Catherine Hennechart-Collette

Océane Dehan

Michel Laurentie

Audrey Fraisse

Sandra Martin-Latil

Sylvie Perelle
email: sylvie.perelle@anses.fr

Detection of norovirus, hepatitis A and hepatitis E viruses in multicomponent foodstuffs

Keywords: Multicomponent foodstuff, Human norovirus, Hepatitis virus (A, E), quantitative real-time RT-PCR, Detection, Process control

Among the enteric viruses implicated in foodborne outbreaks, the human norovirus and hepatitis viruses A and E (HAV and HEV) represent a serious public health concern.

International standard ISO 15216 proposes methods for detecting HAV and norovirus (genogroups I and II) RNA from soft fruit, leaf, stem and bulb vegetables, bottled water or food surfaces. These methods had not previously been validated for detecting the targeted viruses in other foodstuffs such as multicomponent foods, nor for detecting other viruses in foodstuffs. The aim of this study was to characterise a method derived from the vegetable method described in ISO 15216 to detect HAV, HEV and norovirus in artificiallycontaminated multicomponent foodstuffs according to the recent international standard ISO 16140-4.

Results showed that the mean recovery rates for all settings did not differ according to the operator. The mean extraction yields ranged from 0.35% to 40.44% for HAV, 5.19% to 100% for HEV, 0.10% to 40.61% for norovirus GI and 0.88% to 69.16% for norovirus GII. The LOD95 was 10 2 genome copies/g for HAV, HEV and norovirus GII and 10 3 genome copies/g for norovirus GI. The LOQ was 2.90x10 4 , 1.40x10 3 , 1.60x10 4 and 1.30x10 4 genome copies/g for HAV, HEV, norovirus GI and norovirus GII respectively. The MNV-1 process control was detected in 120 out of 128 RNA extracts analysed and was recovered with an efficiency of between 3.83% and 50.22%. The mean inhibition rates of quantitative real-time RT-PCR reaction ranged from 3.25% to 28.70% and varied significantly with the type of food matrix.

The described method could be used to detect viruses in composite food products for routine diagnosis needs.

Introduction

Viruses are a leading cause of foodborne disease worldwide. Human norovirus and hepatitis viruses (hepatitis A (HAV) and hepatitis E (HEV)) are recognised to be the main viruses of public health importance. Enteric viruses are primarily transmitted via the faecal-oral and vomit-oral routes, including direct person-to-person contact, consumption of contaminated food or water, contact with contaminated environmental surfaces [START_REF] Kotwal | Environmental persistence and transfer of enteric viruses[END_REF][START_REF] Matthews | The epidemiology of published norovirus outbreaks: a review of risk factors associated with attack rate and genogroup[END_REF] and for hepatitis E virus direct contact with infected animals (Dalton et al., 2013;[START_REF] Meng | Hepatitis E virus: animal reservoirs and zoonotic risk[END_REF][START_REF] Pavio | L'hépatite E : une zoonose méconnue[END_REF][START_REF] Pavio | Hépatite E: nouvelles connaissances[END_REF]. Various food products, such as bivalve molluscs, fresh fruit and vegetables including different types of lettuce, onions or berries have been involved in foodborne disease outbreaks worldwide [START_REF] Bernard | Large multistate outbreak of norovirus gastroenteritis associated with frozen strawberries, Germany[END_REF][START_REF] Donnan | A multistate outbreak of hepatitis A associated with semidried tomatoes in Australia[END_REF][START_REF] Ethelberg | Outbreaks of gastroenteritis linked to lettuce, Denmark, January 2010[END_REF][START_REF] Fournet | Another possible foodborne outbreak of hepatitis A in the Netherlands indicated by two closely relatedmolecular sequences[END_REF][START_REF] Gallot | Hepatitis A associated with semidried tomatoes, France[END_REF][START_REF] Herman | Outbreaks attributed to fresh leafy vegetables, United States, 1973-2012[END_REF][START_REF] Guyader | Comprehensive analysis of a norovirus-associated gastroenteritis outbreak, from the environment to the consumer[END_REF]Muller et al., 2016;[START_REF] Sarvikivi | Multiple norovirus outbreaks linked to imported frozen raspberries[END_REF][START_REF] Thebault | Infectivity of GI and GII noroviruses established from oyster related outbreaks[END_REF][START_REF] Wadl | Food-borne norovirus-outbreak at a military base, Germany[END_REF]. A wide variety of foodstuffs have been implicated and mixed foods and ready to eat meals have frequently been considered as the carriers. In France, in addition to the high-risk food categories, meat, water, dishes with mixed foods or ready to eat meals have also been found to be responsible for 33% of foodborne illnesses and more specifically for 31% of viral foodborne disease outbreaks (Santé publique France, 2017). Virus outbreaks have been much more frequently reported in settings using catering services and in restaurants (Santé publique France, 2017). Food contamination mainly occurs in restaurants during food preparation by infected workers, and is most often associated with food service settings [START_REF] Baert | Evaluation of viral extraction methods on a broad range of ready-to-eat foods with conventional and real-time RT-PCR for Norovirus GII detection[END_REF][START_REF] Barrabeig | Foodborne norovirus outbreak: The role of an asymptomatic food handler[END_REF][START_REF] Franck | Sources of Calicivirus contamination in foodborne outbreaks in Denmark, 2005-2011the role of the asymptomatic food handler[END_REF]Ronnqvist et al., 2014;[START_REF] Stals | Norovirus transfer between foods and food contact materials[END_REF].

Most foodborne viruses are currently difficult or impossible to cultivate [START_REF] Hamza | Methods to detect infectious human enteric viruses in environmental water samples[END_REF], and sensitive molecular methods are therefore used to detect them in food. International standard ISO 15216 proposes methods for detecting HAV and norovirus (genogroup I and II) RNA from soft fruit, leaf, stem and bulb vegetables, bottled water or food surfaces.

However, these methods have neither been validated to detect other targeted viruses in other foodstuffs, such as multicomponent foods.

The aim of this study was to validate a method adapted from ISO 15216-1 for the detection of HAV, HEV and norovirus in multicomponent foodstuffs based on the recent international standard ISO 16140-4 (Microbiology of the food chain -Method validation -Part 4:

Protocol for method validation in a single laboratory) [START_REF] Anonymous | Microbiology of the food chain-Horizontal method for determination of hepatitis A virus and norovirus using real-time RT-PCR-Part1: Method for quantification[END_REF] to ensure the safety of these products.

Materials and methods

Viruses and cells

The FRhK-4 (foetal rhesus monkey kidney) cell line was purchased from the American Type Culture Collection (ATCC) (ATCC ® CRL-1688 ™ ) (LGC standards SARL, Illkirch, France). These epithelial cells were cultured in Dulbecco's modified Eagle's medium (DMEM, Gibco™) supplemented with non-essential amino acids (NEAA, Gibco™) and 10% of heat-inactivated foetal bovine serum (FBS, Gibco™) (Thermo Fisher Scientific, Waltham, MA, 209 USA). Cells were maintained at 37°C in a humidified atmosphere containing 95% air and 5% CO2.

HAV strain HM175/18f, clone B (VR-1402), was obtained from ATCC. This clone replicates rapidly and has cytopathic effects in cell cultures [START_REF] Lemon | Antigenic and genetic variation in cytopathic hepatitis A virus variants arising during persistent infection: evidence for genetic recombination[END_REF]. HAV stock was produced by propagation in foetal rhesus monkey kidney (FRhK-4) cells (ATCC, CRL-1688) [START_REF] Cromeans | Development of a plaque assay for a cytopathic, rapidly replicating isolate of hepatitis A virus[END_REF]. The virus production titre was determined in HAV RNA genomic copies using a quantitative real-time RT-PCR standard curve obtained from the ten-fold diluted in vitro RNA transcripts as previously described [START_REF] Fraisse | Digital RT-PCR method for hepatitis A virus and norovirus quantification in soft berries[END_REF]. The HAV stock had a titre of 2.90x10 9 genome copies/mL.

A clarified HEV genotype 3e suspension was obtained from faecal samples of infected swine provided by the ANSES Maisons-Alfort Laboratory for Animal Health. The partial sequence of ORF2 had previously been deposited with GenBank (accession number JF718793). The faecal sample was suspended in 10 mM Phosphate Buffered Saline (PBS), pH 7.4, to obtain a final 10% suspension (w/v), vortexed and centrifuged at 4000 g for 20 min at 4°C. Aliquots of supernatants containing viral particles were then stored at -80°C. The number of HEV RNA copies in the faecal suspension was quantified by ausing the standard curve obtained with the quantified in vitro RNA transcripts as previously described (Martin-Latil et al., 2012). The clarified suspension stock of HEV had a titre of 1.40x10 7 genome copies/mL. Stool samples of norovirus GI (E8050) and norovirus GII (E7022) from infected humans were provided by the French National Reference Center for Gastroenteritis Viruses in Dijon, France. The faecal samples were suspended in 10 mM phosphate-buffered saline (PBS), pH 7.4 to obtain a final 10% suspension (w/v), and then vortexed and centrifuged at 4,000 x g for 20 min at 4°C. Aliquots of 100 μL were frozen and kept at -80°C for later use. The genomic titres of the clarified faecal suspensions were determined by quantitative real-time RT-PCR using a quantitative real-time RT-PCR standard curve obtained with the 10-fold diluted in vitro RNA transcripts as previously described [START_REF] Hennechart-Collette | Multiplex real-time RT-qPCR for the detection of Norovirus in bottled and tap water using murine norovirus as a process control[END_REF].

The clarified suspension stocks of norovirus GI and norovirus GII had titres of approximately 1.60x10 7 and 1.30x10 7 genome copies/mL respectively.

