Keywords: xylooligosaccharide, SHIME®, rodent, piglet, clinical trial

Bifidogenic effect of a wheat arabinoxylan (MC-AX) is observed across two animal models, a simulated human intestine model (SHIME®) and a clinical study.

Introduction

Dietary fibres are carbohydrate polymers (average degree of polymerisation (avDP) ≥ 3) of plant origin, which may or may not be associated in the plant with lignin or other noncarbohydrate components (polyphenols, waxes, saponins, cutins, phytates, phytosterols, etc.) or, as is the case for the product described in this paper, a processed (by physical, enzymatic or chemical means) carbohydrate polymer [START_REF] Elleuch | Dietary Fibre and Fibre-Rich by-Products of Food Processing: Characterisation, Technological Functionality and Commercial Applications: A Review[END_REF]; Phillips and Cui 2011; Lee and Prosky 1994). In addition dietary fibre is neither digested nor absorbed in the small intestine. To be considered prebiotics, those non-digestible food ingredients need to beneficially affect the host by selectively stimulating the growth and/or the activity of one or a limited number of bacterial species already resident in the colon, and thus attempt to improve host health [START_REF] Gibson | Dietary Modulation of the Human Colonic Microbiota: Introducing the Concept of Prebiotics[END_REF][START_REF] Gibson | Dietary Modulation of the Human Colonic Microbiota: Updating the Concept of Prebiotics[END_REF]). It has at least one physiological effect [START_REF] Lunn | Carbohydrates and Dietary Fibre[END_REF] on stools production [START_REF] Harvey | EFFECTS OF INCREASED DIETARY FIBRE ON INTESTINAL TRANSIT[END_REF][START_REF] Kurasawa | Plant Residue and Bacteria as Bases for Increased Stool Weight Accompanying Consumption of Higher Dietary Fiber Diets[END_REF], colonic fermentation, pre-prandial cholesterol levels [START_REF] Marlett | Sites and Mechanisms for the Hypocholesterolemic Actions of Soluble Dietary Fiber Sources[END_REF][START_REF] Pereira | Cholesterol Assimilation by Lactic Acid Bacteria and Bifidobacteria Isolated from the Human Gut[END_REF] or post-prandial blood sugar and /or insulin levels [START_REF] Aleixandre | Dietary Fiber in the Prevention and Treatment of Metabolic Syndrome: A Review[END_REF][START_REF] Hallfrisch | Mechanisms of the Effects of Grains on Insulin and Glucose Responses[END_REF][START_REF] Lu | Arabinoxylan Fibre Improves Metabolic Control in People with Type II Diabetes[END_REF]). Arabinoxylan (AX) is the main non-starch polysaccharide of cereal grains [START_REF] Izydorczyk | Cereal Arabinoxylans: Advances in Structure and Physicochemical Properties[END_REF] and can be found in a variety of tissues of cereals such as wheat, rye, barley, oat, rice, sorghum, as well as in other plants (bamboo shoots, rye grass and pangola grass). These complex carbohydrates occur in cell walls of the starchy endosperm cells and the aleurone layer in most cereals (60-70% (w/w) of the total carbohydrate). They can be found in the endosperm cell walls of barley (20% (w/w)) and rice (40% (w/w)). Nonendospermic tissues of wheat, particularly the pericarp and testa, also contain a very high concentration of AX (64% (w/w)) [START_REF] Grootaert | Microbial Metabolism and Prebiotic Potency of Arabinoxylan Oligosaccharides in the Human Intestine[END_REF]; [START_REF] Williams | In Vitro Fermentation Kinetics and EndProducts of Cereal Arabinoxylans and (1,3;1,4)-β-Glucans by Porcine Faeces[END_REF]. Arabinoxylan consists of long backbone chains of anhydro-D-xylopyranosyl residues linked together by β-(1→4) glycosidic bonds. The major substituents are single α-Larabinofuranosyl residues attached at the 2-or 3-position to isolated xylosyl residues of the main chain, sometimes to two adjacent residues, less frequently to three, and not to four or more consecutive residues [START_REF] Andrewartha | Solution Properties of Wheat-Flour Arabinoxylans and Enzymically Modified Arabinoxylans[END_REF]. Xylose can also be directly coupled with acetic acid or methylglucuronides [START_REF] Van Craeyveld | Structurally Different Wheat-Derived Arabinoxylooligosaccharides Have Different Prebiotic and Fermentation Properties in Rats[END_REF][START_REF] Izydorczyk | Cereal Arabinoxylans: Advances in Structure and Physicochemical Properties[END_REF]. Some of the arabinose units are substituted with the phenolic units such as ferulic acid or p-coumaric acid [START_REF] Cleemput | Heterogeneity in the Structure of Water-Soluble Arabinoxylans in European Wheat Flours of Variable Bread-Making Quality[END_REF], which is linked with an ester bond between its carboxyl group and carbon 5 of the arabinose. Ferulic acid can bind different xylose polymeric chains together, creating a jellified, insoluble cross-linked structure [START_REF] Van Craeyveld | Structurally Different Wheat-Derived Arabinoxylooligosaccharides Have Different Prebiotic and Fermentation Properties in Rats[END_REF]). The degree of arabinose substitution (DAS) refers to the arabinose moieties on the xylose backbone and is further described as arabinose/xylose ratio. AX from wheat, rye or barley tend to have a lower DAS than polymers from rice or sorghum [START_REF] Maes | Structural Characterisation of WaterExtractable and Water-Unextractable Arabinoxylans in Wheat Bran[END_REF]. Similarly there are differences in substitution ratio depending on the tissue. AX from wheat endosperm has a DAS of 0.5 to 0.7 while aleurone and seed coat are less substituted at 0.1 to 0.4 and the outer pericarp is highly substituted at 1.1 to 1.3 [START_REF] Broekaert | Prebiotic and Other Health-Related Effects of CerealDerived Arabinoxylans, ArabinoxylanOligosaccharides, and Xylooligosaccharides[END_REF][START_REF] Izydorczyk | Cereal Arabinoxylans: Advances in Structure and Physicochemical Properties[END_REF]. The substitution and distribution of side chains are thus important factors in the physicochemical properties of AX. As for other polymers, the degree of polymerization (DP), defined as the molecular weight of the polymer divided by the molecular weight of the repeating units, is an important factor in the physicochemical properties of AX [START_REF] Andrewartha | Solution Properties of Wheat-Flour Arabinoxylans and Enzymically Modified Arabinoxylans[END_REF]. It is thought that cereals can have polymers constituted of 1500 to 15000 residues [START_REF] Ebringerova | Xylan and Xylan Derivatives -Biopolymers with Valuable Properties, 1. Naturally Occuring Xylans Structures, Isolation Procedures and Properties[END_REF] although it is quite difficult to determine for sure as degradation can occur early during the extraction process [START_REF] Broekaert | Prebiotic and Other Health-Related Effects of CerealDerived Arabinoxylans, ArabinoxylanOligosaccharides, and Xylooligosaccharides[END_REF]). Native AX from wheat endosperm could have a molecular mass up to 800.000g per mol, meaning that the degree of polymerization can reach 6000. Arabinoxylan oligosaccharides (AXOS) can be obtained by enzymatic hydrolysis. Depending on the nature and degree of substitutions, a number of microbial enzymes can be used. They can be found on the cereals themselves [START_REF] Broekaert | Prebiotic and Other Health-Related Effects of CerealDerived Arabinoxylans, ArabinoxylanOligosaccharides, and Xylooligosaccharides[END_REF], added during food processing such as bread or beer making ( Courtin et al. 2009), or ultimately they can be produced by colonic bacteria such as Bacteroides or Roseburia [START_REF] Chassard | Characterization of the Xylan-Degrading Microbial Community from Human Faeces[END_REF]). AX can be degraded by endo-β-1,4-xylanases, α-Larabinofuranosidases, β-xylosidases, αglucuronidases or even ferulic acid esterases [START_REF] Grootaert | Microbial Metabolism and Prebiotic Potency of Arabinoxylan Oligosaccharides in the Human Intestine[END_REF]. Xylanases are the most obvious for AX degradation and cleave the xylose backbone randomly to shorter fragments [START_REF] Courtin | Arabinoxylans and Endoxylanases in Wheat Flour BreadMaking[END_REF]. As well as AXOS or short-chain arabinoxylan (SCAX), which are end-products characterized by DP 2-10, AX hydrolysate can be further divided into longchain arabinoxylans (LCAX) with DP > 50, and medium-chain arabinoxylan (MC-AX) with intermediate DP values, depending on the degree of hydrolysis. Prebiotic activity of dietary fibres has long been investigated using different models. In vitro models, whether using pure bacterial cultures or faecal samples, whether using static or dynamic fermenters, are the first port of entry for prebiotic investigations. Rodents (mice and rats) are then often used as a primary in vivo tool before moving to clinical studies. Piglet, for its size and common physiological features is more and more considered as a pertinent model for investigations of the digestive tract with better extrapolation to human [START_REF] Rowan | Comparison of the Ileal and Faecal Digestibility of Dietary Amino Acids in Adult Humans and Evaluation of the Pig as a Model Animal for Digestion Studies in Man[END_REF]. A quick survey of search engines Pubmed/Medline or ScienceDirect using keywords "oligosaccharides", "dietary fibre", "prebiotic" and "in vitro", rodent", "pig" or "clinical trial" led to hundreds of hits. There is therefore little chance of original data, whatever the dietary fibre chosen. It is important however to note that each publication is using a given model but very few compare between models [START_REF] Koecher | Estimation and Interpretation of Fermentation in the Gut: Coupling Results from a 24 H Batch in Vitro System with Fecal Measurements from a Human Intervention Feeding Study Using Fructo-Oligosaccharides, Inulin, Gum Acacia, and Pea Fiber[END_REF]. There is thus the question of whether all preclinical models are good predictors of human clinical outcomes. Medium chain arabinoxylans have been chosen to investigate this hypothesis. The aim of this work is therefore to compare data from our clinical study [START_REF] Lecerf | XyloOligosaccharide (XOS) in Combination with Inulin Modulates Both the Intestinal Environment and Immune Status in Healthy Subjects, While XOS Alone Only Shows Prebiotic Properties[END_REF] to previously unpublished preliminary tests on in vitro and animal models using the same active ingredient, MC-AX. 2. Materials and Methods

Ethical statements

The clinical trial was conducted according to the guidelines laid down in the Declaration of Helsinki and French regulation. The study protocol (2007-A00273-50) was approved by the regional Ethics Committee for North West France based at the Lille University Hospital and registered on clinicaltrials.gov (NCT02287844). Written informed consent was obtained from all subjects prior to inclusion. Animal studies were carried out in strict accordance with the recommendations in the European Union guidelines and regulations. The protocols for the rodent and piglet studies were approved by the committee on the Ethics of animal experiments of the University of Liege and of Institut polytechnique LaSalle Beauvais, respectively. All efforts were made to minimize animal suffering at any time.

Experimental product MC-AX The raw material is a by-product from wheat endosperm (DF3 SAS France). It was obtained from purification of the side stream of a wheat starch producing factory using a three phase decanter for the separation of the two main streams -starch and gluten. This side stream was purified in order to eliminate most of the starch, proteins, minerals, and fats, arabinoxylan was partially hydrolysed by means of an endoxylanase, and the reaction mixture was concentrated and spray dried. The obtained MC-AX sample, a white powder, was characterized by a dry matter of 95%, an AX content of 80% on DM, an avDP of 25 (range 7-300) and DAS 0.75.

