high concentration (0.1 mg/mL) using ultrasonication (stock solution). Then, few amounts of the nanocrystal suspension (5-50 µl) are added to the cell medium under constant agitation during 10 min. Finally, cells are incubated before the observations to recover.

We adjusted the concentration of nanocrystal, ranging from 0.01 to 0.001 mg/ml. High concentration 0.1 mg/mL appears lethal for neurons, while low concentration reduces significantly the probability to attach nanocrystals on neurites. The final concentration was fixed at 0.01 mg/mL, which enables to keep healthy neurons and several sensing sites where nanocrystals and neurites are colocalized. Most of the large aggregates are removed during media changes (Figure 3a). The remaining large ones provide controls for measuring PL of the nanocrystals, while smaller nanocrystals are rather internalized or attached on the membranes of neurons (Figure 3a-d).

Temperature dependence of the nanocrystals PL in neuronal environment.

We assess here the ability to monitor the PL of the nanocrystals in standard conditions of cells observation using an epifluorescent Olympus BX51 microscope, equipped with FITC-filter cube (excitation 480 nm +/-20nm; dichroic mirror 505LP; emission filter 535 +/-25nm). Nanocrystals (0.01 mg/ml) were seeded on bare coverslips (bath #1) and on neurons cultured five days on glass coverslip (batch #2). To record the PL signals, we used high magnification fluorescent objective (x 60) immersed in the suspension containing the nanocrystals and the neurons. The temperature of the solution is varied from room temperature 20°C to 40°C or 45°C, with and without neurons respectively, by using the same temperature stage (Linkam THMS600) than the one used on the micro-PL setup. Samples are maintained at the desired temperature for 5 min before the measurement. Emitted photons are collected during 50 ms with an sCMOS Camera (Andor Zhyla 5.5) for each temperature steps, while keeping constant all other parameters such the incident illumination (6 mW), the grey scale, or the dynamic range (16 bits). The total intensities emitted from the nanocrystals are then extracted with imageJ software.

PL of the Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals is clearly observed on the optical micrograph as illustrated within Figure 3. The transmitted wavelengths of the FITC cube ranges from 510 and 550 nm (>80%) with a maximum at 525 nm (95%), and enables to collect the main emission spectra of the nanocrystals with two main emission peaks at 522 and 544 nm (Figure 1). Large aggregates identified in dark field (dark clusters within Figure 3a) correlated with fluorescent cluster on FITC-micrograph, as the one highlighted by the arrows to give an example (Figure 3b). Also, smaller crystals that are barely seen in visible, appear clearly in the FITC range, characterized by small fluorescent dots along the neurites and the somas that can be distinguished from the microtubule filaments stained with YL1/2-FITC markers (closer views, Figures 3c-d).

Regarding the PL dependence with temperature, we observe a change of PL intensity when increasing the temperature from 20°C to 45°C. This change is reversible when decreasing the temperature. Without cells (red dots, Figure 4) in liquid, the nanocrystal PL exhibits a rather linear dependence with the temperature, varying of about 15% from room temperature to 45°C with a maximal variation of the PL intensity around 0.5 % K -1 (given by the slope of the red curve, Figure 4). This value is close to the variation measured in micro-PL spectroscopy: ∂LIR/∂T ~ 0.65 % K -1

(Figure 2). The slightly lower sensitivity obtained with the epifluorescence microscope is expected as we cannot distinguish the contribution of the two PL peaks of the nanocrystals, but only the resulting contribution of both. The contribution of the first PL peak (522 nm) is the most sensitive to temperature change and corresponds the maximum of the transmission of the FITC filter. But its variations may be hidden by the second peak which exhibits higher PL signals (almost twice).

In the presence of cells, the PL intensity still decreases when increasing the temperature and the observed variations remain in the expected ranges of intensity, with a maximum sensitivity around 37°C. But the recorded variations are weaker and the dependence with the temperature is unclear around 25°C. At the wavelengths of interest (in the range 488-550 nm), neurons are not optically transparent, they can either absorb or scatter the optical signals. In particular, the scattering coefficient is high in the visible range [47] and could impact the collected intensity in a significant manner, that may explain the non-linear dependence with the temperatures that we observed. Additionally, cells could act as a thermal barrier between the regulated substrate and the nanocrystals, making slightly less efficient the thermal coupling between the bath and the temperature stage. In both cases, such effects may contribute to increase the non-linearities on the recorded PL signals and make more difficult the extraction of the PL signal of interest around room temperatures. These possible artefacts can be overcome using the PL intensity ratio over the absolute PL intensity variation that lies in more sensitive to temperature change. To enhance the thermal sensitivity and repeatability we could isolate the contribution of the two nanocrystals PL peaks at 522 and 544 nm. This could be obtained by adding dichroic mirrors on a second emission line to select the first and/or the second peak only. Its contribution can then be isolated and subtracted from the first emission line to improve the signal on noise ratio.