The murine norovirus MNV-1 (CW1 strain) was provided to the ANSES Fougères Laboratory (Fougères, France) by Dr. H. Virgin from Washington University (Saint Louis, MO, USA), and was propagated in a mouse leukaemic monocyte macrophage (RAW 264.7, ATCC TIB-71) cell line [START_REF] Cannon | Surrogates for the study of norovirus stability and inactivation in the environment: comparison of murine norovirus and feline calicivirus[END_REF]. RAW 264.7 was grown at 37°C in an atmosphere containing 5% CO2 in DMEM supplemented with GlutaMAX™, 1% non-essential amino acids and 10% foetal bovine serum (Life Technologies, Saint Aubin, France). The production stock of MNV-1 had titres of approximately 2.15x10 7 TCID50/mL.

Experimental design and artificial contamination of multicomponent foodstuffs

Twelve multicomponent foodstuff samples (samples 1 to 12 including vegetables, mixed vegetables, meals including meat or fish, soup or sauce) were purchased from a local supermarket. These samples were selected by taking account of food processing factors such as freezing, vacuum packaging or canning. Table 1 describes the selected samples according to matrix type and processing factor. The selected food samples were classified according to the EFSA FoodEx2 classification (https://data.food.gov.uk/codes/foodtype/hierarchy/main/_report).

The experimental design described in ISO 16140-4:2017 was used. Multicomponent foods were artificially contaminated at four contamination levels for all four viruses (HAV, HEV, norovirus GI and GII). The concentration levels for HAV, HEV and norovirus were obtained by using different inoculum dilution levels. The four inoculation levels ranged from 2.90x10 3 to 2.90x10 6 genome copies for HAV, 1.40x10 2 to 1.40x10 5 genome copies for HEV, 1.60x10 3 to 1.60x10 6 genome copies for norovirus GI and 1.30x10 3 to 1.30x10 6 genome copies for norovirus GII. The food samples were randomly allocated to four different settings (R1, R2, R3 and R4). For each setting, four different matrices per contamination level were analysed by two operators (operators A and B). Table 2 describes the allocation of the food matrices to the different settings along with virus level. Thus, for each virus, four settings (R1 to R4) with four inoculation levels and two replicates (operators A and B) were tested corresponding to 32 analyses in total. Tests were conducted on three different days to evaluate the reproducibility of the method. R1 and R2 settings were analysed on the same day, R3 and R4 on two different days.

Each sample was co-inoculated with 2.15x10 4 TCID50 of MNV-1 (process control virus) just before adding elution buffer. For each matrix, one food sample inoculated only with sterile water was used as a negative control during the entire sample processing and viral detection procedure.

Sample processing

The method used to recover the viruses was adapted from the ISO 15216-1 procedure described for vegetables by adding a final purification step with chloroform-butanol. Briefly, each inoculated sample (25 g) placed in a 400 mL polypropylene bag containing a filter compartment was soaked in 40 mL of elution buffer (Tris-HCl 100 mM, glycine 50 mM, 1%

beef extract (Sigma-Aldrich, Saint-Quentin Fallavier, France)) at pH of 9.5. The rinsing fluid was removed via the bag's filter compartment and centrifuged at 10,000 g for 30 min at 4°C to pellet the food particles. The pH of the decanted supernatant was adjusted to 7.2 ± 0.2 by the addition of 5 N HCl while the fluid was swirled constantly. The neutralised supernatant was supplemented with 10% (w/v) polyethylene glycol (PEG) 8,000 and 0.3 M NaCl (Sigma-Aldrich), and was then incubated with constant shaking at 4°C for one hour. The viruses were concentrated by centrifuging the solution at 10,000 g for 30 min at 4°C. The supernatant was discarded then what remained was centrifuged again at 10,000 g for 5 min at 4°C to compact the pellet. The pellet was then suspended in 500 µL of PBS and vortexed with 500 µL of chloroform: butanol, 1:1 (v/v). The suspension was then incubated for 5 min at room temperature, and centrifuged at 8,000 g for 15 min at 4°C. The upper aqueous phase containing viruses was directly processed by the nucleic acid extraction procedure.

Viral RNA extraction

NucliSENS® easyMAG™ lysis buffer (BioMérieux, Marcy l'Etoile, France) was added to the virus suspension (up to 3 mL) and total nucleic acid was extracted using the NucliSENS® easyMAG™ platform with the "off-board Specific A" protocol according to the manufacturer's instructions. Nucleic acids were eluted in 100 µL of elution buffer and stored at -80°C.

Primers and probes

The primers and probes used to quantify HAV, norovirus GI and norovirus GII have already been described in the literature [START_REF] Costafreda | Development, evaluation, and standardization of a real-time taqman reverse transcription-Pcr assay for quantification of Hepatitis A virus in clinical and shellfish samples[END_REF][START_REF] Da Silva | Evaluation of removal of noroviruses during wastewater treatment, using real-time reverse transcription-PCR: different behaviors of genogroups I and II[END_REF][START_REF] Kageyama | Broadly reactive and highly sensitive assay for Norwalk-like viruses based on realtime quantitative reverse transcription-PCR[END_REF][START_REF] Loisy | Real-time RT-PCR for norovirus screening in shellfish[END_REF][START_REF] Pinto | Risk assessment in shellfish-borne outbreaks of hepatitis A[END_REF][START_REF] Svraka | Etiological role of viruses in outbreaks of acute gastroenteritis in the Netherlands from 1994 through 2005[END_REF] and are recommended in ISO 15216 standards. The primers and probe used to quantify HEV were adapted from the model described by Jothikumar et al. (2006) and have been previously described (Martin-Latil et al., 2012b, 2014). The primers and the TaqMan® probe targeting the ORF1 polyprotein of the murine norovirus (MNV-1), which were designed using Beacon Designer software (Bio-Rad, Marnes-la-Coquette, France), have also been described in the literature (Martin-Latil et al., 2012b). All the primers and probes were purchased from Eurofins MWG Operon (Les Ulis, France).

Quantitative real-time RT-PCR conditions

One-step quantitative real-time RT-PCR amplifications were performed in duplicate on the

Statistical analysis

All statistical analyses were performed using the Statgraphics Centurion XVII software (Statgraphics Centurion Version 17.1.04). The influence of the operator factor on virus recovery rates (norovirus, HAV, HEV and MNV) was first assessed by using one-way analysis of variance (ANOVA). The result of the ANOVA is a p value associated with the hypothesis that the mean recovery rates of all groups were the same. The influence of additional factors on extraction yields of pathogenic viruses (norovirus, HAV, HEV and MNV) was studied by using a one-way ANOVA. The effect of virus (norovirus, HAV, HEV and MNV-1) on virus recovery rates was first evaluated. Because the extraction yields were statistically different according to the virus used (ANOVA, p < 0.05), a multiple-comparison procedure (Fisher's least-significant-differences (LSD)) was applied to determine which viruses, provided the highest extraction yields. Graphs plotting the mean and its standard error for each group illustrate the multiple comparison procedure. When confidence intervals for means do not overlap, the difference between two groups of a factor is significant. Next, four factors were tested on extraction yields: (1) dilution of RNA extracts (pure vs. 10-fold diluted), ( 2) the inoculated level of viruses, (3) setting experiments (R1 to R4) and (4) EAC recovery rates according to the type of food matrix.

The limit of detection values (LOD) which correspond respectively to 50% (LOD50) and 95%

(LOD95) of the probability of detection were calculated with the method for estimating POD (probability of detection) function and the LOD of a qualitative microbiological measurement method as described by [START_REF] Wilrich | Estimation of the POD function and the LOD of a qualitative microbiological measurement method[END_REF]. The POD-LOD calculation software was used (version 9, dated 2017-09-23) [START_REF] Wilrich | Estimation of the POD function and the LOD of a qualitative microbiological measurement method[END_REF]. This program can be freely downloaded from www.wiwiss.fuberlin.de/fachbereich/vwl/iso/ehemalige/professoren/wilrich/index.html. The limit of quantification (LOQ) was estimated for each virus using the total error approach based on the accuracy profile (Hubert et al, 2007).

The accuracy profile is calculated by a two-sided β expectation tolerance interval (β-ETI) for each level from the results of the validation experiments expressed, such as recovery (trueness) and precision (repeatability and intermediate precision). Data are log10 transformed and we defined the acceptability limits (λ) at ± 95 % of bias because the expected recovery is low. To calculate the accuracy profile a specified proportion has been defined. This proportion is fixed to 80% (β) and corresponds to future measurements would fall within the acceptability limits. The complete theory and calculation of accuracy profiles are described in the references (Hubert et al, 2007a, Hubert et al, 2007b[START_REF] Hubert | Harmonization of strategies for the validation of quantitative analytical procedures: A SFSTP proposal. Part IV. Examples of application[END_REF] and a fully developed application is presented for microbiological methods by [START_REF] Feinberg | Validation of an alternative method for counting enterobacteriaceae in foods based on accuracy profile[END_REF] and [START_REF] Boubetra | Analysis of drinking water and their validation of alternative methods for their application to escherichia coli[END_REF]. The procedure reported here is simplified and tailored to molecular methods, such as published by [START_REF] Saint-Cyr | Use of accuracy profile procedure to validate a real-time PCR method to quantify bacteria in feces[END_REF].

The results are presented through a graphical representation as follows: the horizontal (x)

axis shows the reference level in log10 concentration level (genome copies/g) and the vertical (y) axis shows at each level the bias, the acceptability limits (λ), and the β-ETI limits.

