Sébastien Banzet
email: sbanzet@crssa.net

Nathalie Koulmann

Nadine Simler

Hervé Sanchez

Rachel Chapot

Bernard Serrurier

André Peinnequin

Xavier Bigard

Banzet Sébastien

Koulmann Nathalie

Simler Nadine

Sanchez Hervé

Chapot Rachel

Serrurier Bernard

Peinnequin André

Sébastien Banzet Crssa

Control of gluconeogenic genes during intense/prolonged exercise: hormone-independent effect of

During intense and prolonged exercise, working skeletal muscles are very dependent on carbohydrates to maintain ATP synthesis and contraction [START_REF] Sherman | Metabolism of sugars and physical performance[END_REF]. In the absence of oral glucose intake, plasma glucose levels are maintained at relatively constant values through changes in hepatic glucose production (HGP). During moderate-intensity exercise, HGP is primarily controlled by portal venous insulin-to-glucagon ratio [START_REF] Wasserman | Hepatic fuel metabolism during muscular work: role and regulation[END_REF][START_REF] Wasserman | Glucagon is a primary controller of hepatic glycogenolysis and gluconeogenesis during muscular work[END_REF], whereas during intense and sustained exercise, several studies have shown that hormonal changes cannot fully account for the increase in HGP, suggesting other unidentified factors are involved [START_REF] Coker | Stimulation of splanchnic glucose production during exercise in humans contains a glucagon-independent component[END_REF][START_REF] Howlett | Glucose production during strenuous exercise in humans: role of epinephrine[END_REF]. A new and original aspect of HGP control during intense exercise as been highlighted recently by a vast series of studies supporting the involvement of muscle-derived factor (for review see [START_REF] Pedersen | Muscle as an endocrine organ: focus on musclederived interleukin-6[END_REF]).

Contracting skeletal muscles produce and release IL-6 in the plasma during prolonged sustained exercise [START_REF] Ostrowski | Evidence that interleukin-6 is produced in human skeletal muscle during prolonged running[END_REF][START_REF] Steensberg | Production of interleukin-6 in contracting human skeletal muscles can account for the exercise-induced increase in plasma interleukin-6[END_REF], and a strong relationship with low muscle glycogen content as been shown [START_REF] Keller | Transcriptional activation of the IL-6 gene in human contracting skeletal muscle: influence of muscle glycogen content[END_REF][START_REF] Steensberg | Interleukin-6 production in contracting human skeletal muscle is influenced by pre-exercise muscle glycogen content[END_REF]. It has been proposed that this cytokine may be a signal of low muscle glycogen stores and, once released in the plasma, could favor hepatic glucose release [START_REF] Febbraio | Muscle-derived interleukin-6: mechanisms for activation and possible biological roles[END_REF].

This hypothesis has been tested in an experiment were subjects exercised with an infusion of saline or recombinant IL-6. The rate of glucose appearance was significantly higher in the IL-6 group with no changes in gluco-regulatory hormones, suggesting IL-6 is involved in HGP increase during intense prolonged exercise [START_REF] Febbraio | Interleukin-6 is a novel factor mediating glucose homeostasis during skeletal muscle contraction[END_REF]. However, the mechanisms involved are unknown and the question was asked whether this effect was due to a direct effect of IL-6 on hepatic tissue or was secondary to IL-6-induced increase in whole-body glucose disposal [START_REF] Febbraio | Interleukin-6 is a novel factor mediating glucose homeostasis during skeletal muscle contraction[END_REF].