Experimental models

The products described above were tested on four different experimental models to investigate their bifidogenic activity. Only markers pertinent to the bifidogenic and prebiotic objectives are presented here. Similarly, rat and human studies included additional test groups that are not reported in this article. As statistical analyses were performed by individual comparison between test and placebo groups, this does not affect the 2.3.1. SHIME® study Description of the SHIME® model has been published elsewhere [START_REF] Joly | Impact of Chronic Exposure to Low Doses of Chlorpyrifos on the Intestinal Microbiota in the Simulator of the Human Intestinal Microbial Ecosystem (SHIME®) and in the Rat[END_REF]; Molly, van de Woestyne, and Verstraete 1993; [START_REF] Possemiers | PCR-DGGE-Based Quantification of Stability of the Microbial Community in a Simulator of the Human Intestinal Microbial Ecosystem[END_REF][START_REF] Reygner | Changes in Composition and Function of Human Intestinal Microbiota Exposed to Chlorpyrifos in Oil as Assessed by the SHIME® Model[END_REF]. Briefly, stool from 5 healthy volunteers were pooled and slurry inoculated into the fermenters 4 to 6 corresponding to the colonic compartments of the system, whilst a specific liquid diet is initially added to fermenter 1 corresponding to the stomach to stimulate transient microflora in the whole system. Once the microbiota is stable, the fibre is added to the twice-daily basal nutriment meal at a final concentration of 10g/day. Samples from fermenters 2 to 6 were collected at baseline and after 7 and 15 days intervention under sterile conditions to measure pH as well as for bacterial quantification. A final set of samples was collected 15 days after the last fibre meal to assess the residual effect.

Rat study Weaned male Wistar rats (Charles River, Brussels) were housed individually in metabolic cages (22°C, 70% humidity, 12h light-cycle). Animals were assigned to placebo or test group (n=8 in each group) through randomized complete block design. Test fibre was fed at a concentration of 7.5% (6g/kg diet) in a semi-purified diet, replaced by 5% starch and 2.5% saccharose in the placebo diet. Initial and final weights of the animals were 108 and 237g, respectively and did not differ among experimental groups. The intake dry matter was limited to 12.6g per day based on the voluntary intake measured during and acclimation period of 5 days prior to the experiment. Twenty-four hour stool samples were collected on day 10 and 20 for bacterial quantification. On day 21, the rats were killed at 5h post prandial by CO2 overdose and caecal content were rapidly collected for pH and short chain fatty acid (SCFA) measurements.

Piglet study Long white F1 weaned male piglets were housed in two bands of 8 animals fed either the placebo or test diet (10g/kg diet). Randomization ensured that weight were homogenous in both groups at the beginning of the intervention. Diets (Trouw Nutrition, France) were produced as powder with energy content of 17% (w/w) proteins, 8% (w/w) lipids and 43% (w/w) carbohydrates. For energy balance, 7.5% MC-AX were replaced by 3% saccharose. Food intake did not vary between groups but weight gain was increased in treated versus placebo group (p=0.024). No pathology not variation in cleanliness of the animals were observed throughout the experimentation. Fresh stool samples were collected in each pen on day 1 and 14 of the intervention and thus represent the production of both animals present in that area.

Human study The complete protocol for this clinical study, including selection and exclusion criteria, has been published elsewhere [START_REF] Lecerf | XyloOligosaccharide (XOS) in Combination with Inulin Modulates Both the Intestinal Environment and Immune Status in Healthy Subjects, While XOS Alone Only Shows Prebiotic Properties[END_REF]). Briefly, healthy volunteers (18-24 years old; 20 per group) were fed either a placebo (maltodextrin) or test fibre (5g/day) in orange juice as a twice-daily meal supplement for 4 weeks. Randomization was stratified for homogeneous distribution of body mass index (BMI) and estimated fibre intake between groups. Stool samples were collected at 14 and 28 days of intervention for pH measurements, bacterial quantification and detection of SCFA. A range of additional markers were measured and are fully described in the study report and related publication.

Measure of biological parameters

Bacterial count by microbiological techniques For SHIME®, 5mL were taken from each reactor 3 to 6 at three times [START_REF] Joly | Impact of Chronic Exposure to Low Doses of Chlorpyrifos on the Intestinal Microbiota in the Simulator of the Human Intestinal Microbial Ecosystem (SHIME®) and in the Rat[END_REF]). Dilutions were plated on various selective and non-selective media for quantitative and qualitative cultures of aerobic and anaerobic microflora. 100 µl of each dilution was cultured on different plates shown table 1. All aerobes plates were incubated at 37°C for 48 hours. All anaerobic plates were incubated in anaerobic chamber (Bactron Anaerobic, Sheldon Manufacturing) and were examined after 4 days of incubation at 37°C. The micro organisms were identified using standard microbiologic techniques. Bacterial viable counts are expressed as log10 ColonyForming-Units CFU/mL. Bacterial detection by qPCR Specific bacterial quantification was conducted using an absolute realtime PCR method. Briefly, bacterial DNA was extracted using the Qiagen DNA stool Kit (Courtaboeuf, France). The 16s rRNA copy number DNA samples were quantified by PCR. Primers and fluorogenic probes are described elsewhere [START_REF] Pouillart | Nutriose, a Prebiotic Low-Digestible Carbohydrate, Stimulates Gut Mucosal Immunity and Prevents TNBS-Induced Colitis in Piglets[END_REF]. Quantitative PCR was performed with 100 ng of DNA, 12.5 µL of 2x Taqman PCR master mix (Applied Biosystems), 300 nmol of each primer and a 200 nmol probe in a final volume of 25 µL. Thermal cycling conditions were as follows: 2 min at 50°C, 10 min at 95°C followed by 45 repeats of 15 s at 95°C, and 1 min at 60°C. A negative control containing distilled water was included.

Quantification of lactate and short chain fatty acids Lactate was measured on in vitro samples according to Talasniemi et al [START_REF] Talasniemi | Analytical Investigation: Assay of D-Lactate in Diabetic Plasma and Urine[END_REF]) with some variants. Samples from the fermenters were diluted 4 fold in triethanolamine (TEA) buffer (0.1M, pH 9.15) and the supernatant incubated with trichloroacetate (TCA) buffer to precipitate protein residues before analysis. The quantification of D-and L-lactate was performed using the enzymatic Lactate assay kit (Sigma-Aldrich, France) with colorimetric detection. Volatile fatty acids were extracted and measured according to Scheppach et al. [START_REF] Scheppach | Fecal ShortChain Fatty Acid (SCFA) Analysis by Capillary Gas-Liquid Chromatography[END_REF] with some variants. Two volumes sample were homogenized in 5 volumes ddH2O then centrifuged at 4500 g for 5 min. The supernatant was acidified to pH 2 with H2SO4 (2 M) and injected into a BP21 gas chromatography column (L: 30 m, id: 530 µm, film: 1 µm) along with an internal standard (4-hydroxy-4-methyl-2pentanone). Hydrogen was supplied as the carrier gas at a flow rate of 1.5 ml/min. The initial oven temperature was 135°C and was kept there for 6 min, and then raised to 180°C by 25°C/min and held there for 1 min, then further increased to 230°C by 25°C/min, and finally held at 230°C for 1 min. Glass wool (Supelco) was inserted in the glass liner of the split injection port. The temperatures of the flame ionization detector (FID) and the injection port were 240 and 280°C, respectively. The flow rates of hydrogen and air as makeup gas were 40 and 400 ml/min, respectively. The injected sample volume for Gas Chromatography (AutoSystem XL, PerkinElmer, France) analysis was 1 µl, and the running time for each analysis was about 10 min. The different SCFA were identified according to the retention time of the various elution peaks. Quantification was obtained by comparison to a standard curve. All data were reported to dry matter.

Data and statistical handling Depending on the model, sample size or sample distribution for the specific markers, parametric or non-parametric tests were used to compare placebo and test groups. Additional statistical tests are described in the result section whenever applicable. All statistical tests were performed using SPSS v17.0 for Windows (SPSS Inc). All data are presented as means and standard deviations whenever possible. 3.

Results

Modulation of bacterial populations

Lactic acid bacteria. One parameter that was included in all four models is quantification of Bifidobacterium. Data from animal and clinical studies were obtained from quantitative real time PCR, while data from the SHIME® model were obtained by culture of live bacteria in selective media. In addition primer sequences were different between models; therefore quantitative data cannot be compared from one species to another. It is however possible to compare qualitative data and trends. All data are plotted in figure 3 and we can observe that AXOS induced an increase in Bifidobacterium in treated groups of at least one log unit, thus highly significant in terms of bacterial modulation. This could be measured as early as 7 days in the treatment (SHIME®, Figure 2). In the rodent model, the difference is clinically pertinent at one log unit against the baseline values for the treatment group at both 10 and 20 days, but variation from placebo was weaker at 0.7 and 0.2 log, respectively (Figure 2). The strongest variation was observed in piglets with nearly 2 log increase in 14 days (Figure 2). Lactobacillus variation was also measured but we observed different patterns for these bacteria. There was no variation between placebo and treatment or over time in the rodent and human studies (Figure 3). An increase of 0.7 and 1.6 log were observed after two weeks in the piglet and SHIME® models, respectively (Figure 3). Different sub-species or distribution of Lactobacillus may be present in ecosystem from the different models, inducing a selective response.

Other bacteria. While Bifidobacterium and Lactobacillus increased in the fermenters, Enterococcus/Streptococcus was stable in the SHIME® model with a possible transient increase in the 4th fermenter (proximal colon) at 7 days. Peptostreptococcus was measured in the clinical study and showed a transient increase at 14 but not 28 days. No data is available from the rat study. Bacteroides and Clostridium did not significantly vary in the SHIME® model. They did not vary significantly from the placebo group in the clinical trial either. They were not measures in the animal models.

Carbohydrate metabolism.

Short chain fatty acids were measured in caecum content of the rodent and in the faeces of human. Figure 4 represents distribution profiles of acetate, propionate and butyrate in the placebo and treatment groups after 21 and 28 days in the rat and human, respectively (Figure 4). Baseline profiles in both groups were not significantly different, nor were they different from the placebo profile at the end of the study (data not shown). In both species, we observed a significant decrease in acetate production, balanced by an increase in both propionate and butyrate. Butyrate production may be favoured in human (+7%, Figure 4A) and rat (+15%, Figure 4) while propionate shows higher increase in the rat (+3%, Figure 4). The difference may also be linked to the sampling site, human measures being done on faeces while the rat measures are done on caecum content. Further degradation or crossfeeding may occur further down the digestive tract. Lactate production was measured in the SHIME® model and showed a time-dependent increase in all fermenters (Figure 4), although concentrations were higher in the first colonic section.