Results

Mean virus recoveries of norovirus, HAV and HEV according to operator

To determine whether the operator factor influenced virus recovery rates, the mean recovery rates obtained for all settings were compared for each operator (A and B) (Figure 1). These results showed that the mean recovery rates for all settings were 8.04% and 13.81% for HAV, 41.25% and 49.59% for HEV, 9.95% and 6.55% for norovirus GI, 15.78% and 14.88% for norovirus GII, 18.18% and 13.38% for MNV-1 for operator A and operator B respectively. The statistical analysis revealed that the operator factor did not influence virus recovery from multicomponent food (one-way ANOVA; p-value=0.3215).

Virus recoveries and limit of detection

Table 3 gives the mean extraction yields obtained for norovirus GI, norovirus GII, HAV, HEV and the process control virus according to the inoculum level and repeat experiments (R1 to R4). As expected, no viral RNA was detected in the uninoculated samples. The average of recoveries with pure RNA extracts ranged from 0.35% to 40.44% for HAV, from 5.19% to 100% for HEV, from 0.1% to 40.61% for norovirus GI and 0.88% to 69.16% for norovirus GII.

The mean recovery rates obtained for HAV, HEV, norovirus GI and norovirus GII showed that the recovery rates from composite matrices vary according to the virus inoculated (one-way ANOVA; p-value<0.001) (Figure 2). More specifically, the multiple comparison tests showed that the recovery rates of HEV were higher and significantly different from other viruses.

By testing the 10-fold diluted RNA extracts, the extraction yields obtained for HAV, HEV and norovirus in composite matrices were improved by a factor that ranged from 0.0 to 25.7

(Table 3).

The process control (MNV-1) was detected in 120 out of 128 RNA extracts analysed and was recovered with an efficiency of between 3.83% and 50.22% for all four food settings.

The limits of detection (LOD) for the method were calculated by the Wilrich approach for each virus in each setting with the twelve samples. The LOD50 and LOD95 values for HAV, HEV, norovirus GI and norovirus GII are shown in Table 4.

The LOD95 values were 10 2 genome copies/g for HAV, HEV and norovirus GII and 10 3 genome copies/g for norovirus GI. The LOD50 values were between 80 and 92 genome copies/g for HAV, HEV and norovirus GII and 360 genome copies/g for norovirus GI.

The limit of quantification (LOQ) was assessed for different viruses using the accuracy profile. Figure 3 shows a typical accuracy profile for HAV. Table 5 summarizes the results for the LOQ and LOD and some performance characteristics (repeatability and intermediate precision). The LOQ was 2.90x10 4 , 1.40x10 3 , 1.60x10 4 , 1.30x10 4 genome copies/g for HAV,

HEV, norovirus GI and norovirus GII respectively. The LOD for HAV, HEV, norovirus GI and norovirus GII was established at 0.97x10 4 , 0.47x10 4 , 0.53x10 4 and 0.43x10 4 genome copies/g respectively.

Influence of experimental factors on virus extraction yield

To assess the influence of the different experimental factors on HAV, HEV and norovirus extraction from multicomponent matrices, the mean recovery rates from virus-spiked samples were compared. Figure 4 shows the recovery rates for HAV, HEV and norovirus according to their inoculation levels. The statistical analysis indicated that the recovery rates were not significantly different for norovirus GII regardless of the inoculation level (one-way ANOVA; p-value=0.2942 for norovirus GII), but were significantly different for norovirus GI, HAV and HEV when different inoculation levels were tested (one-way ANOVA; p-value<0.001

for HAV and norovirus GI, and p-value<0.0243 for HEV).

More specifically, the multiple-comparison tests showed that inoculations with 2.90x10 3 HAV genome copies, 1.40x10 2 HEV genome copies and 1.6x10 3 norovirus GI genome copies differed significantly from other inoculation levels of HAV, HEV and norovirus GI.

Figure 5 shows the mean recovery rates of HAV, HEV, norovirus in the four repeat experiments. The differences between R1 to R4 were not significant for the extraction yield of norovirus GI (one-way ANOVA; p-value=0.0736) but were significant for the extraction yield of norovirus GII, HAV and HEV (one-way ANOVA; p-value=0.0028 for HAV, one-way ANOVA; p-value<0.001 for norovirus GII and for HEV). More specifically, the multiplecomparison tests showed that R3 was significantly different from R1, R2 and R4 for HAV.

Both R1 and R3 were significantly different from R4 for HEV, and R4 was significantly different from R1, R2 and R3 for norovirus GII.

The statistical analysis also revealed that the allocated food setting (factor R1 to R4) influences mean MNV-1 recoveries (one-way ANOVA; p-value=0.0046). The multiplecomparison tests showed that R1 and R2 were significantly different from R3 and R4. Mean MNV-1 recoveries from allocated food settings R1 and R2 were higher than allocated food settings R3 and R4.

Recovery rates for external amplification control (EAC)

An EAC corresponding to the viral target was used to examine quantitative real-time RT-PCR inhibition. Table 6 shows the mean percentages of quantitative real-time RT-PCR inhibition for each of the twelve samples. The mean percentages of quantitative real-time RT-PCR inhibition varied from 3.25% to 28.70%. They vary significantly with the type of food matrix (one-way ANOVA; p-value=0.0209) but not with repeated experiments (R1 to R4) (one-way ANOVA; p-value=0.0554).

Discussion

Enteric viruses are the leading cause of foodborne outbreaks. Their detection is challenging because many of them are either difficult or impossible to replicate in cell culture, and the number of viral particles present in the food may be very low. Today, quantitative real-time RT-PCR is widely used for virus detection because it is sensitive, specific, rapid and can deliver quantitative data. ISO standards 15216-1 and 15216-2 were published in 2017 and 2019 respectively for detecting and quantifying norovirus and HAV in high-risk food categories such as shellfish, bottled water and vegetables. The method used in this study to recover viruses from composite foods is adapted from the ISO 15216 procedure described for vegetables. The LOD95 varied from 10 2 to 10 3 genome copies per g regardless of the virus analysed. Because of the complexity of the food matrices used in our study, the LOD95 values obtained for norovirus and HAV in composite foods and multicomponent foodstuffs were higher than the LOD95 recently reported for lettuce or raspberries (Lowther et al., 2019).

Derived from the LOQ estimated by accuracy profile, the LOD in our study ranged between 4.33x10 3 and 9.67x10 3 genome copies per g. These values are slightly different than the estimated LOD95 but the difference, which is around one log, can be explained by the definition of these parameters. LOD95 is the level at which the probability of detection is equal to 95 % (ISO standard 16140-1), whereas the LOD is estimated using a quantitative approach based on a chemical method. Classically, the LOD is the lowest level that differs from the background noise with an accepted confidence level. [START_REF] Armbruster | Limit of blank, limit of detection and limit of quantitation[END_REF] have previously explained that the LOD is a value as mean + k SD with k, the coverage factor, ranged between 2 and 10. The common value for k is 3 for LOD and if k = 10 it is the LOQ.

However, the LOQ is the lowest level quantifiable with a defined trueness and precision (ISO standard 16140-1). In our case, we defined the LOQ using an accuracy profile with a defined trueness and precision, then derived the LOD from this LOQ. These two approaches are totally different because one is for a qualitative approach and the other for a quantitative approach. The results are nonetheless of the same magnitude and consequently the LOD confirms the LOD95 values.

Various methods have been described for extracting viruses from vegetables, composite foods or dairy products [START_REF] Baert | Evaluation of viral extraction methods on a broad range of ready-to-eat foods with conventional and real-time RT-PCR for Norovirus GII detection[END_REF][START_REF] Blaise-Boisseau | Duplex real-time qRT-PCR for the detection of hepatitis A virus in water and raspberries using the MS2 bacteriophage as a process control[END_REF][START_REF] Cheong | Concentration method for the detection of enteric viruses from large volumes of foods[END_REF][START_REF] Coudray | Comparison of two extraction methods for the detection of hepatitis A virus in lettuces using the murine norovirus as a process control[END_REF][START_REF] Dubois | Modified concentration method for the detection of enteric viruses on fruits and vegetables by reverse transcriptase-polymerase chain reaction or cell culture[END_REF]El Sanousy et al., 2013;[START_REF] Hennechart-Collette | Comparison of three extraction methods to detect noroviruses in dairy products[END_REF][START_REF] Hida | Development of a rapid total nucleic acid extraction method for the isolation of hepatitis A virus from fresh produce[END_REF][START_REF] Hyeon | Rapid detection method for hepatitis A virus from lettuce by a combination of filtration and integrated cell culture-real-time reverse transcription PCR[END_REF][START_REF] Kim | Optimization of methods for detecting norovirus on various fruit[END_REF][START_REF] Morillo | Rapid detection of norovirus in naturally contaminated food: foodborne gastroenteritis outbreak on a cruise ship in Brazil, 2010[END_REF][START_REF] Pan | Detection of human norovirus in cherry tomatoes, blueberries and vegetable salad by using a receptor-binding capture and magnetic sequestration (RBCMS) method[END_REF][START_REF] Sanchez | A single method for recovery and concentration of enteric viruses and bacteria from fresh-cut vegetables[END_REF][START_REF] Scherer | Comparison of two extraction methods for viruses in food and application in a norovirus gastroenteritis outbreak[END_REF][START_REF] Schwab | Development of methods to detect "Norwalk-like viruses" (NLVs) and hepatitis A virus in delicatessen foods: application to a food-borne NLV outbreak[END_REF]Stals et al., 2011aStals et al., , 2011b)).