As an alternative, we assessed the possibility to measure temperature of a neuronal network with the micro-PL setup (previously described). An aqueous solution of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals (c = 0.1 mg/mL) was dispersed onto neuronal network (Figure 5a). Once the nanocrystals have sedimented, the excitation beam is focused onto a small nanocrystal aggregate situated onto a dendrite (circled in black in Figure 5a) in order to record the PL spectra at different temperatures in the physiological range. The PL signal of small aggregates is easily detectable despite the increase of luminescence background, attributed to the microscope cover slide. In agreement with the measurements made using the conventional microscope, the overall PL intensity decreases with the temperature increase. The LIR varies linearly with temperature, in a similar manner as for the nanocrystals in water (Figure 5b), in agreement with previous experiments using Quantum Dots as thermal sensors [START_REF] Tanimoto | Detection of temperature difference in neuronal cells[END_REF]; however, the overall accuracy is decreased due to the presence of a background signal due to the glass coverslip, leading to a temperature uncertainty, δT of 2 °C. This value could be improved by using a different sample container. We demonstrated here that Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals can either enter into neuronal cells or remain adsorbed on cell membrane, enabling dual extra and intracellular measurements over physiological temperature range 20-40°C with a maximum sensitivity at 37°C. The variation of PL intensity over temperature that we reached (0.5 % K -1 ) is enough to capture thermal difference of few degrees. These thermal gradients are expected within cell compartments [51,52] or during neurostimulation [53,54] to give few examples. However further enhancements would be required to reveal weaker changes (10 -4 -10 -5 K) associated to cell activity [START_REF] Baffou | A critique of methods for temperature imaging in single cells[END_REF]55].

Although there are numerous papers reporting the thermal performances of nanothermometers [START_REF] Brites | Lanthanide-Based Thermometers: At the Cutting-Edge of Luminescence Thermometry[END_REF], fewer of them deal with in-vitro experiments [56]. From the ratiometric approach on the micro-PL setup, we measured a relative thermal sensitive Sr of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals of 1.2 % K -1 in neuronal cells, higher than previously reported values for other inorganic materials, including Green Fluorescent Proteins (GFP) [57], Nd 3+ -doped nanocrystals (0.1 % K) [START_REF] Rocha | Subtissue Thermal Sensing Based on Neodymium-Doped LaF3 Nanoparticles[END_REF] or quantum dots [58] ( Scattering and absorption may be non-negligible for neuronal sensing especially when the number of collected photons are low, as it could impact both the number of collected photons or cell metabolism. Although this effect should be less within in-vitro cultured neurons than for in-vivo applications, it could impact the measurements and the temperature dependence of the collected PL intensity. Future works should investigate how absorption and scattering by neurons could introduce significant artefacts on the measurements, as well as the stability of the measurement depending on the position of the emitters within the cell. Extraction of the PL depth-profile within the cellular layer might be a way to perform such analysis that would be required for in-vivo and in-vitro applications.

In addition, future work will investigate the gain that could be reached using a conventional microscope with dedicated set of emission/excitation filters to capture separately the emission of the two excited states at 525 nm and at 542 nm, as using the PL intensity ratio over the absolute PL intensity variation lies in more sensitive to temperature change. Indeed, it allows to get rid of the fluctuations of the experimental conditions (excitation beam power, mechanical drift, etc) and thus reducing possible measurement artefacts.

For all these reasons, Yb 3+ ,Er 3+ -doped La(IO3)3 nanocrystals developed here appears as suitable candidates for contactless (luminescent) thermometers. It is worth noting that in this work, Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals were excited at 488 nm, i.e. directly in an excited level of Er 3+ . However, to envision future experiments, including invivo experiments, which require an excitation in a transparency window of the biological tissue (i.e. in the near infrared), it will be possible to excite these nanocrystals at 980 nm, i.e. in the Yb 3+ ions, which already prove to efficiently transfer their energy to Er 3+ via up-conversion processes.

Conclusions

Amongst the different nanoprobes that are developed for nanomedicine applications, one of the most desired features is their versatility, which allows for their use under different conditions and/or environment and for different purposes (e.g. diagnostics, therapy). Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals already exhibit two different functionalities: the Second Harmonic Generation and the photoluminescence, which are rarely obtained at the nanoscale. These properties enable to visualize them using conventional fluorescence microscopes, commonly available in biology and also with non-linear microscopes along for in-depth imaging. In addition to this dual-light emission, we show here that these nanocrystals, exhibiting a relative thermal sensitivity of 1.2 % K -1 and a PL intensity variation with temperature of 0.5 % K -1 , appear as promising candidates for in-vitro bio-imaging and thermal sensing of neuronal cells. Indeed, they combine both a high spatial resolution and large field of view and enable measurements over physiological temperature range (20-40°C) with epi-fluorescent microscopy which makes the technique ready to use in cell culture laboratory.