Discussion

The object of this paper was to compare the bifidogenic and general prebiotic activity of a single molecule, MC-AX, over four different models and to determine whether the three preclinical studies are good predictors of human outcome. It is unusual to find for the same product complementary information from different models going from in vitro to clinical studies. This is the first time to our knowledge that four different models of prebiotic activity are being directly compared using the same dietary fibre as a reference product. The four studies were designed to investigate the prebiotic and bifidogenic effect of a dietary fibre with specific physicochemical properties. While they were designed and performed independently, it is interesting to compare and contrast the results so each model, with all its specificities, can add to the overall understanding of the prebiotic effect of MCAX. The first parameter that we can consider and which is identical on the four models is the age of the subjects. The animals selected were juvenile at the beginning of the studies, consistent with the target population from the clinical trial and SHIME® (young adults). Looking at other similarities, the in vitro SHIME® model and clinical study share a species, both investigating the human microbiota. On the other hand, animal studies represented global organisms where the interactions between microbiota and the intestinal tract could be taken into account, hence closer to the clinical setting. The samples collected were similar in the human and piglet studies (stool) while caecal and colonic content were collected from the rat and in vitro studies, respectively. Therefore, and despite species variations, a number of comparisons are possible. We will first compare the individual models with data extracted from the literature before moving to a comparison between models. We will conclude on what we learned from this exercise and whether preclinical models are transferable to human functions in the field of prebiotic activity.

In vitro data

Prebiotic effects A number of publications, some of which cited in the introduction, have reported prebiotic effects of arabinoxylans in vitro. The main findings are summarised alongside our data in table 2. A number of information can be deduced from this table. The first one is an overall consistency between our data and published studies on the bifidogenic potential of arabinoxylans. Some structure-activity relationships can also impact the bifidogenic and prebiotic activities. Shortchain arabinoxylans (SC-AX) have a higher fermentation rate compared to longchain arabinoxylans (LC-AX). This is observed by a higher production rate of butyrate and propionate with SC-AX compared to LC-AX [START_REF] Mäkeläinen | XyloOligosaccharides Enhance the Growth of Bifidobacteria and Bifidobacterium Lactis in a Simulated Colon Model[END_REF]). SC-AX are also associated with stimulation of the growth of both Bifidobacterium and Lactobacillus while LC-AX mostly impact Bifidobacterium in a batch culture set-up [START_REF] Hughes | In Vitro Fermentation by Human Fecal Microflora of Wheat Arabinoxylans[END_REF]) but also semi-continuous model [START_REF] Mäkeläinen | XyloOligosaccharides Enhance the Growth of Bifidobacteria and Bifidobacterium Lactis in a Simulated Colon Model[END_REF]). Our product, a medium-chain arabinoxylan (MC-AX) is in agreement with those data in terms of bacterial impact as Bifidobacterium is strongly stimulated and Lactobacillus is only slightly affected in our in vitro model. The degree of arabinose substitution (DAS) refers to the arabinose moieties on the xylose backbone and is further described as arabinose/xylose ratio. AX from wheat, rye or barley tend to have a lower DAS than polymers from rice or sorghum [START_REF] Maes | Structural Characterisation of WaterExtractable and Water-Unextractable Arabinoxylans in Wheat Bran[END_REF]. Similarly there are differences in substitution ratio depending on the tissue. AX from wheat endosperm has a DAS of 0.5 to 0.7 while aleurone and seed coat are less substituted at 0.1 to 0.4 and the outer pericarp is highly substituted at 1.1 to 1.3 [START_REF] Broekaert | Prebiotic and Other Health-Related Effects of CerealDerived Arabinoxylans, ArabinoxylanOligosaccharides, and Xylooligosaccharides[END_REF][START_REF] Izydorczyk | Cereal Arabinoxylans: Advances in Structure and Physicochemical Properties[END_REF]. It is difficult however to assess the exact role of DAS from those data as AX29/0.30 [START_REF] Sanchez | Arabinoxylan-Oligosaccharides (AXOS) Affect the Protein/carbohydrate Fermentation Balance and Microbial Population Dynamics of the Simulator of Human Intestinal Microbial Ecosystem[END_REF]) and AX25/0.75 (us) are not obtained from the same concentration. However the role of arabinose in the Lactobacillus stimulation can be suggested when comparing short-term exposure with long-chain XOS (DP40) and AXOS (DP60), the first one inhibiting Lactobacillus growth [START_REF] Mäkeläinen | XyloOligosaccharides Enhance the Growth of Bifidobacteria and Bifidobacterium Lactis in a Simulated Colon Model[END_REF] while the second has no effect (Van den Abbeele et al. 2009). The substitution and distribution of side chains are thus important factors in the physicochemical properties of AX. Ferulic acid (F) linkage can also play a major role in the prebiotic effect of arabinoxylans. FAX have a lower fermentation rate, bacterial stimulation than free AX [START_REF] Grootaert | Microbial Metabolism and Prebiotic Potency of Arabinoxylan Oligosaccharides in the Human Intestine[END_REF][START_REF] Hopkins | Degradation of CrossLinked and Non-Cross-Linked Arabinoxylans by the Intestinal Microbiota in Children[END_REF]). Propionate production is also higher in free rather than Flinked AX [START_REF] Hopkins | Degradation of CrossLinked and Non-Cross-Linked Arabinoxylans by the Intestinal Microbiota in Children[END_REF]) and some bacterial specificity seem to apply. In the MCS model, insoluble LC-F-AX were less efficient prebiotic than soluble free LC-AX [START_REF] Vardakou | In Vitro Three-Stage Continuous Fermentation of Wheat Arabinoxylan Fractions and Induction of Hydrolase Activity by the Gut Microflora[END_REF]). But xylanase-treated F-AX produced SC-F-AX that presented prebiotic and bifidogenic activity close to that of other reference oligosaccharides [START_REF] Rycroft | A Comparative in Vitro Evaluation of the Fermentation Properties of Prebiotic Oligosaccharides[END_REF], again demonstrating that the degree of polymerization may play a more important role than substitutions. Both SC-AX and LC-AX, when compared to inulin, presented a prebiotic activity that was displaced toward distal compartments of the colon [START_REF] Grootaert | Microbial Metabolism and Prebiotic Potency of Arabinoxylan Oligosaccharides in the Human Intestine[END_REF] that the prebiotic effect such as stimulation of Bifidobacterium and Lactobacillus remains after 2 weeks wash-out of the SHIME® system. Our data suggest that the effect was reversible for MC-AX using the same model. In vitro batch culture studies are also a great tool to better understand the selectivity of bacterial strains to saccharolytic substrates. While the main outcome is an increase in the Bifidobacterium and Lactobacillus strains, these are not propionate or butyrate producers. And eventhough we did not detect any changes in these two metabolites, they are strongly associated with arabinoxylan fermentation. The growth promoting effect on selected Lactobacillus and Bifidobacterium strains has been reported by in vitro culture experiments [START_REF] Grootaert | Microbial Metabolism and Prebiotic Potency of Arabinoxylan Oligosaccharides in the Human Intestine[END_REF][START_REF] Kontula | The Isolation of Lactic Acid Bacteria from Human Colonic Biopsies after Enrichment on Lactose Derivatives and Rye ArabinoxyloOligosaccharides[END_REF]. Some Bacteroides/Eubacterium strains are known to produce propionate or butyrate from arabinoxylans, either directly or through cross feeding [START_REF] Chassard | Bacteroides Xylanisolvens Sp. Nov., a Xylan-Degrading Bacterium Isolated from Human Faeces[END_REF][START_REF] Chassard | Characterization of the Xylan-Degrading Microbial Community from Human Faeces[END_REF][START_REF] Duncan | Roseburia Intestinalis Sp. Nov., a Novel Saccharolytic, Butyrate-Producing Bacterium from Human Faeces[END_REF]. Lactate conversion to propionate by some of these strains is well known [START_REF] Belenguer | Impact of pH on Lactate Formation and Utilization by Human Fecal Microbial Communities[END_REF]). While we did not observe a stimulation of the species, the PCR probes or culture media may not have been selective enough to target the butyrate producers. Similarly, lactate conversion to propionate or butyrate could not be proven from our data. Overall in vitro data obtained from either one of the 4 most prominent colonic simulation models suggest a bifidogenic activity of AX, more pronounced with shorter-chain and noncross-linked molecules. This may be associated with stimulation of Lactobacillus and some butyrate-producing bacteria. The main fermentation site is the distal colon and is expressed as increased propionate and butyrate production. These metabolites were not detected in significant amount in our in vitro study. They were however increased in the clinical study. [START_REF] Lu | Arabinoxylan Fiber from a By-Product of Wheat Flour Processing Behaves Physiologically like a Soluble, Fermentable Fiber in the Large Bowel of Rats[END_REF] suggest that acetate is produced in the caecum while propionate appears further along and accumulates in the faeces. In our study, it is butyrate that is mostly modulated, in agreement with in vitro observations [START_REF] Hughes | In Vitro Fermentation by Human Fecal Microflora of Wheat Arabinoxylans[END_REF]). Similarly to in vitro research, van Craeyveld investigated structure activity relationships [START_REF] Van Craeyveld | Structurally Different Wheat-Derived Arabinoxylooligosaccharides Have Different Prebiotic and Fermentation Properties in Rats[END_REF]). The authors suggested that structure may impact which saccharolytic metabolites will be produced. Butyrate seems to increase as the degree of polymerisation decreased while propionate was associated with increased arabinose substitution. There also seems to be a role for the solubility of the fibre, which has not been taken into account in other models. Damen et al [START_REF] Damen | Prebiotic Effects and Intestinal Fermentation of Cereal Arabinoxylans and Arabinoxylan Oligosaccharides in Rats Depend Strongly on Their Structural Properties and Joint Presence[END_REF] showed that short-chain soluble arabinoxylans and long chain soluble arabinoxylan increased acetate production in the caecum, associated with lower pH and increased Bifidobacterium populations. On the other hand, long-chain unsoluble arabinoxylan increased butyrate production and the butyrate producer Roseburia without affecting Bifidobacterium. Hence the structure-activity relationships work on a number of parameters and their interactions make it more difficult to predict the behaviour of a given molecule. Piglet model are less common but find similar increase in the production of propionate and acetate both in vitro [START_REF] Wang | In Vitro Fermentation of Various Fiber and Starch Sources by Pig Fecal Inocula[END_REF][START_REF] Williams | In Vitro Fermentation Kinetics and EndProducts of Cereal Arabinoxylans and (1,3;1,4)-β-Glucans by Porcine Faeces[END_REF]) and in vivo using cannulated animals [START_REF] Wang | In Vitro Fermentation of Various Fiber and Starch Sources by Pig Fecal Inocula[END_REF]). Neither publication reported on bacterial variations. Each of the two animal studies also showed particularities. Rodents are more reproducible, due to their selected genetic background, while piglets are closer to the human intestinal properties [START_REF] Rowan | Comparison of the Ileal and Faecal Digestibility of Dietary Amino Acids in Adult Humans and Evaluation of the Pig as a Model Animal for Digestion Studies in Man[END_REF]. The two models differ by the number of animals necessary and volume of collected samples, hence affecting the number of complementary analyses that can be done on the samples. Recently a model of polycannulated pig was developed, allowing for investigation on specific sections of the digestive tract [START_REF] Nitrayova | The Pig as a Valuable Model for Monitoring Digestion and the Ecosystem of the Digestive Tract[END_REF]). This will allow cumulating the benefits of the in vitro and in vivo systems. Overall, animal data from the literature are consistent with our findings and within vitro and clinical data reported herein.

Clinical data.