The PEG concentration method was used in our study to concentrate viruses from multicomponent foods. Our results are in agreement with data reported in other studies on food. Virus recovery rates ranged from 0.35% to 40.44% for HAV, 5.19% to 100% for HEV and 0.10% to 69.16% for norovirus. The PEG concentration method was previously developed and applied to recover viruses from suspected food such as shellfish, smooth-surfaced, semidried tomatoes or food samples such as spaghetti, chicken, ham and sausages [START_REF] Martin-Latil | Method for HEV detection in raw pig liver products and its implementation for naturally contaminated food[END_REF], 2012a;[START_REF] Saito | Development of a practical method to detect Noroviruses contamination in composite meals food[END_REF][START_REF] Summa | Evaluation of four virus recovery methods for detecting noroviruses on fresh lettuce, sliced ham, and frozen raspberries[END_REF][START_REF] Rutjes | Real-time detection of noroviruses in surface water by use of a broadly reactive nucleic acid sequence-based amplification assay[END_REF]. The norovirus recovery rates in the literature varied between 0.02% and 47%, while the HEV recovery rate from pig liver sausage ranged from 3.94 to 18.38% and the HAV recovery rate from semidried tomatoes ranged from 2.91% to 50.92% [START_REF] Hennechart-Collette | Comparison of three extraction methods to detect noroviruses in dairy products[END_REF][START_REF] Martin-Latil | Method for HEV detection in raw pig liver products and its implementation for naturally contaminated food[END_REF], 2012a;[START_REF] Saito | Development of a practical method to detect Noroviruses contamination in composite meals food[END_REF][START_REF] Summa | Evaluation of four virus recovery methods for detecting noroviruses on fresh lettuce, sliced ham, and frozen raspberries[END_REF][START_REF] Rutjes | Real-time detection of noroviruses in surface water by use of a broadly reactive nucleic acid sequence-based amplification assay[END_REF]. Taking into account the mean recovery rates obtained with other matrices, our results lie within the same range.

ISO standard 15216-1 was validated for the detection and quantification of HAV and norovirus in seven food matrices: bottled water, food surfaces, Pacific oysters (Crassosttrea gigas), common mussels (Mytilus edulis), raspberries, lettuce and green onions [START_REF] Lowther | Validation of ISO 15216 part 1 -Quantification of hepatitis A virus and norovirus in food martrices[END_REF]. Other methods for the detection of viruses in semi-dried tomatoes, raw pig livers and dairy products have also been characterised [START_REF] Hennechart-Collette | Comparison of three extraction methods to detect noroviruses in dairy products[END_REF][START_REF] Martin-Latil | Method for HEV detection in raw pig liver products and its implementation for naturally contaminated food[END_REF], 2012a). However, to our knowledge, no publications until now have reported the validation of a method for detecting norovirus, HAV or HEV from multicomponent foodstuffs.

The experimental design from ISO standard 16140-4:2017 was applied to characterise a method used to detect norovirus, HAV and HEV in multicomponent foodstuffs. In addition to the factors studied in the ISO 16140 standard, part 4 describes the calculation of repeatability and reproducibility. The experimental design allowed us to study a method without a reference method. Twelve different foodstuffs (vegetables, mixed vegetables, meals including meat or fish, soup or sauce) were selected and analysed in this study. This selection allowed us to take into account a wide selection of multicomponent foods representative of samples analysed in the laboratory when investigating viral foodborne outbreaks. The items selected are consumed in a variety of forms and are major components of meals.

To calculate repeatability, foods were randomly allocated to four different settings (R1, R2, R3 and R4) and results show that the differences between repeated experiments (R1 to R4)

were significant for the extraction rate of norovirus GII, HAV and HEV. Different studies have

shown that the composition of food products can affect virus extraction [START_REF] Blaise-Boisseau | Duplex real-time qRT-PCR for the detection of hepatitis A virus in water and raspberries using the MS2 bacteriophage as a process control[END_REF][START_REF] Butot | Procedure for rapid concentration and detection of enteric viruses from berries and vegetables[END_REF][START_REF] Summa | Evaluation of four virus recovery methods for detecting noroviruses on fresh lettuce, sliced ham, and frozen raspberries[END_REF][START_REF] Yavarmanesh | Impact of milk components on recovery of viral RNA From MS2 bacteriophage[END_REF][START_REF] Yavarmanesh | Impact of milk components in recovery of the MS2 bacteriophage as an indicator of enteric viruses[END_REF]. Due to the presence of substances that can inhibit PCR amplification, the implementation of different controls such as the virus process control and EAC are necessary to validate results.

According to the recommendations in ISO 15216, the inhibition rates for RNA extracted from food samples have to be lower than 75% and virus process control extraction yields higher than 1%. In this study, the mean percentages of quantitative real-time RT-PCR inhibition were always lower than 75% and the recovery rate of MNV-1 was more than 1% in 93% of the RNA extracts analysed. In our study, MNV-1 was successfully tested as a process control virus for detecting HAV, HEV and norovirus in multicomponent foodstuffs. Its use had previously been described in the literature for different types of water, shellfish, milk products, meat such as sausages, soft red fruits, lettuces and ready-to-eat foods [START_REF] Coudray | Comparison of two extraction methods for the detection of hepatitis A virus in lettuces using the murine norovirus as a process control[END_REF][START_REF] Hennechart-Collette | Determination of which virus to use as a process control when testing for the presence of hepatitis A virus and norovirus in food and water[END_REF][START_REF] Martin-Latil | Method for HEV detection in raw pig liver products and its implementation for naturally contaminated food[END_REF][START_REF] Sanchez | A single method for recovery and concentration of enteric viruses and bacteria from fresh-cut vegetables[END_REF]Stals et al., 2011aStals et al., , 2011b)).

According to the ISO 16140 procedure, analyses were conducted by two operators. An evaluation of operator influence showed no significant difference between the two operators. The calibration and maintenance of equipment should also limit the difference between operators working at the same laboratory. To evaluate the quality of virus detection from multicomponent foodstuffs for other laboratories, it should be necessary to carry out an inter-laboratory assay.

To conclude, the method described herein could be used to detect viruses in composite food products for routine diagnosis needs. This research underscores the importance of further research to develop, standardise and validate methods for detecting viruses from other food matrices or other viruses than those described in standard ISO 15216.

Virus

  CFX96™ real-time PCR detection system (Bio-Rad). Reactions were performed in a 25 μL reaction mixture containing 1X of RNA UltraSense™ master mix and 1.25 μL of RNA UltraSense™ enzyme mix, which are components of the RNA UltraSense™ One-Step Quantitative RT-PCR System (Life Technologies), 2 U RNase inhibitor (Life Technologies), 1.25µg of bovine serum albumin (Life Technologies), 500 nM of forward primer, 900 nM of reverse primer, 250 nM of probe and 5 μL of RNA extract.Positive controls containing RNA extracted from virus suspensions and a negative control containing all the reagents except the RNA template were included with each set of reaction mixtures. The one-step quantitative real-time RT-PCR programme involved 60 min of reverse transcription of RNA at 55°C, followed by a 5 min denaturation step at 95°C, and finally 45 cycles of 15 s at 95°C, 1 min at 60°C and 1 min at 65°C. Fluorescence was recorded by the apparatus at the end of the elongation steps (1 min at 65°C) for each amplification cycle. All the samples were characterised by a corresponding cycle threshold (Ct) values. Negative samples gave no Ct value. A standard curve for each viral target was generated with RNA extracts resulting from the serial dilution of viral stock suspension in distilled water. The slopes (S) of the regression lines were used to calculate the amplification efficiency (E) of the quantitative real-time RT-PCR reactions according to the formula E=10 |1/s| -1 to determine the performance of quantitative real-time RT-PCR assays. HAV, HEV, norovirus GI, norovirus GII, and MNV-1 recovery rate percentages from spiked samples were calculated by using the standard curves obtained with viral inoculum dilution and the following formula: quantity of virus recovered after spiking experiments/quantity of viral inoculum X100. The HAV, HEV, norovirus GI or norovirus GII RNA transcript was used as an external amplification control (EAC) to monitor RT-PCR inhibition in samples. This approach has been described in ISO 15216-1, where an external control RNA (i.e. an RNA species carrying the target sequence of interest) is added to an aliquot of RNA sample. The degree of RT-PCR inhibition in each tested sample is obtained by comparing these results with the results of EAC RNA in the absence of sample RNA (i.e. in water).

Figure 1 :Figure 2 :

 12 Figure1: Comparison of mean recovery rates for HAV, HEV, norovirus GI, norovirus GII and MNV according to the operator factor (operator A and operator B). For each virus, 16 analyses were performed by each operator (four foodstuffs for all settings with all levels of inoculation) and RNA extracts were analysed in duplicate (pure and ten-fold diluted) with a quantitative real-time RT-PCR assay

Figure 4 :Figure 5 :

 45 Figure4: Recovery rates for HAV, HEV and norovirus with respect to artificial inoculation levels

Table 2 :

 2 Experimental design for detection of each enteric virus (HAV, HEV and norovirus) in multicomponent foodstuffs adapted from the ISO standard 16140-4:2017 procedure performed by two operators