Figure 1 .

 1 Figure 1. (a) Low-temperature TEM image of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals. The inset shows the powder X-ray diffraction pattern associated with these nanocrystals. (b) Room-temperature PL and PL excitation spectra of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals. The PL spectrum was recorded under a 380 nm excitation, whereas the PL excitation was recorded while monitoring the emission at 545 nm spectrum. (c) Second Harmonic Scattering signal from Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals in ethylene glycol under a 1064-nm excitation.

with 1 /

 1 T attests from a Boltzmann distribution of the two excited state levels. One can note that the value of LIR varies according to the environment, which can be explained by a change of thermal conductivity at the nanocrystals interface. Recent works report the key role of the environment and interfaces of nanothermometry: A.R.N.Bastos et al. evidence a significant variation of Sr values for nanocrystals dispersed in water and inD2O [44]. It should be specified that all our experiments were performed using a micro-PL set-up involving high excitation power density (~ 100 kW cm -2 ) at 488 nm, where heating of the liquid medium is very likely. In order to assess the reliability of these nanocrystals as nanothermometers, further work would be required, measuring PL dependence according to on the environment (pH, nature of the tissues, etc) and the operating conditions (excitation power densities, excitation wavelength, etc)[45].

Figure 2 .

 2 Figure 2. (a)Normalized PL spectrum of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals dried on a silicon wafer, normalized at 544 nm, as a function of temperature (λexc = 488 nm, 100 kW.cm -2 ). (b) Evolution of the LIR = I522/I544 ratio with temperature, for Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals in different media.

Figure 3

 3 Figure 3 illustrates the typical shape of 5 days old neurons incubated 3-4 hours with nanocrystals. The neurons appear healthy despite the presence of nanocrystals, with the expected cell density, number of neurites per cell and neurite length for this maturation stage. The spreading of neurites over all the substrate indicates also the healthy state of cells. No neurite bundles neither large soma aggregates are observed which are first signs of stress or cytotoxic effects within in-vitro neural networks. Also, there is no sign of membrane disruption.

Figure 3 .

 3 Figure 3. Bright field and fluorescent micrographs of hippocampal neurons (after five days in culture) loaded with Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals (concentration of 0.01 mg/mL). Neurons are stained with DAPI (blue), YL-1/2-FITC (green) and synapsin-TRITC (red) to locate the soma, neurites and synapses. FITC fluorescent signals of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals and microtubules are superimposed (λexc=488 nm, λem,= 525 ± 15 nm, 6 mW). We can clearly distinguish the large clusters of nanocrystals by comparing optical and fluorescent images (a-b), as well as smaller nanocrystals along the somas (c) and the neurites (d) as highlighted with the arrows. Scales bares (a-b) 30 µm, (c-d) 10 µm.

Figure 4 .

 4 Figure 4. Temperature dependence of the PL of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals monitored with an epifluorescent microscope (Olympus BX51), without and with neurons (red and green/blue dots respectively). The intensity per pixel is collected with × 60 water-immersion objective, as function the substrate temperature adjusted with a Linkam controller. FlTC-filters cube enables to select the incident illumination (Xenon lamp) at the nanocrystal excitation wavelength λexc= 488 ± 20 nm and to collect their emission around λem = 535 ± 25 nm. High pass dichroic mirror at λdic = 510 nm suppresses the contribution of the incident photons on the recorded signals. Signal is monitored with sCMOS Andor Zhyla 5.5 camera at constant exposure time 50 ms.

Figure 5 .

 5 Figure 5. (a) Optical image of Yb 3+ ,Er 3+ -doped α-La(IO3)3 nanocrystals loaded onto neuronal cells with a concentration of 0.1 mg/mL, in water (Scale bar: 12 µm). (b) Evolution of LIR of the nanocrystals adsorbed on neurons as a function of temperature.