Human trials are by definition the optimal study design for human nutritional applications. Investigators also have the opportunity to work on large sample volumes and run a wide range of analyses. This was the case here and the full report can be found in a previous publication [START_REF] Lecerf | XyloOligosaccharide (XOS) in Combination with Inulin Modulates Both the Intestinal Environment and Immune Status in Healthy Subjects, While XOS Alone Only Shows Prebiotic Properties[END_REF]. We investigated at the same time a wide range of related biomarkers, going from the classical microbiota and carbohydrate metabolites to bacterial enzymatic activity, pH or protein putrefaction metabolites. This gave us more detailed information on the effect of the test product. Compared to other prebiotics like inulin, only a few clinical trials have been investigating the prebiotic activity of arabinoxylans. Matteuzzi et al [START_REF] Matteuzzi | Prebiotic Effects of a Wheat Germ Preparation in Human Healthy Subjects[END_REF]) fed volunteers 10g/day of a prebiotic product containing up to 10% arabinoxylans and 10% raffinose for 3 weeks. No clear effect was observed. Post-hoc analyses of the results demonstrated that only for the individuals with a baseline level below a set threshold (8log10 CFU/g and 4.5log10 CFU/g dry matter for Bifidobacterium and Lactobacillus, respectively) was it possible to increase the population of lactic acid bacteria. No indication on the product could help define the nature of the arabinoxylans. A number of other studies reported the effect of xylans on healthy populations with doses ranging from roughly 2.5g/d (Damen et [START_REF] Cloetens | Tolerance of Arabinoxylan-Oligosaccharides and Their Prebiotic Activity in Healthy Subjects: A Randomised, PlaceboControlled Cross-over Study[END_REF]) fed volunteers 10g/day SC-AX (avDP=6; DAS=0.26) for 3 weeks. Bifidogenic activity was barely significant as the placebo product also induced bacterial growth. Francois et al [START_REF] François | Effects of Wheat Bran Extract Containing Arabinoxylan Oligosaccharides on Gastrointestinal Parameters in Healthy Preadolescent Children[END_REF]) detected an increase in Bifidobacterium after 3 weeks of treatment at 5g/d with another SC-AX (avDP=5; DAS=0.19). Finegold et al [START_REF] Finegold | Xylooligosaccharide Increases Bifidobacteria but Not Lactobacilli in Human Gut Microbiota[END_REF]) did not show any stronger activity after 8 weeks treatment at 8g/d with XOS. Finally a dose of 4.8g/d for 3 weeks showed similar low bifidogenic activity [START_REF] Maki | Digestive and Physiologic Effects of a Wheat Bran Extract, Arabino-Xylan-Oligosaccharide, in Breakfast Cereal[END_REF]). Another article using 2.4g/d AXOS produced in situ in white bread did not show any bifidogenic effect [START_REF] Walton | A Randomised, Double-Blind, Placebo Controlled Cross-over Study to Determine the Gastrointestinal Effects of Consumption of ArabinoxylanOligosaccharides Enriched Bread in Healthy Volunteers[END_REF]. This is all lower than our data, showing an increase over 1 log at a dose of 5g/d. One reason may be the structure of the arabinoxylan, our product MC-AX being a higher DP, the bacterial stimulation may be longer-lasting, in agreement with in vitro and preclinical findings. A stimulation of Lactobacillus is reported by only 2 articles (Cloetens et [START_REF] Maki | Digestive and Physiologic Effects of a Wheat Bran Extract, Arabino-Xylan-Oligosaccharide, in Breakfast Cereal[END_REF]). Other metabolites were not affected in any of the reports. A number of publications can be found on the prebiotic activity of arabinoxylans or their derivatives. They range from short-term in vitro batch culture to in vitro intestinal simulation models, in vivo animal models and clinical studies. Our clinical study is the only of the 4 models where all markers have been measured (Figure 5). MC-AX stimulates lactic acid bacteria, including Bifidobacterium. By extension, other bacterial families are stimulated and increase production of butyrate and lower the pH. This in turn inhibits proteolytic bacteria (Bacteroides, Clostridium) and their metabolites (p-cresol). These observations are mostly consistent with the literature and with our SHIME® model and the 2 preclinical models using the same product. This is the first time to our knowledge that the Comparison with some other prebiotic molecules is summarized in table 3. At an equivalent dose of 5g/day, only FOS showed an equivalent bifidogenic activity [START_REF] Bouhnik | The Capacity of Nondigestible Carbohydrates to Stimulate Fecal Bifidobacteria in Healthy Humans: A Double-Blind, Randomized, PlaceboControlled, Parallel-Group, DoseResponse Relation Study[END_REF][START_REF] Mitsou | Prebiotic Potential of Barley Derived β-Glucan at Low Intake Levels: A Randomised, Double-Blinded, Placebo-Controlled Clinical Study[END_REF]. Galactooligosaccharides (GOS) on the other hand could have moderate stimulation [START_REF] Bouhnik | The Capacity of Nondigestible Carbohydrates to Stimulate Fecal Bifidobacteria in Healthy Humans: A Double-Blind, Randomized, PlaceboControlled, Parallel-Group, DoseResponse Relation Study[END_REF][START_REF] Davis | A Dose Dependent Impact of Prebiotic Galactooligosaccharides on the Intestinal Microbiota of Healthy Adults[END_REF][START_REF] Mitsou | Prebiotic Potential of Barley Derived β-Glucan at Low Intake Levels: A Randomised, Double-Blinded, Placebo-Controlled Clinical Study[END_REF] or strong stimulation of Bifidobacterium [START_REF] Depeint | Prebiotic Evaluation of a Novel Galactooligosaccharide Mixture Produced by the Enzymatic Activity of Bifidobacterium Bifidum NCIMB 41171, in Healthy Humans: A Randomized, Double-Blind, Crossover, Placebo-Controlled Intervention Study[END_REF] or total lactic bacteria [START_REF] Alander | Effect of GalactoOligosaccharide Supplementation on Human Faecal Microflora and on Survival and Persistence of Bifidobacterium Lactis Bb-12 in the Gastrointestinal Tract[END_REF]). 5. Conclusions same product has been tested and compared on such a range of complementary models. 6. Financial Disclosure All four studies were privately funded by DF3 SAS. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. 7. Conflict Of Interest The authors have the following competing interests to declare. All four studies were privately funded by DF3 SAS. This does not alter the authors' adherence to all MRA policies on sharing data and materials. 8.

Figures and tables

  Animal modelsBifidogenic activity of AX (whether SC-AX, MC-AX or LC-AX) has been reported in vivo in different rodent models[START_REF] Van Craeyveld | Structurally Different Wheat-Derived Arabinoxylooligosaccharides Have Different Prebiotic and Fermentation Properties in Rats[END_REF][START_REF] Hsu | Xylooligosaccharides and Fructooligosaccharides Affect the Intestinal Microbiota and Precancerous Colonic Lesion Development in Rats[END_REF]; Neyrinck et al. 2011; Van den Abbeele et al. 2011) and is a consistent benefit of arabinoxylan consumption. The major site of Bifidobacteium production seems dependent upon the degree of polymerisation, with higher DP shifting the fermentation towards the distal parts of the tractus (Van Craeyveld et al. 2008; Damen et al. 2011). Lactobacillus stimulation is not as clear. It was observed with XOS (Christensen et al. 2014; Santos, San Mauro, and Díaz 2006) but not with AXOS (Damen et al. 2011). The impact on SCFA production and distribution seems to be less consensual. Van Craeyveld (Van Craeyveld et al. 2008) and van den Abbeele (Van den Abbeele et al. 2011) have shown an increase in propionate production in the caecum. This is in agreement with observations on in vitro models presented previously (Grootaert et al. 2009; Van den Abbeele et al. 2013) where propionate was the main bacterial metabolite. Lu et al (Zhong X.

1 :

 1 Figures and tablesFigure1: Summary of the four prebiotic protocols Study design, duration and dose (expressed as g per day equivalent of pure MC-AX) of the test product in the four models are represented. Dotted lines represent run-in and non-interventional periods, while full and hashed arrows represent test and placebo treatment periods, respectively.

Figure 2 :

 2 Figure 2: Bifidogenic activity of MC-AX Bifidobacterium was semi-quantified by culture (SHIME®) or qPCR (the other 3 models) and expressed as log unit averages. Data for the SHIME® represent the average value (and standard deviation) of the 3 colonic fermenters. Data for the rat, human and piglet models are expressed as mean and SEM. Models were fed MC-AX treatment (black line) or placebo treatment (dotted line) according to the protocols described in the text. * and § represent statistical variation from baseline and placebo group, respectively (p<0.05).

Figure 3 :

 3 Figure 3: Lactic bacteria stimulation of MC-AX Lactobacillus was semi-quantified by culture (SHIME®)or qPCR (the other 3 models) and expressed as log unit averages. Data for the SHIME® represent the average value (and standard deviation) of the 3 colonic fermenters. Data for the rat, human and piglet models are expressed as mean and SEM. Models were fed MC-AX treatment (black line) or placebo treatment (dotted line) according to the protocols described in the text. * and § represent statistical variation from baseline and placebo group, respectively (p<0.05).

Figure 4 :

 4 Figure4: Lactate and short chain fatty acid modulation of MC-AX Lactate was quantified in vitro and short chain fatty acids (SCFA) were quantified in the rat and human studies. Data for the SHIME® represent the average value (and standard deviation) of the 3 colonic fermenters. Data for the rat and human models are expressed as mean and SEM. Models were fed MC-AX treatment (dark grey) or placebo treatment (light grey) according to the protocols described in the text. * and § represent statistical variation from baseline and placebo group, respectively (p<0.05).

Figure 5 -

 5 Figure 5-Bifidigenic activity of MC-AX. Diagram of bifidogenic activity and its consequences in terms of intestinal environment. The different relations are validated by at least 2 of the study models presented herein. In grey are links that have been shown by our clinical trial but not measured in the other models.

  ; Van den Abbeele et al. 2013), in agreement with our findings on MC-AX. Results from van den Abbeele et al (Van den Abbeele et al. 2013) on LC-AX indicate

  al. 2012; Finegold et al. 2014; Maki et al. 2012; Walton et al. 2012) to about 5g/d (François et al. 2014; Lecerf et al. 2012; Maki et al. 2012), 10g/d (Childs et al. 2014; Cloetens et al. 2010) and up to 30g/d (Windey et al. 2014). Bifidogenic activity was detected, although not to a clinical level of 1 log in 4 of the articles. Cloetens et al

  al. 2010; Walton et al. 2012), again showing the mixed responses toward this bacterial gender. Most publications did not report on additional bacterial activity, in accordance with our own findings. As observed in vitro (Van den Abbeele et al. 2013), Lactobacillus modulation, when observed, remained significant after a period of wash-out (Matteuzzi et al. 2004). A reduction of proteolytic activity is consistent with our findings (Lecerf et al. 2012). This was observed by a reduction in phenol (Damen et al. 2012), cresol (Cloetens et al. 2010; Damen et al. 2012; Windey et al. 2014) or branchedchain fatty acids (François et al. 2014; Walton et al. 2012). Eventhough pH drop was not measured in the different studies, this may be a consequence in luminal environment unfavourable to proteolytic fermentation following the acid bacteria stimulation (Bifidobacterium, Lactobacillus). Finally, consumption of 2.2g/day of a MC-AX (avDP=18) over 3 weeks increased butyrate production as compared to LC-AX control group (avDP>170). Butyrate stimulation was observed by some authors(Damen et al. 2012; Walton et al. 2012), in agreement with our own data, but not by others (Childs et al. 2014; François et al. 2014; Windey et al. 2014), while another one observed a reduction in butyrate

Acknowledgements The authors would like to acknowledge all researchers, engineers and technicians who have collaborated on the different models. We would also like to thank Yann Dugenet (DF3 SAS) for supplying the test product, and Sam Possemiers (Bio-Actor) for his support and discussion in the design of this article.