	Virus	Number of genome	RNA extracts	Repeated		Repeated		Repeated		Repeated
		copies		experiment R1	(F)	experiment R2	(F)	experiment R3	(F)	experiment R4	(F)
				(%±SD)		(%±SD)		(%±SD)		(%±SD)
	HAV	2.90x10 6	pure	1.56±1.31 (4/4)	1.6	2.80±1.05 (4/4)	1.2	32.53±3.99 (4/4)	1.5	4.39±0.72 (4/4)	0.8
			10-fold diluted	2.58±2.54 (4/4)		3.28±1.80 (4/4)		48.81±3.96 (4/4)		3.50±0.80 (4/4)
		2.90x10 5	Pure	3.72±4.01 (4/4)	1.5	3.62±2.34 (4/4)	1.1	7.26±8.27 (4/4)	1.7	0.41±0.67 (4/4)
			10-fold diluted	5.57±4.52 (4/4)		4.05±2.91 (4/4)		12.16±2.18 (3/4)		nd
		2.90x10 4	Pure	3.11±3.49 (4/4)	1.5	3.45±3.36 (4/4)	0.9	4.75±4.06 (4/4)	0.9	4.02±2.54 (4/4)	1.0
			10-fold diluted	4.56±5.29(3/4)		3.21±2.67 (4/4)		4.33±6.70 (2/4)		0.40±0.63 (4/4)
		2.90x10 3	Pure	24.99±21.17 (4/4) 2.3	19.06±16.29 (3/4)	1.2	40.44±14.48 (2/4) 0.1	0.35 (1/4)
			10-fold diluted	57.90±49.40 (4/4)		22.55 (1/4)		3.13±0.43 (2/4)		nd
	MNV-1	MNV		8.49±9.84		8.12±6.22		12.28±17.52		3.83±4.85
		Total samples with				27/32
		recovery rates >1%
	Virus	Number of genome	RNA extracts	Repeated		Repeated		Repeated		Repeated
		copies		experiment R1	(F)	experiment R2	(F)	experiment R3	(F)	experiment R4	(F)
				(%±SD)		(%±SD)		(%±SD)		(%±SD)
	HEV	1.40x10 5	pure	29.14±13.87 (4/4) 2.1	6.24±4.92 (4/4)	2.7	59.04±9.99 (4/4)	1.7	16.65±18.31 (4/4) 0.3
			10-fold diluted	60.22±40.64 (4/4)		16.92±10.99 (4/4)		100.00±0.00 (4/4)		4.62±2.21 (3/4)
		1.40x10 4	pure	33.64±45.25 (4/4) 2.8	57.34±38.49 (3/4)	1.0	27.49±16.87 (4/4) 2.2	5.19±6.09 (2/4)	0.9
			10-fold diluted	94.45 (1/4)		57.63±51.29 (2/4)		59.87±14.20 (2/4)		4.87±5.96 (3/4)
		1.40x10 3	pure	16.09±11.75 (2/4) 6.2	17.90 (1/4)	11.1	100.00±0.00 (3/4)		nd
			10-fold diluted	100.00±0.00 (3/4)		99.67±99.50 (2/4)		nd		14.62±10.48 (3/4)
		1.40x10 2	pure	55.79±62.52 (2/4)		11.64 (1/4)	8.6	100.00±0.00 (3/4)		50.64±53.76 (2/4) 1.7
			10-fold diluted	nd		100.00±0.00 (2/4)		nd		86.35
	MNV-1	MNV		18.43±11.74		15.40±16.00		12.78±8.44		12.33±5.39
		Total samples with				30/32
		recovery rates >1%

Table 3 :

 3 Mean percentage recovery calculated for four inoculum levels of HAV, HEV, norovirus GI or GII in the presence of MNV-1The mean of operator A and B replicates was used for each inoculation level sample. Results are expressed as means +/-SD. The number of positive Ct determinations is mentioned for HAV, HEV, norovirus GI and GII. RNA extracts were tested twice for each operator, resulting in four cycle threshold (Ct) values for each sample. The ratio between the mean values for extraction yields obtained with undiluted RNA extracts and those obtained with 10-fold diluted RNA extracts were calculated to determine whether the dilution of RNA extracts enhanced mean extraction yields (F). nd: not detected.

	Virus	Number of genome	RNA extracts	Repeated		Repeated		Repeated		Repeated
		copies		experiment R1	(F)	experiment R2	(F)	experiment R3	(F)	experiment R4	(F)
				(%±SD)		(%±SD)		(%±SD)		(%±SD)
	norovirus	1.60x10 6	pure	0.10±0.08 (4/4)	0.6	1.77±0.12 (4/4)	0.5	10.95±6.63 (4/4)	1.5	1.42±0.41 (4/4)	1.3
	GI		10-fold diluted	0.06±0.05 (2/4)		0.93±0.36 (4/4)		16.29±11.10 (4/4)		1.91±0.63 (4/4)
		1.60x10 5	pure	3.36±0.64 (4/4)	1.4	8.24±2.95 (4/4)	0.4	5.59±2.79 (4/4)	2.2	5.44±0.96 (4/4)	0.9
			10-fold diluted	0.48 (1/4)		2.96±3.33 (3/4)		12.01±10.46 (3/4)		5.18±4.23 (3/4)
		1.60x10 4	pure	2.55±3.34 (2/4)	4.6	nd		3.89±3.14 (4/4)	25.7	11.68±9.93 (3/4)	0.1
			10-fold diluted	11.96 (1/4)		9.76 (1/4)		100.00 (1/4)		1.68 (1/4)
		1.60x10 3	pure	1.14±0.54 (2/4)		nd		40.61 (1/4)		nd
			10-fold diluted	nd		nd		nd		100.00 (1/4)
	MNV-1	MNV		11.99±12.50		14.75±12.65		9.38±8.34		15.49±8.83
		Total samples with				31/32
		recovery rates >1%
	Virus	Number of	RNA extracts	Repeated		Repeated		Repeated		Repeated
		genome copies		experiment R1	(F)	experiment R2	(F)	experiment R3	(F)	experiment R4	(F)
				(%±SD)		(%±SD)		(%±SD)		(%±SD)
	norovirus	1.30x10 6	pure	1.85±0.67 (4/4)	1.0	2.08±1.59 (4/4)	1.0	10.16±6.97 (3/4)	1.8	21.63±1.76 (4/4)	1.5
	GII		10-fold diluted	1.78±0.79 (4/4)		2.09±2.09 (4/4)		18.60±4.04 (4/4)		32.06±2.60 (4/4)
		1.30x10 5	pure	8.10±1.89 (4/4)	0.4	7.98±1.47 (4/4)	0.4	14.24±0.79 (4/4)	0.9	49.94±12.26 (4/4)	1.2
			10-fold diluted	3.39±2.35 (4/4)		3.46±4.11 (4/4)		12.95±3.93 (3/4)		58.92±17.38 (4/4)
		1.30x10 4	pure	10.32±1.70 (4/4)	0.0	0.88±0.63 (4/4)		8.04±6.01 (4/4)	0.1	69.16±6.90 (4/4)	0.04
			10-fold diluted	0.34 (1/4)		nd		0.37 (1/4)		2.56±2.06 (3/4)
		1.30x10 3	pure	4.35 (1/4)		nd		8.89±8.67 (3/4)		nd
			10-fold diluted	nd		nd		nd		nd
	MNV-1	MNV		50.22±41.50		35.41±35.44		12.08±4.45		11.84±5.04
		Total samples				32/32
		with recovery
		rates >1%

Table 4 :

 4 LOD 50 and LOD 95 calculated by Wilrich approach for HAV, HEV and norovirus (*LOD 50 and LOD 95 values were estimated. It was not possible to determine values because the LOD was not reached)

Table 5 :

 5 performance characteristics for all viruses

		Virus
	HAV	HEV	norovirus GI norovirus GII

Table 6 :

 6 Mean percentages of quantitative real-time RT-PCR inhibition for each sample

	Sample number	Mean inhibition recovery rates (%±SD)
		15.71±8.56 (N= 8)
		6.44±9.15 (N= 8)
		3.25±5.77 (N= 8)
		8.75±9.29 (N= 16)
		12.99±23.26 (N= 16)
		13.67±25.95 (N= 16)
		6.27±11.60 (N= 8)
		28.70±27.00 (N= 16)
		22.00±13.36 (N= 8)
	10	16.51±10.33 (N= 8)
	11	15.44±13.08 (N= 8)
	12	26.84±13.39 (N= 8)

Acknowledgements

We would like to thank P. Pothier (CNR des virus des gastroentérites, Dijon, France) for providing contaminated stool samples. We are also grateful to N. Pavio (ANSES, Animal Health Laboratory, Unité Mixte de Recherche (UMR) 1161 Virology, Institut National de la Recherche Agronomique (INRA), École Nationale Vétérinaire d'Alfort (ENVA), Maisons-Alfort, France) for providing HEV-contaminated pig faeces.

R1 R2 R3 R4 Very low Sample 8 Sample 4 Sample 5 Sample 6 Low Sample 1 Sample 11 Sample 6 Sample 3 Medium Sample 2 Sample 10 Sample 7 Sample 5


Fuensanta Aroca

Julie Decaup
email: julie.decaup@im.unam.mx

Guillaume Rond
email: guillaume.rond@univ-amu.fr

FINITENESS RESULTS CONCERNING ALGEBRAIC POWER SERIES

Keywords: 2010 Mathematics Subject Classification. 11D88, 11G50, 13F25, 14G05 algebraic power series rings, constructible set

We construct an explicit filtration of the ring of algebraic power series by constructible sets, measuring the complexity of these series. As an example of use of this, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.

The ring of polynomials over a field k is a k-vector space filtered by finite dimensional vector spaces, namely the k-vector spaces of polynomials of degree less than d, for every d ∈ N. In many situations, the description of the ring of polynomials as a ring filtered by finite dimensional vector spaces is very useful, both for theoretical and applied aspects. The main advantage of this is that the set of polynomials of degree bounded by d is fully described by a finite amount of data. One common use of this fact is when one tries to approximate objects by polynomials. For example, this is the case in analysis, by the use of Weierstrass approximation theorem, or in algebra when one deals with approximations by polynomials of solutions of differential or functional equations, or transcendental estimations of such solutions.

In several cases, dealing with polynomials is not practical or effective enough, in particular because the implicit function theorem does not hold in the polynomial setting. To avoid this problem, one can replace the ring of polynomials by the ring of algebraic power series over k. This is the set of formal power series that are algebraic over the ring of polynomials. This set is a ring (satisfying the implicit function theorem), in particular it is a k-vector space, but there is no explicit or effective description of a filtration of this ring by finite dimensional spaces.