  In this work, we demonstrate the possibility to use two different optical microscopy techniques to get a local temperature measurement within cultured neurons networks: fluorescent-based imaging and micro-PL spectroscopy. This includes possible observations with epi-fluorescent microscope mounted with conventional FITC-filter, which makes the technology ready to use in cell culture laboratory. Fluorescence-based imaging is a promising strategy for cell thermometry because it can be non-invasive and can have a high spatial resolution (diffraction limited level, around 200 nm) on a wide field of view (mm²). When associated with highly sensitive luminescent thermometers, it could enable to follow tiny temperature change along single neuron and large network. As luminescent nanoprobes, we investigate here Yb 3+ ,Er 3+ -doped La(IO3)3. Many rare-earth-doped nanocrystals are known as Boltzmann-based nanothermometers, whose temperature variation is monitored by following the PL intensity ratio of two emission lines of Er 3+ . Though developed a few decades again, many challenges still remain for their use in biology, including the control of the nanocrystal size, morphology, stability, optical efficiency and thermal sensitivity.Under a conventional microscope, the temperature-dependent PL intensity of the nanoprobes deposited on a glass substrate follows a typical Boltzmann distribution (Figure4, red data), which is as expected for temperature-sensitive fluorescent materials. Nonradiative processes increases with the temperature, while radiative transition rate remains constant which results in a decrease of the emission from the excited state[48,49]. Inversely, the PL intensity increases when reducing the temperature. As a result, the chronograms of PL intensity provide a direct, local and fast temperature measurement (between 0.2 s and 2 ms depending on the PL intensity) over millimeter scale network area.In comparison, quantum dots based on red-shift emission require multiple acquisition of the PL spectrum which are usually too long (typ. 0.5 s -1 s) and limited to one specific location per spectrum, preventing larger scale thermal mapping[50].

Table

Table 1 .

 1 Comparison of relative thermal sensitivity (Sr) and thermal resolution (δT) of different fluorescent thermal sensors used in cells.

	ln(I 522 /I 544 )

). This value is comparable to that of other Er 3+ -doped materials

[59,60]

or hybrid materials [61], as a consequence of the energy gap between the two emitting levels of Er 3+ ( 4 S3/2 and 2 H11/2) whose splitting does not differ much with the host matrix. More complex systems, such as the

(b)

Acknowledgments: G.D. acknowledges Y. Mugnier and R. Le Dantec from Symme Laboratory, Annecy, France for the Second Harmonic characterization.

Funding: This research was funded by the French-Swiss Interreg V (Project OncoNanoscreen) and by the French Research Agency (ANR Nanomesh, grant number: ANR-18-CE42-0003)

Author Contributions: All authors contributed to the experimental work presented in this paper and have read and agreed to the published version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.


PhD B Trévisan

Associate professor K Ege

B Laulagnet

Vibroacoutsic modelling of piano soundboards through analytical approaches in frequency and time domains

Keywords: Orthotropic materials, Vibroacoustic of non-rectangular ribbed structures, Frequency and time domains, Coupled systems, Piano soundboard

Extended Abstract

The vibratory behavior and radiation of complex structures are a real challenge for many industrial domains. The increasing requirements of users and manufacturers justify the interest of the scientific community about this subject, particularly about ribbed structures. Initially, the design of such a structure is led by structural reasons and offers in the same time the possibility to reduce the weight and to reinforce the conception. Thus, they are common in many industrial domains, but also building and crafting sector. Among them, mention automotive, shipbuilding and aerospace industries or musical instruments too (see Figure 1). However, their conception makes difficult the understanding of their vibroacoustical behavior contrary to simplest structures as beams or plates that are now well described. Since several decades, some studies tried to find elements of response to industrial needs and now allow understanding the behavior of more complex structures but remain far enough from real industrial case: finite rectangular ribbed plate, infinite ribbed cylindrical shells for examples. Moreover, the democratization of composite materials offers new possibilities of improvement, which is in good agreement with industrial considerations (like about reducing the weight of vehicles for example). So, taking into account the behavior of these new materials in addition to structures designs becomes a real challenge. For example, interactions between layers in the case of laminated composites or also honeycomb panels due to their reduced weight.

Actually, computing resources allow to describe various designs and to take into account material specificities with accurate results. However, these purely numerical methods need important hardware and imply long time computations for the most complex structures. Thus, they do not allow a step of comprehension of the phenomena and are rather adapted to evaluate performances after a first phase of design. Hence, it misses rules that allow to indicate the good practices to adopt as well as to anticipate their performances during the first outlines of design and at the time of optimization. Their modeling through lower cost methods remains a major challenge in order to do parametrical studies and so, highlight main tendencies that can be used in various domains. In that way, analytical approaches are particularly well adapted but imply simplified modeling as presented in [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF]. However, simplifications applied to the method do not impact the main tendencies of the results. We are interested in orthotropic non-rectangular plates. To have a general method, the principal axis of orthotropy can make an angle with the edges of the structure. In the other case, we talk about a special orthotropy. Moreover, we Piano soundboard consider some stiffeners / superstructures tied to the plate that can be oriented in the directions of plate's Young moduli, i.e. in two perpendicular directions.