Yann Bugeaud

Andrej Dujella

Wenjie Fang

Tomislav Pejković

Bruno Salvy

Absolute root separation

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic.

Separation and absolute separation

The absolute separation of a polynomial P ∈ C[X] is the minimal nonzero distance between the absolute values of its complex roots: abs sep(P ) := min

P (α)=P (β)=0, |α| =|β| |α| -|β| .
Having good lower bounds on this quantity for polynomials with integer coefficients is of interest in the asymptotic analysis of linear recurrent sequences.

To the best of our knowledge, the first published bound on this problem [START_REF] Gourdon | Effective asymptotics of linear recurrences with rational coefficients[END_REF] was abs sep(P ) ≫ H(P ) -d(d 2 +2d-1)/2 where, here and below, the constant implicit in the ≫ sign depends only on the degree d, while H(P ), the height of the polynomial P , is the maximum of the absolute values of its coefficients. This exponent was later [START_REF] Dubickas | Counting and testing dominant polynomials[END_REF][START_REF] Sha | Effective results on the Skolem problem for linear recurrence sequences[END_REF] improved to -d 3 /2 + d 2 + d/2 -2 and even more recently [START_REF] Dubickas | On the distance between two algebraic numbers[END_REF] to -d 3 /2 + d 2 + d/2 -1. In this work, we improve this exponent and that of related problems. Still, we do not know how far the exponent we obtain is from being optimal. Thus an important part of this article is devoted to experiments in low degree, from where we can infer families of polynomials exhibiting a behaviour in H(P ) -d-1 for d ∈ {3, 4, 5, 6}.

In the much more classical case of the separation sep(P ) := min

P (α)=P (β)=0, α =β |α -β|, 1
the best available bound for a polynomial with integer coefficients goes back to Mahler [START_REF] Mahler | An inequality for the discriminant of a polynomial[END_REF]: sep(P ) ≫ H(P ) -d+1 .

Even in that case, the tightness of the exponent -d + 1 is still unknown, with best known upper bounds -(2d -1)/3 for general d and -2 for d = 3 [START_REF] Bugeaud | Root separation for reducible integer polynomials[END_REF][START_REF] Evertse | Distances between the conjugates of an algebraic number[END_REF][START_REF] Schönhage | Polynomial root separation examples[END_REF].

(A consequence noted by Mahler is that the right-hand side of (1) also gives a lower bound on the absolute value of the imaginary parts of nonreal roots of P .) This work consists of two parts. In the first one, we improve the known exponents for the absolute separation and related problems with the following.

Theorem 1. Let P ∈ Z[X] be a polynomial of degree d and let α and β be two of its roots such that |α| = |β|, then 1. if α and β are real, then |α| -|β| ≫ H(P ) -(d-1) ; 2. if α is real and β is not, then |α| -|β| ≫ H(P ) -2(d-1)(d-2) ; 3. if neither of them is real, then |α| -|β| ≫ H(P ) -(d-1)(d-2)(d-3)/2 .

We proved a more precise version of the first bound in a previous work [START_REF] Bugeaud | Absolute real root separation[END_REF], where we showed that the exponent of H(P ) is optimal in that case. More detailed but less precise bounds for the second case can be found in [4, Lemma 2.5], [START_REF] Melczer | Symbolic-numeric tools for analytic combinatorics in several variables[END_REF]Lemma 3.6] and [START_REF] Melczer | Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems[END_REF]Lemma 53]. Note that the third case requires d ≥ 4 to be meaningful, since a cubic polynomial with real coefficients cannot have two non-real roots with distinct absolute values.

The proof of Theorem 1 is based on constructing auxiliary polynomials with integer coefficients of controlled height whose roots contain the desired difference. This is a very versatile approach. We illustrate it to rederive Mahler's exponent -d + 1 in Section 2.1. Next, we apply it to the three cases covered in Theorem 1. Similar bounds for the difference between the real or the imaginary parts of roots of integer polynomials are derived in Sections 2.4 and 2.5.

In the second part of this work (Section 3), we describe experiments leading to lower bounds on the absolute separation for small degrees, which can be summarized as follows.

Theorem 2. For each d ∈ {3, 4, 5, 6}, there exists a sequence

(P d,M ) of polyno- mials of degree d in Z[X], such that as M → ∞, the polynomial P d,M has two roots α M , β M with |α M | = |β M | and |α M | -|β M | ≪ H(P d,M ) -d-1 , M → ∞.
Thus apart from its first part, d = 3 is the only case where we know Theorem 1 to be optimal. It is interesting to note that, in our examples, the growth of H(P 3,M ) is exponential in M , while that of H(P d,M ), for d ∈ {4, 5, 6}, is linear in M .

General bounds from symmetric functions of roots

Here and in the rest of this article, we consider a polynomial P (X) ∈ Z[X],

P (X) = d i=0 a i X i = a d d i=1 (X -α i ),
of degree d (a d = 0) with coefficients of absolute value bounded by H and complex roots α 1 , . . . , α d . Our bounds on various types of separations that are asymptotic in the height of the polynomial are obtained from the following two classical results, of which we sketch the proofs for completeness.

Proposition 1 (Effective Version of the Fundamental Theorem of Symmetric Functions). [START_REF] Yap | Fundamental Problems in Algorithmic Algebra[END_REF]Thm. 6.21]

Let P (X) = d i=0 a i X i = a d d i=1 (X -α i ) ∈ Z[a 0 , . . . , a d , X] and let G ∈ Z[X 1 , . . . , X d ] be a symmetric polynomial of degree at most k in each X i . Then a k d G(α 1 , . . . , α d ) is a polynomial of total degree at most k in Z[a 0 , . . . , a d ].
This is a standard result on symmetric functions [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF]Ch. 1]. We now give an elementary self-contained proof.

Proof. Let b i = a d-i /a d for 1 ≤ i ≤ d. These are algebraically independent symmetric functions of the α j 's. We need to prove that G(α 1 , . . . , α d ) is a polynomial of total degree at most k in Z[b 1 , . . . , b d ]. By linearity, it is sufficient to consider the case where G is homogeneous of total degree k. Let Λ k be the linear space of such symmetric polynomials in α 1 , . . . , α d .

Since b i is of total degree i in the α j 's, all products of the form

e µ = b µ1 1 b µ2 2 • • • b µ d d
indexed by µ = (µ i ) satisfying d i=1 iµ i = k are linearly independent and in Λ k . Such a product has total degree d i=1 µ i ≤ k in the b i 's. However, Λ k is also linearly generated by the monomials of the form

m λ = σ∈S d α λ1 σ(1) • • • α λ d σ(d) indexed by λ = (λ j ) with λ 1 ≥ λ 2 ≥ • • • ≥ λ d and d j=1 λ j = k.
The dimension is given by the number of distinct such (λ j ), which is also the number of (µ i ), where each µ i = λ i -λ i+1 . Therefore, the e µ 's also linearly span Λ k and G can be written as a linear combination of e µ 's with rational coefficients.

To prove that the coefficients are integers, we order the monomials in α j 's lexicographically. Since b i is the sum of j∈S α j over all subsets S of {1, . . . , d} with i elements, the largest monomial in e µ is given by α

µ ′ 1 1 • • • α µ ′ d d , where µ ′ j = d i=j µ i . The map µ → µ ′ = (µ ′ j
) is a bijection, which implies that every e µ has a distinct monic leading monomial and all other monomials have integral coefficients. Induction in the lexicographical order finally shows that the coefficients of G in the basis (e µ ) µ are all integers.

Proposition 2 (Cauchy Bound). [START_REF] Mignotte | Mathematics for Computer Algebra[END_REF]Thm. 4.2 (ii)] Let P (X) = d i=0 a i X i ∈ Z[X] with |a i | ≤ H for all i and let α = 0 ∈ C be one of its roots. Then

|α| ≥ 1 1 + H .
Proof. We may assume |α| < 1, otherwise the result is obvious, and a 0 = 0. Then, we get

1 ≤ |a 0 | ≤ d i=1 |a i | • |α i | ≤ H |α| 1 -|α| ,
giving at once the lower bound |α| ≥ 1 1+H .

Motivating example: Mahler's bound

Mahler's bound is usually deduced from Hadamard's bound on Sylvester's matrix applied to the discriminant of the polynomial. When only the asymptotic exponent of H(P ) in the estimate is needed, Propositions 1 and 2 are sufficient. Indeed, consider the polynomial

M (X) = a 2(d-1) d i<j (X -(α i -α j ) 2 ).
It is symmetric in the α i 's, with degree in each α i that is twice the number of j = i, i.e., 2(d -1). Thus, by Proposition 1, the polynomial M (X) has integer coefficients of height bounded by cH 2(d-1) for some constant c that depends only on d. By Proposition 2, we thus get that for any (i, j) such that α i = α j ,

|α i -α j | 2 ≥ 1 1 + cH 2(d-1) .
Choosing (i, j) that minimizes |α i -α j | and taking square roots thus gives sep(P ) ≫ H -(d-1) , recovering the exponent in Mahler's bound.

This method could be refined to produce a bound rather than an asymptotic estimate, by a more precise analysis of the coefficients of the polynomial involved in Proposition 1.

Note also that the polynomial obtained by considering the roots α i + α j in place of α i -α j in M satisfies the same bounds. It follows that the sum of non-opposite roots of P is also lower bounded by H -(d-1) , giving part 1 of Theorem 1.

Absolute real-complex gap

Part 2 of the theorem is obtained by considering the polynomial

R(X) = a 2(d-1)(d-2) d i<j,k / ∈{i,j} X -(α 2 k -α i α j ) .
This polynomial is symmetric in the α i 's, of degree 2(d -1)(d -2) in each of them. Thus by Propositions 1 and 2, its smallest nonzero root has magnitude at least 2) . In particular, if α is a real root of P and β a nonreal root such that |α| = |β|, taking α k = α, α i = β and α j = β gives 2) .

H -2(d-1)(d-
|α 2 -|β| 2 | ≫ H -2(d-1)(d-
If |α| + |β| ≤ 2,
+ |α| + 1 ≥ |α| + |β| > 2 and so |α| > 1/2 while 2|β| ≥ |α| + |β| > 2 gives |β| > 1.
Thus, 1/|αβ| < 2 and 1/|α| + 1/|β| < 3. The same bounds are obtained when supposing |α| ≥ |β|. Now, since α -1 and β -1 are roots of the reciprocal polynomial X d P (1/X) that has the same height as P , we also have

|α -2 -|β| -2 | = |α| -|β| |α||β| 1 |α| + 1 |β| ≫ H -2(d-1)(d-2) ,
whence the conclusion by dividing out by the factor smaller than 6.