In this note, we give an explicit description of a filtration of the ring of algebraic power series by constructible sets which are not vector spaces. A constructible set is a subset of an affine space k n which is described by polynomial equalities and inequalities. Here, the sets of this filtration are the sets A(d, h) of algebraic power series whose minimal polynomial P (x, T ) over k[x], is such that deg x (P ) ≤ h, deg T (P ) ≤ d.

We prove that these sets can be seen as constructible subsets of k N (d,h) for some constant N (d, h) depending only on d and h (see Corollary 2.8), and the dimension of this constructible set is computed. Roughly speaking, the idea is to try to identify an algebraic power series with its minimal polynomial. But this cannot work directly, since two distinct algebraic power series may have the same minimal polynomial. Moreover, an irreducible polynomial in x and T has no power series root in general. Therefore there is no correspondence between algebraic power series and irreducible polynomials. To overcome this problem, we first describe the subset of A(d, h) of algebraic power series whose minimal polynomial satisfies the implicit function theorem as a constructible subset of some k N (see Theorem 2.6). And from this we deduce the same kind of result for A(d, h). Let us mention that this kind of approach has already been used in [START_REF] Flajolet | Coefficients of algebraic series[END_REF] and [START_REF] Hickel | On the algebraicity of Puiseux series[END_REF].

As an example of use of this filtration by constructible sets, we give a bound on the dimension of the set of solutions of polynomial equations with coefficients in k((x)) whose entries are in A(d, h), when k is algebraically closed (see Theorem 3.1). This problem is a non-archimedean analogue of the problem of bounding the number of Q-points of a complex algebraic variety.

Preliminaries

In the whole paper k will always denote a field, and x will denote a single indeterminate. The ring of algebraic power series will be denoted by k x . For every integer n, A n k will denote the affine space of dimension n over k. Definition 1.1. Let f ∈ k x . The morphism ψ : k[x, T ] -→ k x defined by ψ(P (x, T )) = P (x, f ) is not injective and its kernel is a height one prime ideal of k[x, T ]. Therefore it is generated by one polynomial. If P (x, T ) is such generator, any other generator of this ideal is a multiple of P (x, T ) by a non-zero element of k. Any such a generator is called a minimal polynomial of f . But, by abuse of language, we will always refer to such an element by the minimal polynomial of f .

Definition 1.2. [AB13][Ro15] Let P (T ) ∈ k[x, T ].
The maximum of the degrees of the coefficients of P (T ) seen as a polynomial in T is called the height of P . For f ∈ k x , the height of its minimal polynomial is called the height of f , and is denoted by H(f ). The degree of f is the degree of the field extension k(x) -→ k(x, f ) or, equivalently, the degree of its minimal polynomial seen as a polynomial in T . It is denoted by Deg(f ).

Definition 1.3. We define the following sets:

• A(d, h) denotes the set of algebraic power series of degree ≤ d and height ≤ h. • A(d, h) 0 denotes the set of algebraic series of A(d, h) that vanish at 0. • I(d, h) denotes the set of algebraic power series of A(d, h) 0 whose minimal polynomial satisfies the implicit function theorem.

That is, f ∈ I(d, h), if and only if its minimal polynomial P (x, T ) satisfies P (0, 0) = 0 and ∂P ∂T (0, 0) = 0.

In particular we have I(d, h) ⊂ A(d, h) 0 ⊂ A(d, h) for every d, h.

Recall that for an algebraic power series f ∈ k x , the (vanishing) order of f is

ord(f ) := sup{n ∈ N | f ∈ (x) n } ∈ N ∪ {∞}. Remark 1.4. It is straightforward to check that A(d, h) = A k × A(d, h) 0 .
Remark 1.5. Let f be an algebraic power series with f (0) = 0 and assume that there is P ∈ k[x, T ] such that P (0, 0) = 0, ∂P ∂T (0, 0) = 0 and P (x, f ) = 0. Then the minimal polynomial of f satisfies the implicit function theorem. Indeed such a minimal polynomial is denoted by Q and should divide P : P = QR for some polynomial R.

Then ∂P ∂T = ∂Q ∂T R + Q ∂R ∂T .
Since f (0) = 0 and Q(x, f (x)) = 0 then Q(0, 0) = 0. Hence ∂Q ∂T R (0, 0) = 0 and

f = f (0) + x e f (1)
where f (0) is a polynomial in x of degree ≤ e and vanishing at 0, and f (1) is an algebraic power series vanishing at 0. Then f (1) ∈ I(d, h + e(d -2)) and f (0) ∈ A e k by identifying the set of polynomials of degree ≤ e vanishing at 0 with A e k . Therefore we define ϕ d,h,e (f ) := (f (0) , f (1) ).

Proof. let f ∈ A(d, h, e) and let

P = a d T d + • • • + a 1 T + a 0
be its minimal polynomial. We can write

f = f (0) + x e f (1)
where f (0) is a polynomial in x of degree ≤ e and f (1) is an algebraic power series vanishing at 0. We have that

P (f (0) + Z) = P (f (0) ) + ∂P ∂T (f (0) )Z + Q(x, Z)
where Q(x, Z) is a polynomial in x and Z, divisible by Z 2 . Moreover if we write

P (f (0) + Z) = b d Z d + • • • + b 0 we have that deg(b i ) ≤ h + e(d -i). Let us set R(T ) := P (f (0) + x e T ) x 2e = P (f (0) ) x 2e + 1 x e ∂P ∂T (f (0) )T + Q(x, x e T ) x 2e .
We have Q(x,x e T )

x 2e ∈ k[x, T ] since Q(x, T ) is divisible by T 2 . Moreover since f (0) - f ∈ (x e+1 ) we have ∂P ∂T (f (0) ) -∂P ∂T (f ) ∈ (x e+1
), and because ∂P ∂T (f ) is of order e then ∂P ∂T (f (0) ) has order e. Finally R(f (1) ) = 0 so P (f (0) )

x 2e ∈ k[x]. This proves that R(T ) ∈ k[x, T ]. We have that R(T ) = c d T d + • • • + c 0 with deg(c i ) = deg(b i ) + ie -2e ≤ h + e(d -2) and ∂R ∂T (0, 0) = c 1 (0) = 0. So f (1)
is the only power series solution of R = 0 vanishing at 0 by the implicit function theorem and

f (1) ∈ I(d, h + e(d -2)).

This construction gives an injective map

A(d, h, e) -→ A e k × I(d, h + e(d -2)) as claimed.
This proves that

k x = e,d,h A e k × I(d, h)
where A e k is identified with polynomials in x of degree ≤ e vanishing at 0 and the inclusion

A e k × I(d, h) ⊂ k x is given by (f, g) -→ f + x e g.
Lemma 2.4. We have

A e k + I(d, h) ⊂ A(d, h + ed) for all integers d, h, e. Proof. Let e ∈ N and let us consider a polynomial R(Z) ∈ k[x, Z], of degree d ≤ d, such that ∂R ∂Z (0, 0) = 0. Assume that deg x R ≤ h.
By the implicit function theorem, R = 0 has a unique algebraic power series solution denoted by f (1) . Now let f (0) ∈ k[x] be a polynomial of degree ≤ e vanishing at 0. We set

P (T ) = x ed R T -f (0)
x e .

Then P (f (0) + x e f (1) ) = 0. Moreover if

R(T ) = c d T d + • • • + c 0 then P (T ) = c d (T -f (0) ) d + c d -1 x e (T -f (0) ) d -1 + • • • + c 0 x d e = a d T d + • • • + a 0 where deg(a i ) ≤ max j≥i deg(c j ) + e(d -j) + deg(f (0) )(j -i) ≤ h + e(d -i). So f := f (0) + x e f (1) ∈ A(d , h + ed ) ⊂ A(d, h + ed).
Lemma 2.5. Let k be a field. Let f ∈ I(d, h) and let P = i≤h,j≤d a i,j x i T j be a polynomial satisfying the implicit function theorem and vanishing at f . Let us consider

P d,h := i≤h,j≤d A i,j x i T j
where the A i,j are new indeterminates and A 0,0 is assumed to be 0. Then P d,h has a unique power series solution

f d,h ∈ (x)k A i,j A 0,1 , x
where (i, j) runs over {0, . . . , d} × {0, . . . , h}\{(0, 0), (0, 1)}. Moreover, the coeffi-

cients of the x k in the expansion of f d,h (x) are in k Ai,j
A0,1 , and we have

f d,h a i,j a 0,1 , x = f (x).
Proof. The existence and the unicity of f d,h comes from the implicit function theorem.

First we prove that

f d,h ∈ k Ai,j A0,1 [[x]
]. Indeed we have that

P (d, h) A i,j A 0,1 , x, 0 ∈ (x) and ∂P ∂T A i,j A 0,1 , x, 0 / ∈ (x).
So by Hensel's Lemma the root f d,h belongs to the completion of k Ai,j A0,1 , x with respect to the ideal (x), i.e. f d,h ∈ k Ai,j A0,1 [[x]]. This implies that f d,h ai,j a0,1 , x is well defined. Finally we conclude that f d,h ai,j a0,1 , x = f (x) since f d,h is the unique solution of P d,h = 0 vanishing at x = 0 and f is the unique solution of P = 0 vanishing at x = 0.

Since there are (d + 1)(h + 1)

-2 = dh + d + h -1 indeterminates Ai,j A0,1 the proposition defines a surjective map f d,h : A dh+d+h-1 k -→ I(d, h).
This map is not injective since different polynomials can have the same power series solution.