The piano soundboard is a typical example of such a structure. Musical instruments making is a particularly interesting domain considering constrains due to conception and their consequences on the perceived sound. In the case of piano, many parameters must be considered from the wood used to the own design of the instrument. Among the mechanical aspects, we can quote: the design of the soundboard itself that have been studied at several times, the strings, the hammer and also interactions between them.

Thus, this is a complete field of investigations that involves subjective and perceptive aspects, which still are uncommon in industrial domains. The piano soundboard is the centerpiece of the instrument because it allows an effective acoustic radiation. Indeed, the strings are too small to radiate by themselves and they are tied to the soundboard through the bridges, exchanging energy. So, the soundboard has an essential function, justifying the several studies about it that have highlighted, all the complexity of its vibroacoustic behavior: its non-rectangular edges, its crown, the use of wood, stiffeners, tapering or the downbearing induced by the strings are particular characteristics of its design. Indeed, its empirical conception is led by structural and dynamical constrains and the rules used by piano makers do not allow to anticipate the consequences of any changes on the quality of the perceived sound. Moreover, with limited financial resources, attempts of innovation are risky.

Despite that, piano makers often report a non-satisfaction about the timber and a good compromise sustain / radiated power is difficult to obtain in the treble zone. The major scientific studies focus on frequency response of the soundboard (modal shapes, mobility at the bridge for the most common) and cannot represent the perceptive aspect of the problem. To answer to purely musical questions, it is relevant to introduce the concept of coupled systems. So, by coupling the soundboard to a string and solving it in time domain, it is possible to listen to the sound produced and appreciate the difference between two soundboards. This way of solving offers new perspective of analysis and new indicators, which can be employed to complete frequency analysis.

In view to solve these issues, a modeling is developed using a variational approach that take its inspiration in [START_REF] Guyader | Sound radiation by finite cylindrical ring stiffened shells[END_REF], [START_REF] Guyader | Structural acoustic radiation prediction: Expending the vibratory response on a functional basis[END_REF] and allows to describe orthotropic non-rectangular ribbed plates. This is done using an extended simply supported plate with special orthotropy and a modal decomposition on its modes. By an addition of several springs in order to block the transversal displacement, it is possible to describe any edges the user want (not limited to soundboard and musical domain) and angle of otrhotropy (see Figure 2). Then, the stiffeners are added as straight stiffeners oriented in the directions of the Young moduli of the extended plate (see Figure 2). The method takes into account the offset from the middle plane of the plate. By this way, we get a matrix formulation of the problem, which can be solved in frequency and time domains. In the frequency domain, the method we propose for the acoustic radiation is an alternative to purely numerical method as for example finite element boundary method, Rayleigh integral or Perfectly Matched Layers. In that way, we calculate the acoustic radiation of the structure using the radiation impedances of the extended rectangular un-ribbed baffled plate, which have been studied several times [START_REF] Wallace | Radiation Resistance of a Rectangular Panel[END_REF], [START_REF] Stepanishen | Modal coupling in the vibration of fluid-loaded cylindrical shells[END_REF], but remains limited to rectangular ribbed plate [START_REF] Guyader | Structural acoustic radiation prediction: Expending the vibratory response on a functional basis[END_REF]. It is now extended to the case of non-rectangular structure and finally, the method allows for example: solving the eigen-problem and so classified modal shapes into four particular families of modes (see Figure 3); solving forced response of a ribbed structure with or without fluid/structure interactions and evaluate the acoustical radiation easily while being representative of real phenomenon on a large frequency band. Thus, it is simple to make parametrical studies and so to determine the main tendencies in order to As mentioned previously, perceptive aspects lead musical questions and we focus on strings / soundboard interactions. To answer these questions, the soundboard model is coupled to strings at the bridges and solved in time domain. The strings are modeled with transversal and longitudinal waves and it is also possible to take into account geometrical non-linearity of the strings. In such a calculi, the interaction efforts that ensure the continuity of displacements at the interface are unknown factors, as impact or friction problems [START_REF] Laulagnet | Rattle Noise Generated by a Beam Tip Impacting a Baffled Plate[END_REF]. That allows evaluating the influence of any structural changes on the coupling between the sub-systems. Then, it is possible to know the influence of the damping of the wood, the dead length of strings on the perceived sound and to highlight many coupling phenomenon and energy exchanges that were not easy to anticipate. We also show the influence of non-linearity (see Figure 4), which are considered as small and so allow keeping notions of modes, on the timber of the instrument injecting several short stresses in the strings.

Figure 1 :

 1 Figure 1: Examples of ribbed structures in various domains.

Figure 2 :

 2 Figure 2: Schematics of method outlines: adding of springs to describe non-rectangular edges and then, stiffeners in both directions.

  get the rules useful to improve the vibro-acoustic behaviour of piano soundboards and more generally to any non-rectangular orthotropic ribbed plates.