Absolute complex-complex gap

We now analyse the polynomial

S(X) = a (d-1)(d-2)(d-3) d i<j, k<ℓ, {i,j}∩{k,ℓ}=∅ X 1/2 -(α i α j -α k α ℓ ) .
Exchanging the pairs (i, j) and (k, ℓ) shows that this is indeed a polynomial in X. It is invariant under permutations of the α i 's. Its degree in X is ∼ d 4 /4, but its degree in each of the α i 's is only (d -1)(d -2)(d -3), corresponding to all the possible choices of the other α j 's. If α and β are two non-real roots of P with |α| = |β|, then taking α i = α, α j = α, α k = β, α ℓ = β and using Propositions 1 and 2 again gives

(|α| 2 -|β| 2 ) 2 ≫ H -(d-1)(d-2)(d-3) .
Taking square roots divides the exponent by 2, and then with the help of the reciprocal polynomial X d P (X -1 ), the same argument as in the case of absolute real-complex gap leads to

abs sep(P ) ≫ H -(d-1)(d-2)(d-3)/2 ,
which concludes the proof of Theorem 1.

Gap on the real parts of roots

The same approach gives bounds on gaps between real parts of roots. There are again three cases: real-real, real-complex, complex-complex. The real-real case is simply the corresponding case in root separation, which is already known. For the real-complex case, we consider the following polynomial

T 1 (X) = a 3 2 (d-1)(d-2) d i<j,k / ∈{i,j} X -(α i + α j -2α k ) ,
while the complex-complex case relies on

T 2 (X) = a (d-1)(d-2)(d-3) d i<j, k<ℓ, {i,j}∩{k,ℓ}=∅ X 1/2 -(α i + α j -α k -α ℓ ) .
The analysis is as before. Both T 1 and T 2 can be seen to be polynomials in X. They are symmetric polynomials in the α i 's. Their degrees in the α i 's is the one used in the exponent of a d . The smallest nonzero root of T 1 divided by 2 is a lower bound of the real-complex real-part gap, while T 2 gives the complexcomplex case. Propositions 1 and 2 then give the following.

Theorem 3. Let P ∈ Z[X] be a polynomial of degree d and let α and β be two of its roots such that ℜα = ℜβ, then

|ℜα -ℜβ| ≫      H(P ) -(d-1) , if ℑα = ℑβ = 0, H(P ) -3(d-1)(d-2)/2 , if ℑα = 0, H(P ) -(d-1)(d-2)(d-3)/2 ,
otherwise.

Gap on the imaginary parts of roots

The situation for imaginary parts is similar. First, if one of the roots is real and the other is not, then as already mentioned, Mahler's bound applies to the imaginary part. If both roots are nonreal but not conjugates, then the same polynomial T 2 as in the case of the real parts can be used, with (α i , α k ) and (α j , α ℓ ) taking the roles of conjugate roots. In the case when one of the roots is purely imaginary, then the following generalization of T 1 can be used

T 3 (X) = a 3(d-1)(d-2) d i<j,k / ∈{i,j} X -(α i -α j -2α k ) 2 .
This discussion leads to the following.

Theorem 4. Let P ∈ Z[X] be a polynomial of degree d and let α and β be two of its roots such that ℑα = ℑβ, then

|ℑα -ℑβ| ≫      H(P ) -(d-1) , if ℑα = 0, H(P ) -3(d-1)(d-2)/2 , if ℜα = 0, H(P ) -(d-1)(d-2)(d-3)/2 ,
otherwise.

Experiments and bounds in low degree

As already mentioned, even in the case of Mahler's bound, the tightness of the exponent is unknown. The situation is similar for the bounds obtained in the previous section. We now turn to experiments that lead to lower bounds on the asymptotic separation. In order to obtain such an asymptotic result, we search for families of polynomials exhibiting a small absolute separation. In particular, we would like to approach a tight estimate in the case of low degrees, as a first step for a better understanding of the actual growth of these bounds with the degree. We use two complementary types of experiments.

Exhaustive search First, we perform an exhaustive search for polynomials of small absolute separation given a degree and a bound on the height. More specifically, given a degree, we search through all polynomials with integer coefficients within a given height, in case a pattern can be discerned in the polynomials with small absolute separation. In some cases indeed, the roots of these polynomials seem to concentrate in certain locations, letting us refine the experiment and search more closely. This is successful in degree 3, where we find a family of polynomials letting us prove the tightness of the exponent -4 in that case (Proposition 3 below). However, even if some time is saved by taking into account various symmetries, the number of polynomials to be tested is too large for this approach to be used for large height and even more so for large degree.

Perturbations While these exhaustive searches are purely numerical, our second type of experiments relies heavily on symbolic computation. In degree d, we consider polynomials of the form

P d (X, ǫ) = R d (X) + ǫQ d (X).
Here, for a given r ∈ N + , R d (X) is a polynomial with several roots of absolute value r, having factors of the type X ±r and X 2 +aX +r 2 , with a an integer such that |a| < 2r. The polynomial

Q d (X) is subject to max(deg(R d ), deg(Q d )) = d so that P d (X, ǫ) has degree d in X.
For X 1 (ǫ) and X 2 (ǫ) two roots of P d (X, ǫ) such that X 1 (0) and X 2 (0) are roots of selected factors of R d , we compute a series expansion of these roots of P d in powers of ǫ, from which we deduce

|X i (ǫ)| 2 = k≥0 c i,k ǫ k .
This computation is purely symbolic, meaning that the c i,k 's are computed as explicit polynomials in the coefficients of R d and Q d . We then look for integer solutions of the polynomial system

{c 1,k = c 2,k , 1 ≤ k ≤ h -1} that do not satisfy c 1,h = c 2,h . Such solutions give a polynomial P d (X, ǫ) such that |X 2 (ǫ)| 2 -|X 1 (ǫ)| 2 is not zero but is O(ǫ h ), so that the polynomial P * d (X) = M P d (X, M -1 )

Results

The rest of this section reports on these experiments. The results of an exhaustive search are first presented in §3.1. Next, we discuss a family of polynomials of degree 3 proving the tightness of the exponent -4 by an argument we have been unable to generalize. In degrees 4, 5 and 6, the perturbative approach lets us find exponents -5, -6, -7.

Exhaustive search for small degree and height

The results of an exhaustive search are displayed in Table 1. It appears that the classical separation seems to be larger than the three other ones (absolute separation, separation of the real or imaginary parts), whose orders of growth seem similar.

For the same degrees and larger height, a fully exhaustive search becomes too time-consuming. Instead, we performed extensive experiments. The resulting record values are reported in Table 2. For degree 3 (resp. degree 4, degree 5), we computed up to height 200 (resp. height 120, height 30). As expected, polynomials with small separation tend to have larger height. To balance this bias, according to the form of the bounds on the separation, we filter the polynomials of interest by their quality, defined as -ln(S)/ ln(H) for a polynomial of height H and separation S, and display only polynomials of "high quality".

P (X) abssep

log abssep log H(P )

2X 3 + X 2 -X - 1
X 4 -X 2 -2X -3 8.615e-4 6.42 X 4 -X 2 + 2X -3 8.615e-4 6.42 3X 4 + 3X 3 + X 2 -2X -4 4.585e-5 7.21 4X 4 + 2X 3 -X 2 -3X -3 3.655e-5 7.37 5X 4 + 3X 3 + X 2 -X -3 5.893e-5 6.05 X 4 + 6X 3 -7X 2 -5X + 6 2.276e-6 6.68 6X 4 + 5X 3 -7X 2 -6X + 1 2.497e-6
6.63 11X 4 + 7X 3 -X 2 -10X -16 2.671e-8 6.29 16X 4 + 10X 3 + X 2 -7X -11 2.266e-8 6.35 3X 4 + 6X 3 -4X 2 + 3X -18 1.799e-8 6.17 18X 4 + 3X 3 + 4X 2 + 6X -3 1.095e-8 6.34 Observing that cubic polynomials of high quality in Table 3, namely 13x 3 + 11x 2 +8x+5 and 181x 3 +153x 2 +112x+71, have roots in similar locations pushed us to refine our search in this vicinity and eventually led us to an unexpected family leading to a proof of optimality in the next section.

X 4 + 40X 3 + 11X 2 -14X -55 3.384e-11 6.02 55X 4 + 14X 3 -11X 2 -40X -1 2.724e-11 6.07 X 5 + X 4 + 2X 3 + 3X 2 -2X + 2 2.697e-5 9.58 2X 5 + X 4 -X 3 + X 2 -X -1 1.051e-3 9.89 2X 5 + X 4 + X 3 -2X 2 + 2X -2 2.790e-3 8.49 2X 5 + X 4 + X 3 -X 2 -X -1 3.800e-3 8.04 2X 5 + X 4 + 2X 3 -2X 2 + X -2 2.130e-3 8.88 2X 5 + X 4 + 2X 3 -X 2 -X -2 3.350e-3 8.22 2X 5 + X 4 + 2X 3 + 2X 2 -2X -2 2.130e-3 8.88 2X 5 + 2X 4 + X 3 -X 2 -2 6.473e-4 10.59 4X 5 + 2X 4 -4X 3 + 3X -2 1.463e-6 8.03 8X 5 + 5X 4 -4X 3 + 4X 2 -5X -4 5.185e-8 8.07
For polynomials of degree 4 up to height 120, the results in Table 2 seem to suggest a bound of at most O(H(P ) -6 ), which should be reached by a realcomplex gap, as the complex-complex gap only gives O(H(P ) -3 ). This is far from the theoretical bound O(H(P ) -12 ) from Theorem 1. The best exponent we obtain by perturbations is -5, see §3.4.

For polynomials of degree 5, we only have results up to height 30 and even those have been obtained by focusing in some cases of real-complex gap. Here, the bound on the height seems to be too small to observe a family. We have two polynomials with good quality that are relatively similar, namely 2x 5 + x 4x 3 + x 2 -x -1 and 8x 5 + 5x 4 -4x 3 + 4x 2 -5x -4. No other polynomial close to these two are observed with height at most 30.

The case of degree 3

Proposition 3. The family of cubic polynomials

P n (X) = p n (3X 3 -2X 2 + 4X -6) + 6q n (X 3 -X 2 + 1) ∈ Z[X],
where (p n /q n ) n is the sequence of convergents of the continued fraction expansion of √ 3, has the property that

abs sep(P n ) ≪ H(P n ) -4 , n → ∞.
Theorem 1 shows that abs sep(P n ) ≫ H(P n ) -4 , so that the exponent -4 is optimal in degree 3. The absolute separations of a few polynomials in that family are displayed in Table 3. The height of the polynomials P n increases exponentially with n.

Proof. It is readily checked that the bivariate polynomial

P (X, Y ) = - 1 3 X 2 + 1 2 X 3 + 2 3 X -1 Y + X 3 -X 2 + 1 (2)
is such that for Y = √ 3, its 3 roots have absolute value exactly √ 3 -1. A small perturbation Y = √ 3 + ǫ sends the real root to

√ 3 -1 + (2 - √ 3)ǫ + O(ǫ 2 ),
while the complex roots have absolute value

√ 3 -1 + (2 - √ 3)ǫ + O(ǫ 2 ),
with a different constant in the O() term so that the difference of the absolute values is O(ǫ 2 ). Evaluating Y at a convergent p/q of the continued fraction expansion of √ 3 leads to ǫ = |p/q -√ 3| < 1/q 2 , so that the difference of the absolute values of roots is O(q -4 ). The polynomial with integer coefficients obtained by normalizing P (X, p/q) has coefficients growing asymptotically like q, giving -4 as a bound in the exponent.