In fact let us identify the set of polynomials of k[x, T ] (up to multiplication with a nonzero constant of k) of degree in x less than h and of degree in T less than d with P (d+1)(h+1)-1 k with homogeneous coordinates A i,j . Then A dh+d+h-1 k is identified with the set of polynomials P (x, T ) such that P (0, 0) = 0 and ∂P ∂T (0, 0) = 0. Thus here A dh+d+h-1 k is the intersection of the affine open chart A 0,1 = 0 with the hypersurface A 0,0 = 0. We can state our first main result concerning the structure of I(d, h):

Theorem 2.6. Assume that k is algebraically closed. We have the following properties:

1) For every d ≥ 1 and every h ≥ 1, there is an injective map

ψ d,h : I(d, h) -→ A dh+d+h-1 k whose image is a constructible subset C d,h that contains a non empty open subset of A dh+d+h-1 k .
Here we identify A dh+d+h-1 k with the set of polynomials P (x, t) such that

P (0, 0) = 0, ∂P ∂t (0, 0) = 1, deg x (P ) ≤ h, deg t (P ) ≤ d,
and the map ψ d,h is defined by identifying I(d, h) with the subset of irreducible polynomials in A dh+d+h-1 k . Moreover, we have

f d,h • ψ d,h = id I(d,h) . 2) For every d ≥ d, h ≥ h, we denote by π (1) d ,d,h ,h : I(d , h ) -→ I(d, h) and by π (2) d ,d,h ,h : A d h +d +h -1 k -→ A dh+d+h-1 k the canonical projection maps.
Then we have that

ψ d,h • π (1) d ,d,h ,h = π (2) d ,d,h ,h • ψ d ,h .
Remark 2.7. We can see in the proof that the map ψ d,h is the map sending a series of I(d, h) onto the vector of coefficients of its minimal polynomial, which is normalized in the sense that the coefficient of x 0 T 1 is 1.

Proof. The map f d,h : A dh+d+h-1 k -→ I(d, h) is not injective since different polynomials may have the same root. To get an injective map, we need to restrict the map to the set of irreducible polynomials. Indeed, by Remark 1.5, the minimal polynomial of f ∈ I(d, h) satisfies the implicit function theorem. Therefore we have to remove from A dh+d+h-1 k the points corresponding to the reducible polynomials. We can do it as follows: For every integers

d 1 , d 2 , h 1 , h 2 with d 1 + d 2 ≤ d and h 1 + h 2 ≤ h set Q d1,h1 = i≤h1,j≤d1 q i,j x i T j , R d2,h2 = i≤h2,j≤d2 r i,j x i T j ,
for some variables q i,j and p i,j . Then the product Q d1,h1 R d2,h2 is a polynomial P = i≤h,j≤d a i,j x i T j where the a i,j are polynomials in the q i,j and r i,j . The product defines a rational map Φ d1,d2,h1,h2 :

A (d1+1)(h1+1) k × A (d2+1)(h2+1) k -→ A (d+1)(h+1) k
whose image can be identified with the polynomials P , with deg T (P ) ≤ h 1 + h 2 and deg x (P ) ≤ d 1 + d 2 , that are the product of 2 polynomials of degrees less than (h 1 , d 1 ) and (h 2 , d 2 ). In fact it is straightforward to check that this map is defined by bi-homogeneous polynomials so it induces a rational map

P k Φ d1,d2,h1,h2 : P (d1+1)(h1+1)-1 k × P (d2+1)(h2+1)-1 k -→ P (d+1)(h+1)-1 k
Here we want to consider only polynomials P whose constant term is zero. If such a polynomial is the product of two polynomials Q and R then the constant term of Q or of R has to be zero. We set

C d1,d2,h1,h2 = Im(P k Φ d1,d2,h1,h2 |P (d 1 +1)(h 1 +1)-2 k ×P (d 2 +1)(h 2 +1)-1 k )∪ ∪ Im(P k Φ d1,d2,h1,h2 |P (d 1 +1)(h 1 +1)-1 k ×P (d 2 +1)(h 2 +1)-2 k ) where P (d1+1)(h1+1)-2 k (resp. P (d2+1)(h2+1)-2 k
) is identified with the projective subspace of P (d1+1)(h1+1)-1 k of polynomials Q (resp. of P (d2+1)(h2+1)-1 k of polynomials R) whose constant term is zero. Therefore the intersection of C d1,d2,h1,h2 with A dh+d+h-1 k corresponds exactly to the set of polynomials P with P (0, 0) = 0 and ∂P ∂T (0, 0) = 0 that are the product of two polynomials whose degrees are coordinatewise less than or equal to (d 1 , h 1 ) and (d 2 , h 2 ). Let us remark that the dimension of

P (d1+1)(h1+1)-2 k × P (d2+1)(h2+1) k is (d 1 + 1)(h 1 + 1) + (d 2 + 1)(h 2 + 1) -3. But we have that (d 1 + d 2 )(h 1 + h 2 ) + d 1 + d 2 + h 1 + h 2 -1 -((d 1 + 1)(h 1 + 1) + (d 2 + 1)(h 2 + 1) -3) = d 1 h 2 + d 2 h 1 ≥ 0 (2.1) So if d 1 + d 2 < d or h 1 + h 2 < h then dim(C d1,d2,h1,h2 ) < (d + 1)(h + 1). If d 1 + d 2 = d and h 1 + h 2 = h we have equality in (2.1) if and only if (d 1 , h 1 ) or (d 2 , h 2 ) = 0, (d 1 , d 2 ) = 0, or (h 1 , h 2 ) = 0.
In the first case Q or R is a nonzero constant of k. In the second case we have that P = QR is of degree 0 in T and so it does not correspond to a point in A dh+d+h-1 k .

In the last case P = QR is of degree 0 in x so its roots are in the algebraic closure of k, and the only algebraic power series vanishing at 0 which is in k is 0, and this case cannot occur if h ≥ 1. Hence, in all the cases we need to consider, we have

(2.2) dim(C d1,d2,h1,h2 ) < dh + d + h -1.
So, when h ≥ 1, we can identify I(d, h) with

C d,h := A dh+d+h-1 k \ d 1 ,d 2 ,h 1 ,h 2 d 1 +d 2 ≤d,h 1 +h 2 ≤h d 1 h 2 +d 2 h 1 >0 C d1,d2,h1,h2
which is a constructible set by Chevalley's Theorem. Finally this former set contains an open subset of A dh+d+h-1 k by (2.2). This proves 1).

Let d ≥ d and h ≥ h. From the construction of f d ,h and ψ d ,h , it is straightforward to see that

f d ,h |A dh+d+h-1 k = f d,h and ψ d ,h |A dh+d+h-1 k = ψ d,h .
This proves 2).

Our second result describes the structure of A(d, h) 0 as a constructible set.

Theorem 2.8. Assume that k is algebraically closed. Let d, h, and e ∈ N. Then we have the following properties: i) The image C d,h,e of the injective map

Id A e k ×ψ d,h+e(d-2) • ϕ d,h,e : A(d, h, e) -→ A e k × C d,h+e(d-2) is constructible in A e k × C d,h+e(d-2) . ii) The set A(d, h) 0 can be identified with the constructible set e≤2dh C d,h,e ⊂ A N (d,h) k for some N (d, h) ∈ N.
iii) The dimension of this constructible set is dh

+ d + h -1.
Remark 2.9. In particular, the map

Id A e k ×ψ d,h+e(d-2) • ϕ d,h,e -1 : C d,h,e -→ A(d, h, e)
is a regular map in the sense that all the coefficients of the series f ∈ A(d, h, e) are polynomial functions into the coordinates on A

N (d,h) k
(by Lemma 2.5) and are well defined on C d,h,e .

Remark 2.10. In the proof we show that we can choose

N (d, h) = 2dh(d -2)(d + 1) + 3dh + d + h -1.
Proof of Theorem 2.8. We do the following: let f (0) ∈ k e , f (1) ∈ I(d, h + e(d -2)), and f := f (0) + x e f (1) . We want to find necessary and sufficient conditions for f to be in A(d, h, e), that is we want to describe the image of the map ϕ d,h,e defined in Lemma 2.3.

Let R(Z) ∈ k[x, Z] be a polynomial such that deg Z (R) ≤ d, deg x (R) ≤ h + e(d -2), R(f (1) ) = 0, and ∂R ∂Z (0, 0) = 0.
We assume that R(Z) is irreducible. For such a f ∈ A(d, h, e), we set

P (T ) = x ed R T -f (0)
x e .

Then P (f (0) + x e f (1) ) = 0, i.e P vanishes at f . Moreover deg

x (P ) ≤ h + ed. The map (f (0) , R(T )) -→ P (T ) := x ed R T -f (0) x e is a polynomial map from A e k × P (d+1)(h+e(d-2)+1)-1 k to P (h+ed+1)(d+1)-1 k .
Let S(T ) be an irreducible polynomial vanishing at f . Since the following field extensions degrees are equal:

[k((x))(f ) : k((x))] = [k((x))(f (1) ) : k((x))],
we have deg

T (S) = deg T (R). Since deg T (P ) = deg T (R), there is a polynomial a(x) ∈ k[x] such that (2.3) P (x, T ) = a(x)S(x, T ), (2.4) ∂P ∂T (x, 0) = a(x) ∂S ∂T (x, 0).
Then, f ∈ A(d, h, e) if and only if S and R are irreducible, deg x (S) ≤ h and ord ∂S ∂T (x, f (0) ) = e.