Figure 3 :

 3 Figure 3: Classification of modal shapes into different families.

Figure 4 :

 4 Figure 4: Influence of the non-linearities into the string on the D1# radiated pressure: non-linearities induce efforts in the string that contribute to energy spectrum, especially in high frequencies.


de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

We prove the two theorems below.

Theorem 1 (Number fields have small 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 over Q. There exist integers r ⩽ Q log n and d ⩽ Q log n such that d+r r ⩽ Qn log n and there exists r polynomials E 1 , E 2 , . . . , E r of degree ⩽ d in Z[x 1 , . . . , x r ] all having coefficients bounded in absolute value by (nδ K ) Qlog n such that the (smooth and zero-dimensional affine) scheme with equations E 1 = E 2 = ⋅ ⋅ ⋅ = E r = 0 and det (∂E i ∂x j ) 1⩽i, j⩽r = 0 contains Spec K as one of its irreducible components.

Theorem 2 (Number fields with bounded discriminant). There exists a positive constant Q such that the following is true. Let n ⩾ Q be an integer. Let H ⩾ 1 be an integer. The number of isomorphism classes of number fields with degree n and discriminant ⩽ H is ⩽ n Qn log 3 n H Q log 3 n .

The meaning of Theorem 1 is that we can describe a number field using few parameters in some sense. We have a short description of it as a quotient of a finite algebra : the smooth zerodimensional part of a complete intersection of small degree and small height in a projective space of small dimension.

Theorem 2 improves on previous results by Schmidt [START_REF] Schmidt | Number fields of given degree and bounded discriminant[END_REF] and Ellenberg-Venkatesh [START_REF] Ellenberg | The number of extensions of a number field with fixed degree and bounded discriminant[END_REF].

Schmidt uses Minkowski theorem on successive minima of the lattice of integers and obtains a bound H n+2 4

times a function of n for the number N n (H) of number fields with degree n and discriminant bounded by H in absolute value.

Ellenberg and Venkatesh consider r-uples of small integers (α i ) 1⩽i⩽r in K and define the mixed trace χ σ = Tr(∏ i α σ i i ) for every r-uple σ = (σ i ) i of rational integers. They show how to reconstruct the multiplicative structure of K from a collection of χ σ where σ runs over a subset Σ ∈ N r of managable size. As a result they prove N n (H) ≪ n H exp(Q √ log n) with a subexponential (halfway between n and polynomial in log n) exponent in H. The semigroup N r of all possible σ plays an important role in their proof.

We too make use of a few small integers in K. We find small algebraic relations between them using Minkowski theorem. Proving that the ideal of relations can be generated by a few of such small relations seems out of reach to us but we can cope with local equations. Familiar generic well-posedness results in multivariate Lagrange interpolation shows through Ellenberg and Venkatesh's work e.g. in the proof of [START_REF] Ellenberg | The number of extensions of a number field with fixed degree and bounded discriminant[END_REF]Lemma 2.3]. Similarly the key to find local equations is a generic well-posedness result in multivariate Hermite interpolation. Such a result has been proven by Alexander and Hirschowitz. We use it to produce a small projective model of Spec K in dimension and degree O(log n) and with bit size of order n(log n) 3 (log δ K + log n). As a consequence one can lower the exponent of H in an upper bound for N n (H) down to a polynomial in log n, namely O(log 3 n).

Following the pre-publication of the present text Lemke and Thorne [START_REF] Robert | Upper bounds on number fields of given degree and bounded discriminant[END_REF] improved the latter exponent to O(log 2 n) by applying the Alexander-Hirshowitz theorem to prove the local smoothness of the multisymmetric map of Ellenberg and Venkatesh.

It is sometimes conjectured that N n (H) ∼ c n H for n ⩾ 2 where c n does not depend on H. This is known for 2 ⩽ n ⩽ 5 according to work by Davenport and Heilbronn [START_REF] Davenport | On the density of discriminants of cubic fields[END_REF] for n = 3, and Bhargava [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] for n = 4, 5. Cohen, Diaz and Olivier have collected experimental data e.g. in [START_REF] Cohen | A course in computational algebraic number theory[END_REF][START_REF] Cohen | Advanced topics in computational number theory[END_REF][START_REF] Cohen | Counting discriminants of number fields[END_REF] suggesting that N n (H) should grow linearly in H for fixed n ⩾ 2. Malle has stated in [START_REF] Malle | On the distribution of Galois groups[END_REF] a more general and accurate conjecture on the distribution of Galois groups of number fields that would confirm this intuition.