Further experiments In an unsuccessful attempt to generalize the nice family of cubic polynomials in Proposition 3 to higher degrees, we analyzed more precisely the properties of the bivariate polynomial of Equation [START_REF] Bugeaud | Absolute real root separation[END_REF]. A polynomial a 3 X 3 + • • • + a 0 with simple roots, all of the same nonzero absolute value, has coefficients that satisfy

a 3 1 a 3 -a 0 a 3 2 = 0. (3)
The set of such polynomials is therefore contained in an algebraic set of dimension 3 in R 4 . The question is to find a good point on this set whose perturbations behave well with respect to the absolute separation. Polynomials in that set factor as (a 2 X + a 1 ) times a quadratic polynomial, whose discriminant has to be negative for its roots to have identical absolute value. This gives necessary and sufficient conditions: a 1 a 2 < 0 ≤ a 1 a 3 ≤ a 2 2 ≤ 3a 1 a 3 . These conditions define a region on our algebraic set where all polynomials have 3 roots of identical absolute value.

A very special property possessed by the polynomial of Equation ( 2) is that in its case, the polynomial from Equation (3) factors as a square (1 -Y 2 /3) 2 . Adding this condition and forcing the perturbation of Y to cancel the constant and linear coefficients of the expansion of the absolute separation finally leads to a 3-dimensional set of polynomials of which that of ( 2) is an instance:

a(X 3 + 10c 3 ) + b(3X 2 + 6cX) -((X 2 + 4cX)b + 6ac 3 ) √ 3.
Polynomials in that family all have three roots of identical absolute value, and replacing √ 3 by convergents to its continued fraction expansion lead to an exponent 4 for their asymptotic absolute separation.

Degrees 4 and 6

For d ∈ {4, 6}, we consider the polynomial P = M (X d -1) -Q d (X), with M an integer and Q d a polynomial of degree d-1 to be made precise later. As M tends to infinity, the roots of P tend to those of X d -1. In particular, one of them tends to -1 and another one tends to c d a root of the d-th cyclotomic polynomial. The asymptotic expansion in powers of 1/M of the difference between the absolute values of these roots can be obtained as follows. First the equation P = 0 is rewritten as

X d -1 Q d (X) = 1 M . (4)
In the neighbourhood of a root ω of X d -1, the left-hand side behaves like

d ωQ d (ω) (X -ω) + d(d -1) 2ω 2 Q d (ω) - dQ ′ d (ω) ωQ d (ω) 2 (X -ω) 2 + • • • .
Power series inversion then gives the asymptotic behaviour of the corresponding root of Equation ( 4):

X ω = ω + ωQ d (ω) 1 dM + ω 2 Q d (ω)Q ′ d (ω) - (d -1) 2 ωQ d (ω) 2 1 d 2 M 2 + • • •
Substituting ω by 1/ω gives the expansion of the conjugate root and multiplying them gives the expansion of |X ω | 2 . Finally, subtracting the values of this expansion for ω = -1 and ω = c d gives an expansion of the distance between the squares of these absolute values with coefficients that are polynomials in c d in the coefficients of Q d . Cancelling those coefficients up to order 1/M d gives a system of d equations in the d coefficients of Q. Up to multiplying M by a constant, when d ∈ {4, 6}, there is only one case when this system has integer solutions that do not correspond to Q d having a common factor with X d -1, leading to the following.

P d,M = M (X d -1)-Q d (X), with Q 4 (X) = X 3 -X 2 + X -5, Q 6 (X) = 9X 5 -9X 4 -26X 3 -9X 2 + 9X -28,
As M tends to infinity, these polynomials have height M and two roots α, β

satisfying 0 < |α| -|β| ≪ H(P d,M ) -d-1 .
Absolute separations for a few polynomials in these families are given in Table 4.

Proof. Once these polynomials have been found, the proof can be carried out by hand (and more easily with the help of a computer algebra system). The polynomial P 4,M has roots Taking the absolute values and subtracting shows that

z 1 = 1 - 1 M - 2 M 2 - 11 2M 3 - 71 4M 4 + O 1 M 5 , z i = i - i M - 1 + 4i 2M 2 - 4 + 11i 2M 3 - 66 + 143i 8M 4 + O 1 M 5 .
|z 1 | -|z 2 | ∼ |z 1 | 2 -|z i | 2 2 = O 1 M 5 .
The same reasoning applies to the polynomial P 6,M , with roots

z 1 = 1 - 19 3M - 1235 18M 2 - 240445 162M 3 + • • • z -1+i √ 3 2 = -1 + i √ 3 2 - 4(1 -i √ 3) 3M - 238 -166i √ 3 9M 2 - 33383 -13511i √ 3 81M 3 + • • • leading to |z 1 | -z -1+i √ 3 2 ∼ |z 1 | 2 -z -1+i √ 3 2 2 2 = O 1 M 7 .
Perturbations of the other roots, namely (1 ± i √ 3)/2, lead to another system of equations for the coefficients of Q 6 , but this system does not have integer solutions.

Further experiments Looking for similar polynomials using different products of cyclotomic polynomials also yields M (X 2 -1)(X 2 + X + 1) -X 3 + 3X + 4 with absolute separation in H -5 .

For d = 8, the same computation only produces an exponent -d + 4: the polynomial system obtained when trying to cancel one more coefficient does not have any integer solution.

For odd degree d, the use of cyclotomic polynomials in this method only seems to reach exponent d instead of d + 1, with examples like

M (X 2 + 1)(X -1) + X 2 -2X + 3, M (X 2 -X + 1)(X -1) + 2X 2 -4X -3, in degree 3 and M (X -1)(X 2 + X + 1)(X 2 -X + 1) + 3X 4 -2X 3 + 5X 2 -4X + 7, M (X -1)(X 2 + 1)(X 2 -X + 1) + 2X 4 -8X 3 + 13X 2 -12X + 7
in degree 5.

Degree 5

For polynomials of degree 5, exhaustive search cannot reach heights sufficient to exhibit the beginning of an asymptotic behaviour.

The perturbative approach remains possible, although the polynomial systems become rapidly too large for the Gröbner basis engine we use, which is Faugère's FGb [START_REF] Faugère | A Library for Computing Gröbner Bases[END_REF]. We consider the polynomial P

= M R 5 (X) -Q 5 (X), with R 5 (X) = (X 2 + aX + r 2 )(X 2 + bX + r 2 ), Q 5 (X) = X 5 + q 4 X 4 + q 3 X 3 + q 2 X 2 + q 1 X + q 0 ,
where all coefficients are unknowns, and expected to take rational values, with a and b in the interval (-2r, 2r) so that R 5 has four roots of absolute value |r|. As in the previous case, the equation P = 0 rewrites

R 5 (X) Q 5 (X) = 1 M .
Expanding the left-hand side in the neighbourhood of solutions of the factors of R 5 and inverting the power series expansion gives expansions of the corresponding perturbed roots X 1 (ǫ), X 2 (ǫ) of P . From there, the coefficients c 1,k , c 2,k of |X 1 (ǫ)| 2 and |X 2 (ǫ)| 2 for k from 1 to 5 are obtained. These coefficients are polynomials in r, a, b and q i for 0 ≤ i ≤ 4. The polynomial system {c 1,k = c 2,k , 1 ≤ k ≤ 5} is unfortunately too big for a direct computation by Gröbner bases. Instead, we run a loop over possible integer values of r, a, b in a given search range, and use Gröbner bases to determine whether each specialized system has (not necessarily integral) solutions and to determine the solution. If the solution happens to be rational, this gives a family with separation O(H(P ) -6 ). This is how the following family was found.

Proposition 5. Let M be a positive integer and P 5,M be defined by Again, given the polynomials, the proof is a direct computation of the asymptotic absolute separation. Absolute separations for a few polynomials in this family are given in Table 5.

P 5,M = M R 5 (X)-Q 5 (X), with R 5 (X) = (X 2 -9X + 36)(X 2 -11X + 36), Q 5 (X) = X 5 -213X 3 + 2404X
Note that actually, we get two one-parameter families of such polynomials with the same R 5 as above and The family in the proposition is the special case q = 0 of the second one.

Conclusion

From our experiments in low degree, it would be tempting to conjecture that a tight bound on the exponent of the absolute separation in degree d is -d -1, but we have not been able to prove this, even for degrees 7 or 8. Even the degree of the exponent of our lower bounds in Theorem 1 seems too high. The bounds in Theorem 1 are derived by considering arbitrary triples or quadruples of roots, without taking into account that some of them are conjugate, so this approach could make them pessimistic. More deeply, the cubic exponent seems to be inevitable for a purely algebraic approach: if P = a 0 + • • •+ a d z d is written a d (z -(x k + iy k )), then x k , y k and x 2 k + y 2 k are generically algebraic of degree d(d -1)/2 over Q(a 0 , . . . , a d ), with minimal polynomials having degree d -1 in the a i 's, whence again a cubic exponent. Obtaining better bounds thus probably needs more analytic tools.

Another question which is of a more analytic nature and more directly relevant for the complexity of algorithms on linear recurrences would be to determine a bound on the minimal distance between the two largest distinct absolute values of the roots. For any polynomial, this quantity is at least as large as the absolute separation, but does it have a different asymptotic behaviour?

On a related matter, Koiran recently used analytic arguments (Rolle's theorem and Baker's theory of linear forms in the logarithms of algebraic numbers) to give a bound on the (classical) root separation for trinomials, with a very small dependency on the degree [START_REF] Koiran | Root separation for trinomials[END_REF]. It is not clear to us whether similar results also hold for the absolute separation.

Proposition 4 .

 4 Let d ∈ {4, 6} and let M be a positive integer. Consider the polynomials P d,M of degree d defined by

Q 5 ( 2 -

 52 X) = X 5 + qX 4 -(20q + 229)X 3 + (2700 + 171q)X (13104 + 720q)X + (25920 + 1296q), Q 5 (X) = X 5 + qX 4 -(20q + 213)X 3 + (2404 + 171q)X 2-(11088 + 720q)X + (20736 + 1296q).

  with integer M has integer coefficients and absolute separation O(H(P * d ) -h ).

	degree, max height	separation type	P (X)	separation
	3,10 3,20 4,10 4,20 5,10 3,10 3,20 4,10 4,20 5,10 3,10 3,20 4,10 4,20 5,10 3,10 3,20 4,10 4,20 5,10	|α -β| |α -β| |α -β| |α -β| |α -β| ||α| -|β|| ||α| -|β|| ||α| -|β|| ||α| -|β|| ||α| -|β|| 9X 5 -5X 4 -4X 3 -2X 2 -2X -9 1.459e-7 5X 3 + 8X 2 -9X + 2 1.421e-2 14X 3 + 17X 2 -13X + 2 4.938e-3 3X 4 -9X 3 -10X 2 + 7X -1 4.187e-3 9X 4 -13X 3 -14X 2 + 17X -4 5.974e-4 9X 5 + X 4 -4X 3 -9X 2 -3X + 7 4.656e-4 10X 3 -3X 2 -2X + 3 5.394e-4 17X 3 -9X 2 -7X + 8 1.233e-5 X 4 -6X 3 -7X 2 + 5X + 6 2.276e-6 5X 4 -17X 3 -20X 2 + 11X + 12 1.034e-7 7X 3 + 5X 2 + 5X + 1 5.952e-4 |ℜα -ℜβ| 19X 3 + 8X 2 + 15X + 2 2.218e-5 |ℜα -ℜβ| |ℜα -ℜβ| 9X 4 + 5X 3 -X 2 + 5X -1 1.472e-6 |ℜα -ℜβ| 13X 4 + 3X 3 + 5X 2 + 19X -7 1.669e-7 |ℜα -ℜβ| 7X 5 -6X 4 -6X 3 -5X 2 + X + 1 2.511e-7 |ℑα -ℑβ| 10X 3 + 6X 2 -6X + 1 2.403e-2 |ℑα -ℑβ| 19X 3 + 9X 2 -19X + 5 5.082e-3 10X 4 + X 3 + 10 6.250e-5 |ℑα -ℑβ| 20X 4 + X 3 + 20 7.813e-6 |ℑα -ℑβ| |ℑα -ℑβ| 5X