We denote by P d1 k (resp. P (d2+1)(d+1)-1 k

) the projective space of nonzero polynomials of k[x] (resp. k[x, T ]) of degree ≤ d 1 (resp. ≤ d 2 in x and ≤ d in T ) modulo multiplication by elements of k * . For every d 1 , d 2 , d, e, h, we denote by V d1,d2,d,e,h the algebraic set of elements (a(x), S(x, T ),

f (0) , R(T )) ∈ P d1 k × P (d2+1)(d+1)-1 k × A e k × P (d+1)(h+e(d-2)+1)-1 k such that (2.5) a(x)S(x, T ) -x ed R T -f (0) x e = 0. Recall that f ∈ A(d, h, e) is identified with (f (0) , R(x, T )) ∈ A e k ×P (d+1)(h+e(d-2)+1)-1 k . Therefore f ∈ A(d, h, e) if and only if there is (a(x), S(x, T )) ∈ P d1 k ×P (d2+1)(d+1)-1 k , for some d 1 , d 2 with d 1 + d 2 ≤ h + ed and d 2 ≤ h, such that        R(x, T ) and S(x, T ) are irreducible, R(0, 0) = 0, ∂R ∂T (0, 0) = 0, (a(x), S(x, T ), f (0) (x), R(x, T )) ∈ V d1,d2,d,e,h
and ord ∂S ∂T (x, f (0) ) = e. The set of irreducible polynomials S(x, T ), with deg

x (S) ≤ d 2 , deg T (S) ≤ d, is a constructible subset C 1 d2,d ⊂ P (d2+1)(d+1)-1 k
as shown in the proof of Lemma 2.6. Moreover, by Lemma 2.6, the set of irreducible polynomials R(x, T ) such that R(0, 0) = 0, ∂R ∂T (0, 0) = 0, deg × A e k . Now, we consider the following projections:

x (R) ≤ h, deg T (R) ≤ d, is a constructible subset C 2 d,
π 3,4 : P d1 k ×P (d2+1)(d+1)-1 k ×A e k ×P (d+1)(h+e(d-2)+1)-1 k -→ A e k ×P (d+1)(h+e(d-2)+1)-1 k π 2,3 : P d1 k × P (d2+1)(d+1)-1 k × A e k × P (d+1)(h+e(d-2)+1)-1 k -→ P (d2+1)(d+1)-1 k × A e k
and we set

E d1,d2,d,e,h := P d1 k × C d2,d,e × P (d+1)(h+e(d-2)+1)-1 k ∩V d1,d2,d,e,h ∩ P d1 k × C 1 d2,d × A e k × C 2 d,h .
Then we have that A(d, h, e) is equal to

d 1 +d 2 ≤h+ed d 2 ≤h π 3,4 (E d1,d2,d,e,h ) that is a constructible subset C d,h,e of A e k × A (d+1)(h+e(d-2)+1)-1 k .
We claim that π 3,4 | E d 1 ,d 2 ,d,e,h is injective. Indeed, for (f (0) , R(x, T )) ∈ A(d, h, e), by (2.5) there is a unique couple (a(x), S(x, T )), where S(x, T ) is irreducible, such that (a(x), S(x, T ), f (0) , R(x, T )) ∈ E d1,d2,d,e,h .

On the other hand, the fiber of π 2,3 over (S(x, T ),

f (0) ) ∈ C d2,d,e ∩ C 1 d2,d × A e k is finite. Indeed, if there is (a(x), R(x, T )) ∈ P d1 k × P (d+1)(h+e(d-2)+1)-1 k such that a(x)S(x, T ) -x ed R T -f (0)
x e = 0, then, f (0) +x e f (1) is a root of S(x, T ) where f (1) is the unique solution of R(x, T ) = 0 vanishing at x = 0. Therefore π -1 2,3 ((S, f (0) )) is finite because S(x, T ) has a finite number of roots. Therefore the dimension of E d1,d2,d,e,h is equal to the dimension of its image under π 2,3 . First let us assume that (S(x, T ), f (0) ) ∈ π 2,3 (E d1,d2,d,e,h ). Indeed, by (2.5), we have that S(x, f (0) + x e f (1) ) = 0 where f (1) is the unique solution of some polynomial equation R(x, T ) = 0 with R(0, 0) = 0 and ∂R ∂T (0, 0) = 0. Therefore S(0, 0) = 0. We denote by C 3 d2,d the set of polynomials S such that S(0, 0) = 0. Then π 2,3 (E d1,d2,d,e,h ) is included in C d2,d,e ∩ (C 3 d2,d × A e k ). We are going to bound the dimension of C d2,d,e ∩ (C 3 d2,d × A e k ) as follows: We denote by F k the coefficient of x k in the expansion of f (0) , for 1 ≤ k ≤ e. We denote by S i,j the coefficient of x i T j in S(x, T ), for 0 ≤ i ≤ d 2 and 0 ≤ j ≤ d. The coefficient of x l in S(x, f (0) ) is a polynomial G l with integer coefficients depending on the indeterminates F k for k ≤ l, and S i,j for i ≤ l -1. Therefore C d2,d,e ∩ (C 3 d2,d × A e k ) is defined by the equations :    S 0,0 = 0 G l (F k , S i,j ) k≤l,i≤l = 0, l < e G e (F k , S i,j ) k≤e,i≤e-1 = 0 and the ideal defining the Zariski closure of C d2,d,e ∩ (C 3 d2,d × A e k ) is the radical ideal of the ideal generated by S 0,0 and the G l for l < e. For any p < e, we set I p := S 0,0 , G 0 , . . . , G p and I -1 = S 0,0 . These are ideals depending only on the following indeterminates (if p ≥ 0): F k , for k ≤ p, and S i,j for i ≤ p.

Moreover G p has the form

G p = S p,0 + G p (F k , S i,j ) k≤p,i≤p-1 .
Therefore we have, for all p ≤ e:

ht(I p-1 ) ≤ ht(I p ∩ Q[F k , S i,j ] k≤p-1,i≤p-1 ) < ht(I p ).
In particular we have that ht(I e ) ≥ e + 2. Hence

dim(C d2,d,e ∩ (C 3 d2,d × A e k )) ≤ [e + (d + 1)(d 2 + 1) -1] -e -2 ≤ (d + 1)(h + 1) -3. Therefore dim(A(d, h, e
)) < (d + 1)(h + 1) -2. Moreover, when e = 0, we know that A(d, h, 0) = I(d, h) and by Theorem 2.6, this set is a constructible set of dimension (d + 1)(h + 1) -2. Thus,

A(d, h) 0 = e≤2dh A(d, h, e)
is a constructible set of dimension equal to (d + 1)(h + 1) -2.

Points of bounded complexity in varieties over k((x))

Let E be a subset of k((x)) n . For all non negative integers d and h, we set Remark 3.2. This result is an analogue of Lemma [CCL15, 5.1.1] (that gives the same kind of bound on the dimension of the polynomial solutions of degree ≤ d).

The proof is based on the use of linear projections. But, on the contrary of the polynomials of degree ≤ d, the difficulty comes from the fact that the sets A(d, h) are not stable by linear change of coordinates with coefficients in k (since these sets are not k-vector spaces).

Proof of Theorem 3.1. We will prove the statement by induction on m. If m = 0, then X is finite and n(d, h) = 0. Now assume that the statement is true for every integer less than m and let X be of dimension m. For every E ⊂ {1, . . . , n} with Card(E) = m, we denote by π : A n -→ A m the projection defined by π E (x 1 , . . . , x n ) := (x i ) i∈E . For such a set E, we define B E to be the subset of points of π E (X) where π E|X is not finite.

Then π E (X)\B E contains an open set of the Zariski closure of π E (X) by the upper semi-continuity of the dimension of the fibers of π E . Therefore the dimension of the Zariski closure of the set of A(d, h)-points of X \ π -1 E (B E ) has dimension less or equal to dim(A(d, h) m ), which is equal to m(dh + d + h) by Remark 1.4 and Theorem 2.8. Thus we can replace X by X ∩ π -1 E (B E ). We repeat this operation for every E as above and we assume that X ⊂ E⊂{1,...,n}, Card(E)=m π -1 E (B E ).

By considering the different irreducible components of the B E separately, we may assume that all the B E are irreducible. Therefore the ideal of π -1 E (B E ) is a prime ideal J E generated by polynomials depending only on the x i for i ∈ E.

We fix such a set E that we denote by E 1 . We also fix an index i 1 ∈ E 1 such that J E1 is generated by polynomials of K[x i , i ∈ E 1 ] but not by polynomials of

K[x i , i ∈ E 1 \ {i 1 }].
Then we pick E 2 ⊂ {1, . . . , n} \ {i 1 } such that Card(E 2 ) = m, and we fix an index i 2 ∈ E 2 such that J E2 is generated by polynomials of K[x i , i ∈ E 2 ] but not by polynomials of K[x i , i ∈ E 2 \ {i 2 }]. We repeat this process and construct a sequence Ei (B Ei )) ≤ m -1. Therefore, the result follows by induction.

  ∂S ∂T (x, f (0) ) = e defines a constructible set C d2,d,e ⊂ P (d2+1)(d+1)-1 k

E

  d,h := E ∩ A(d, h) n and we denote by n d,h (E) the dimension of the Zariski closure of E d,h in the affine space A nN (d,h) k where N (d, h) is given in Remark 2.10. Theorem 3.1. Let k be an algebraically closed field and let X be an algebraic subvariety of A n k((x)) of dimension m. Then n d,h (X) ≤ m(dh + d + h).

E 1

 1 , . . . , E n-m+1 ⊂ {1, . . . , n} of subsets of cardinal m, and a sequencei 1 ∈ E 1 \ (∪ n-m+1 i=2 E i ), i 2 ∈ E 2 \ (∪ n-m+1 i=3 E i ), . . . , i n-m+1 ∈ E n-m+1 such that J E k is generated by polynomials of K[x i , i ∈ E k ] but not by polynomials of K[x i , i ∈ E k \ {i k }].Hence, by Krull's principal ideal theorem we have ht J E1 + J E2 + . . . + J En-m+1 ≥ n -m + 1.

Acknowledgment. We are very grateful to Michel Hickel and Mickaël Matusinski for their comments and suggestions on a previous version of our paper. In particular they pointed out that there was an essential gap in the proof of Theorem 2.8. We are also grateful to an anonymous referee for their helpful comments.