In Section 2 we recall notation, definitions and elementary results from the geometry of numbers. In Section 3 we construct models for number fields as irreducible components of complete intersections with small height in low dimensional projective spaces. The last section is devoted to the proof of Theorem 1 and Theorem 2.

The author thanks Pascal Autissier, Karim Belabas, Georges Gras and Christian Maire for their comments and suggestions. This study has been carried out with financial support from the French State, managed by the French National Research Agency(ANR) in the frame of the Programmes FLAIR (ANR-17-CE40-0012 ANR-10-IDEX-03-02) and CLapCLap (ANR-18-CE40-0026).

SHORT INTEGERS

Let K be a number field and let n be the degree of K over Q. Let O be the ring of integers of K. Let (ρ i ) 1⩽i⩽r be the r real embeddings of K. Let (σ j , σj ) 1⩽j⩽s be the 2s complex embeddings of K. We also denote by (τ k ) 1⩽k⩽n the n = r + 2s embeddings of K. Let

K R = K ⊗ Q R = R r × C s
be the Minkowski space. We follow the presentation in [16, 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 .
In particular the contribution of complex embeddings is counted twice

< x, x >= 1⩽i⩽r x 2 i + 2 1⩽j⩽s x j 2 .
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.
ω i ⩽ v O = δ n 2 K .
We assume that the sequence i ↦ ω i is non-decreasing and deduce that

ω i ⩽ v 1 (n+1-i) O
for every 1 ⩽ i ⩽ n. This inequality is a bit unsatisfactory because it provides little information on the largest ω i . To improve on this estimate we use the fact that O is an integral domain. We let m = ⌈(n + 1) 2⌉ be the smallest integer bigger than n 2. On the one hand

ω i ⩽ δ K
for every 1 ⩽ i ⩽ m. On the other hand the products (ω i ω j ) 1⩽i,j⩽m generate a Z-module of rank n. Otherwise there would exist a non-zero linear form f ∶ O → Z vanishing on these products. So the m forms f ○ ω i would be orthogonal to the m vectors ω j . Then m + m ⩽ n. A contradiction. We deduce that all the successive minima of O are ⩽ δ 2 K . In other words O is well balanced.

Proposition 1 (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 (α i ) 1⩽i⩽n over Z such that all the absolute values of all the α i are ⩽ δ 2 K . 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 which is somewhat stronger but less accurate.

SMALL MODELS

Let K C = K ⊗ Q C = C n . 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. Let (P τ ) τ be n pairwise distinct points in

C r = A r (C).
The P τ are indexed by the n embeddings of K. These n points form a set (a reduced zerodimensional subscheme of P r C ) called P . We call I the corresponding ideal sheaf on P r C . We denote by 2P the scheme associated with I 2 . It consists of n double points. We say that the scheme 2P is well poised (or non-special) in degree d if it imposes n(r + 1) independent conditions on degree d homogeneous polynomials. Equivalently, the map

H 0 (O P r (d)) → H 0 (O 2P (d))
is surjective. This is the case if and only if the n(r + 1) × d+r d matrix M 1 P = [(m(P τ )) τ, m∈M , (∂m ∂x 1 (P τ )) τ, m∈M , (∂m ∂x 2 (P τ )) τ, m∈M , . . . , (∂m ∂x r (P τ )) τ, m∈M ] 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 a theorem of Alexander [START_REF] Alexander | Singularités imposables en position générale à une hypersurface projective[END_REF], generalized by Alexander and Hirschowitz [START_REF] Alexander | Polynomial interpolation in several variables[END_REF]. Chandler [7, Theorem 1] provides a simpler statement and proof. The recent exposition and simplification by Brambilla and Ottaviani [START_REF] Chiara | On the Alexander-Hirschowitz theorem[END_REF] is very useful also.

We now let (α i ) 1⩽i⩽n be n linearly independant short elements in O as in Proposition 1. 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 α i for 1 ⩽ j ⩽ r. Let Q ∶ Q[x 1 , . . . , x r ] → K be the homomorphism of Q-algebras sending x j to κ j for 1 ⩽ j ⩽ r. Let e Q ∶ Spec K → A r Q ⊂ P r Q
be the corresponding morphism of schemes. Tensoring Q by R we obtain an homomorphism

R ∶ R[x 1 , . . . , x r ] → K R sending x j to ((ρ(κ j )) ρ , (σ(κ j )) σ ). We call e R ∶ Spec K R → A r R ⊂ P r R
the corresponding morphism of schemes. We define

C ∶ C[x 1 , . . . , x r ] → K C and e C ∶ Spec K C → A r C ⊂ P r C
similarly. In particular C maps x j onto (τ (κ j )) τ . We now consider the points (P τ ) τ such that x 0 (P τ ) = 1 and

(x j (P τ )) τ = 1⩽i⩽n u i,j τ (α i ) τ ,
for 1 ⩽ j ⩽ r or equivalently

P τ = ( 1⩽i⩽n u i,j τ (α i )) 1⩽j⩽r ∈ C r = A r (C) ⊂ P r (C).
The maximal minors of the corresponding matrix 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 [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. We assume that the u i,j meet these conditions.