5

-8X 4 + 6X 3 + 5X 2 -5X + 8 1.061e-7 1: Record polynomials for small degree and height

Table 2 :

 2 Polynomials with small absolute separation

	n	height abssep	-log abssep log H(P )
	2	12 5.093e-3	2.12
	5	123 2.447e-6	2.68
	10	2,340 4.643e-11	3.36
	20	1,694,157 1.690e-23	3.66
	50 642,934,702,584,732 8.146e-58	3.86

Table 3 :

 3 Absolute separation for cubic polynomials P n from Proposition 3

Table 4 :

 4 Absolute separation for polynomials P d,M from Proposition 4

	d height abssep	-log abssep log H(P )
	4	10 3.716e-5	4.43
	4	20 4.183e-7	4.90
	4	50 2.653e-9	5.05
	4	100 7.175e-11	5.07
	4	500 2.055e-14	5.07
	4	1000 6.335e-16	5.07
	6	10 2 3.336e-7	3.24
	6	10 3 1.373e-14	4.62
	6	10 4 1.267e-21	5.22
	6	10 5 1.257e-28	5.58
	6	10 10 1.256e-63	6.29
	6	10 20 1.256e-133	6.65
	6	10 30 1.256e-203	6.76

Table 5 :

 5 Absolute separation for polynomials P 5,M from Proposition 5 As M tends to infinity, this polynomial has height O(M ) and two roots α, β satisfying 0 < |α| -|β| ≪ H(P 5,M ) -6 .

	height abssep	-log abssep log H(P )
	10 10 7.165e-38	3.81
	10 20 7.164e-98	4.91
	10 50 7.164e-278	5.56
	10 100 7.164e-578	5.78
	10 200 7.164e-1178	5.89
	10 500 7.164e-2978	5.96

2

-11088X + 20736.

Ackowledgements W. Fang and B. Salvy were supported in part by FastRelax ANR-14-CE25-0018-01. A. Dujella and T. Pejković were supported by the Croatian Science Foundation under the project no. IP-2018-01-1313 and the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund -the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). Y. Bugeaud, A. Dujella, T. Pejković, and B. Salvy were supported in part by the French-Croatian bilateral COGITO project 'Approximation diophantienne et courbes elliptiques'.


Heat Pain Threshold Modulation Through Experiencing Burning Hands in Augmented Reality

Keywords:

Figure 1: Participant wearing the VST-HMD, experiencing burning hands in VR and measuring

INTRODUCTION

AR allows the superimposition of visual and auditory cues on the real world, including the user's own body. The cross-modal interaction of these cues can lead to interesting phenomena. For example, in an AR experience, which allows the user to see their own hands burning when they are looking through a VST-HMD, makes some users feel an involuntary thermal sensation [START_REF] Weir | BurnAR: Feel the heat[END_REF], resulting even in some real thermoregulatory responses [START_REF] Eckhoff | Investigation of microcirculatory effects of experiencing burning hands in augmented reality[END_REF]. These results show complex crossmodal interactions in users, suggesting that it would be possible to also manipulate how they are perceiving pain.

Pain modulation has been effectively researched in Virtual Reality (VR). However, in VR, participants are not looking at their real hand, but at an embodied virtual hand. There is little research about pain modulation through AR experiences, where users are looking at their real hand.

Thus, we want to investigate to what extent these cross-modal interactions in AR can influence the perception of pain. In particular, we wanted to address the following research question: RQ1: Does the display of virtual flames on the user's hand lead to a change in HPT?

RELATED WORK

HPT is the temperature at which a thermal stimulus begins to evoke pain. The HPT in humans is usually determined by applying a thermal stimulator to the skin that increases in temperature until the HPT is reached. Typically, the HPT is reached at temperatures of 42-48°C [START_REF] Defrin | Quantitative Somatosensory Testing of Warm and Heat-Pain Thresholds: The Effect of Body Region and Testing Method[END_REF][START_REF] Pertovaara | Influence of skin temperature on heat pain threshold in humans[END_REF]. These values are influenced by many different factors such as the ambient temperature, rate of heating, location on the skin, skin temperature. The HPT also varies between individuals, place of measurement, and time.

It has been shown that cognitive factors or cross-modal interactions can greatly module the perception of pain. Longo et al. reported that the vision of one's own body part in pain reduces the pain intensity [START_REF] Longo | Visually Induced Analgesia: Seeing the Body Reduces Pain[END_REF]. In their experiments, they induced painful stimuli to the participant's hand while they were looking at it. Participants felt less pain compared to when they were looking at a box or somebody else's hand.

The Rubber Hand Illusion (RHI) [START_REF] Botvinick | Rubber hands 'feel' touch that eyes see [8[END_REF] is an important experiment in the area of body ownership. There have been several studies examining a change of pain perception during the RHI. Two separate experiments by Valenzuela-Moguillansky et al. showed how pain ratings to painful heat stimuli decreased immediately after the RHI [START_REF] Valenzuela-Moguillansky | The Role of Body Awareness in Pain[END_REF].

VR has been effectively used as a form of analgesia [START_REF] Hoffman | Virtual reality as an adjunctive pain control during burn wound care in adolescent patients[END_REF]. In a later study, the authors could show using fMRI that their VR experience significantly reduces pain-related brain activity. This has been mostly accredited to the distraction factor. Martini et al. [START_REF] Martini | What Color is My Arm? Changes in Skin Color of an Embodied Virtual Arm Modulates Pain Threshold[END_REF] used a VR system that overlaid a red, blue, or green color on a virtual embodied arm. In their experiment, the reddened arm decreased, whereas the blue skin increased the HPT.

With a rubber hand or in VR, participants are not looking at their real hand, but at an embodied rubber or virtual hand. AR can directly manipulate how the user is perceiving their own body, possibly creating different effects than with a foreign embodied hand.

EXPERIMENT

To answer our research question (See RQ1), we have designed and conducted the following experiment. The experiment was approved by the ethical committee of the university.

Participants We recruited 7 healthy (3 female, 4 male, 24 ± SD age: 2.5 years) right-handed participants from the student population. None suffered from a loss of stereo vision, which has been assessed before the experiment using a randot stereo test.

Platform Our interactive AR platform simulates realistic fire and smoke effects taking into account the shape and movement of the user's hand. For this, we have implemented a real-time fire synthesis and volumetric rendering running on the Unreal engine, which also handled spatial fire sound effects. We used a Varjo XR-3 as a VST-HMD. The display has a pixel density of 3000 PPI, a refresh rate of 90Hz, and a field of view of 115°. The dual front-facing cameras of the XR-3 feature a 12-megapixel video pass-through. The Valve Lighthouse tracking system was used to track the headset.

Thermal Stimulator Thermal heat stimuli were delivered by a PID-controlled Peltier device. Pain thresholds were assessed with the Method of Limits. The probe temperature was increased from a constant baseline temperature (30°C) at a steady temperature increase. To prevent tissue damage, the maximal temperature was set at 50°C. Every participant experienced the same temperature increase.

1 Daniel Eckhoff, Alvaro Cassinelli, Christian Sandor Outcome Measures We conducted the experiment in a university research laboratory. Before the experiment began, all participants were provided with information sheets and consent forms. After the participant gave consent, the first round of questionnaires was given through an online web interface provided by Psy-Toolkit [START_REF] Stoet | Psytoolkit: A software package for programming psychological experiments using linux[END_REF][START_REF] Stoet | Psytoolkit: A novel web-based method for running online questionnaires and reaction-time experiments[END_REF]. This included a Visual Analogue Mood Scale to assess the mood before as well as after the experiment as well as a question as to whether they have experienced AR or VR on HMDs before.

Before the start of the real experiment, we made participants familiarize themselves with the thermal stimulator as well as the procedure by going through one HPT identification run. Participants were asked to place their right index finger on the thermal stimulator. The temperature then increased steadily from 30°C to 50°C. We asked participants to press a button with their left hand as soon as they perceived the stimulation as being painful. Immediately after pushing the button, the HPT was identified and the temperature rapidly decreased back to the baseline temperature of 30°C.

The HPT identification experiment is using the Method of Limits design. We identified the HPT without the participant wearing the VST-HMD two times and two times when experiencing their right hand burning in AR when looking through the VST-HMD. When using a within-subjects design, carry-over effects sometimes occur when the effect of the first condition persists and influences the responses of the second treatment [START_REF] Dwan | Consort 2010 statement: extension to randomised crossover trials[END_REF]. To prevent this, we have pseudo-randomized the order of the conditions for each participant.

After the experiment participants filled out additional three additional questionnaires to assess motion sickness [START_REF] Kennedy | Simulator sickness questionnaire: An enhanced method for quantifying simulator sickness[END_REF], another Visual Analogue Mood scale as well as an AR presence questionnaire inspired by Gandy et al. [START_REF] Gandy | Experiences with an AR evaluation test bed: Presence, performance, and physiological measurement[END_REF].

ANALYSIS

The analysis was carried out in JASP [START_REF][END_REF]. We averaged the two HPT measurements for each condition (AR and no AR). A paired student t-test revealed a significant difference in HPT with no AR (45.091°C±2.755°C) and when the participants were experiencing their own hands burning in AR (43.841°C±3.590°C); t(6) = 2.459, p = 0.025 (See Figure 2).

The Simulator Sickness Questionnaire revealed that no participant suffered from motion sickness after the experiment. The VAS mood scale revealed that participants were significantly feeling less tense (p = 0.005) and sleepy (p = 0.009) after the experiment. The mean presence score was 4.49 out of 7.

DISCUSSION AND CONCLUSIONS

The purpose of this pilot study was to elucidate to what extent our burning hand experiment can influence the perception of pain. We were able to show that the HPT was significantly lower during the AR exposure. This is to our best knowledge the first experiment using AR for HPT modulation.

In light of these results, it would be interesting to conduct a largescale study to include the investigation of whether the HPT could be modulated with different augmentations (water, ice) or change of skin color (red, blue). These could increase the HTP, therefore reducing the perception of pain and be used as a non-invasive form of analgesia. Hofmann et al. already used VR to reduce pain the pain of patients with severe burn injuries [START_REF] Hoffman | Virtual reality as an adjunctive pain control during burn wound care in adolescent patients[END_REF]. However, compared to VR, AR can be easily integrated into the daily life.

Figure 2 :

 2 Figure 2: Heat pain threshold (HPT) in degree Celsius measured with the Method of Limits in the baseline without the use of AR (left) and when participant's were experiencing their own hands burning in AR (right).