Since 2P is well poised, P is well poised also. So e Q , e R and e C are closed immersions. In order to describe them efficiently 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. The lattice of relations with integer coefficients and degree ⩽ d is the intersection between Ker R and

V Z = Z[x 1 , . . . , x r ] d . This is a free Z-module L ⊂ V R 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 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

L ⊥ = L ⊥ ∩ V Z . Since V Z is
det(T M 0 P (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 divided by d K . Recall that the coefficients in M 0 P are degree ⩽ d monomials in the κ j = ∑ 1⩽i⩽n u i,j α i . The coefficients u i,j are bounded form above by dn(r + 1). All the absolute values of the α i are bounded from above by δ 2 K . So the coefficients in M 0 P are bounded from above by

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

D = d + r d (n 2 d(r + 1)) 2d δ 4d K .
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 the size of the i-th equation is bounded from above

E i ⩽ 2( +1-i) D n 2( +1-i)
for every 1 ⩽ i ⩽ . Again, this inequality is a bit unsatisfactory because it provides little information on the largest equations. This time we see no other way around than forgetting the last n -1 equations. On the one hand

E i ⩽ 2n D 1 2 for every 1 ⩽ i ⩽ + 1 -n.
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. In more geometric terms the C-vector space generated by the + 1 -n first equations (E i ) 1⩽i⩽ +1-n surjects onto the cotangent space to P r C at the geometric point P τ for at least one τ . This means that there exist r integers

1 ⩽ i 1 < i 2 < ⋅ ⋅ ⋅ < i r ⩽ + 1 -n such that the minor determinant det ((∂E i k ∂x j )(P τ )) 1⩽k, j⩽r
is non-zero for some τ and thus for all τ by Galois action.

× d + r d 1 2 (n 2 d(r + 1)) d δ 2d K where = d + r d -n,
and such that the (smooth and zero-dimensional affine) scheme with equations

E 1 = E 2 = ⋅ ⋅ ⋅ = E r = 0 and det (∂E i ∂x j ) 1⩽i, j⩽r = 0
contains Spec K as one of its irreducible components.

PROOF OF MAIN RESULTS

In this section, 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.

We specialize the values of the parameters r and d in Proposition 2. We will take d = r. It is evident that 2r r ⩾ 2 r so 1 r + 1 We choose r to be the smallest positive integer such that n(r + 1) ⩽ 2r r . We have This proves Theorem 1. Theorem 2 follows because there are r 2r r coefficients to be fixed. We note also that there may appear several number fields in the smooth zero dimensional part of the complete intersection E 1 = E 2 = ⋅ ⋅ ⋅ = E r = 0. However the Chow class of this intersection is r r ⩽ (log n) Q log n and the number of isolated points is bounded by this intersection number [START_REF] Fulton | Intersection theory[END_REF]Chapter 13].

  unimodular, L and L ⊥ have the same volume. See [15, 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. We call Z,d ∶ Z[x 1 , . . . , x r ] d → O the evaluation map in degree ⩽ d. We observe that L ⊥ contains the image of Ô by the transpose map ˆ Z,d ∶ Ô → Z[x 1 , . . . , x r ] d where 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 matrix M 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

( 1 )

 1 n(r + 1) ⩽ 2r r ⩽ 4n(r + 1) and r ⩽ 3 log n for n large enough. We deduce that= 2r r -n ⩽ 4n(r + 1) ⩽ Qn log n. So 2n ⩽ n Q log n .From Equation (1) we deduce that 2r r ⩽ Qn log n. Also n 2 d(r + 1) ⩽ Qn 2 log 2 n and n 2 d(r + 1) r ⩽ n Q log n . So the coefficients of equations E i are bounded in absolute value byn Q log n δ Q log n K .

  The volume of the sphere {x ∈ L, x ⩽ 1} is ⩾ 2 -2 . Applying Minkowski's second theorem [19, 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⩽

  There exists r polynomials E 1 , E 2 , . . . , E r of degree ⩽ d in Z[x 1 , . . . , x r ] having coefficients bounded in absolute value by

	n(r + 1) ⩽	d + r d	.
	2n

Proposition 2 (Number fields have small models). Let K be a number field of degree n and root discriminant δ K over Q. Let d ⩾ 5 and r ⩾ 1 be rational integers such that