Exercise-induced increase in HGP results from coordinated changes in glycogenolysis and gluconeogenesis (GNG) under complex hormonal and nervous control [START_REF] Coker | Glucoregulation during exercise : the role of the neuroendocrine system[END_REF][START_REF] Kjaer | Hepatic metabolism during exercise[END_REF]. Liver glycogenolysis starts at the onset of exercise and is the main source of glucose during the first 30 min of exercise [START_REF] Kjaer | Hepatic metabolism during exercise[END_REF]. Thereafter, hepatic gluconeogenic flux increases and GNG becomes a key contributor to HGP [START_REF] Wasserman | Hepatic fuel metabolism during muscular work: role and regulation[END_REF]. GNG is controlled at the transcriptional level by unidirectional key enzymes, phosphoenolpyruvate-caboxykinase (PEPCK), fructose 1,6 bisphosphatase and glucose-6 phosphatase (G6Pase) the activities of which increase during exercise [START_REF] Dohm | Time course of changes in gluconeogenic enzyme activities during exercise and recovery[END_REF]. PEPCK, the most studied gluconeogenic enzyme, plays an important role during exercise since its pharmacologic inhibition reduces endurance time in rats [START_REF] Hb | Reduced running endurance in gluconeogenesis-inhibited rats[END_REF][START_REF] Turcotte | Glucose kinetics in gluconeogenesis-inhibited rats during rest and exercise[END_REF]. PEPCK activity is hormonally controlled at the transcriptional level [START_REF] Hanson | Regulation of phosphoenolpyruvate carboxykinase (GTP) gene expression[END_REF] and exercise induces an important increase in hepatic PEPCK mRNA levels and enzyme activity [START_REF] Friedman | Role of glucocorticoids in activation of hepatic PEPCK gene transcription during exercise[END_REF][START_REF] Wasserman | Hepatic fuel metabolism during muscular work: role and regulation[END_REF]. Among the welldescribed regions of PEPCK gene promoter, the glucocorticoid response unit (GRU) and two cAMP regulatory element (CRE) are involved in exercise-related gene expression, since their mutation substantially decreased hepatic PEPCK mRNA levels [START_REF] Friedman | Role of glucocorticoids in activation of hepatic PEPCK gene transcription during exercise[END_REF][START_REF] Nizielski | Involvement of transcription factor C/EBP-beta in stimulation of PEPCK gene expression during exercise[END_REF]. However, the molecular mechanisms controlling PEPCK gene expression in response to exercise is still not fully understood. Recently, peroxisome proliferator-activated receptor γ coactivator-1! (PGC-1!) has been shown to play a pivotal role in hepatic GNG regulation and PEPCK gene transcription in response to food deprivation [START_REF] Puigserver | Peroxisome proliferator-activated receptorgamma coactivator 1 alpha (PGC-1 alpha): transcriptional coactivator and metabolic regulator[END_REF][START_REF] Yoon | Control of hepatic gluconeogenesis through the transcriptional coactivator PGC-1[END_REF]. An increase in PGC-1! mRNA levels has been reported recently in exercised mice [START_REF] Hoene | Acute regulation of metabolic genes and insulin receptor substrates in the liver of mice by one single bout of treadmill exercise[END_REF], however its role in hepatic GNG is unknown.

IL-6 binds to a specific receptor, IL-6 receptor ! (IL-6R!) that cooperates with a nonspecific subunit, glycoprotein 130 (gp130). The signal transduction occurs through either the janus kinase (JAK)/signal transducer and activator of transcription (STAT) pathway, or a Ras/extracellular signal-regulated protein kinase (ERK)/CAAT enhancer binding protein (C/EBP) ß pathway (for review see [START_REF] Kamimura | IL-6 signal transduction and its physiological roles: the signal orchestration model[END_REF]). C/EBP family members can trans activate PEPCK promoter via the CRE binding sites [START_REF] Hanson | Regulation of phosphoenolpyruvate carboxykinase (GTP) gene expression[END_REF][START_REF] Park | The role of the CCAAT/enhancer-binding protein in the transcriptional regulation of the gene for phosphoenolpyruvate carboxykinase (GTP)[END_REF]. C/EBPß, also known as nuclear factor of IL-6 (NF-IL6), has been proposed to play a role in PEPCK gene transcription during exercise [START_REF] Nizielski | Involvement of transcription factor C/EBP-beta in stimulation of PEPCK gene expression during exercise[END_REF].

Because this transcription factor is regulated by IL-6 at the transcriptional level and is a downstream signaling molecule of activated IL-6 receptor (reviews in [START_REF] Akira | IL-6-regulated transcription factors[END_REF][START_REF] Ramji | CCAAT/enhancer-binding proteins: structure, function and regulation[END_REF]), muscle-can act directly on hepatic tissue to favor PEPCK gene transcription during exercise has to be explored.

The present study focused on prolonged and intense exercise to further understand glucose homeostasis in very demanding physiologic situations. Because plasma IL-6 concentration is a function of exercise intensity and duration [START_REF] Pedersen | Muscle as an endocrine organ: focus on musclederived interleukin-6[END_REF] and because muscle-IL-6 release is concomitant of GNG increase, we hypothesized that this cytokine could act directly on hepatic tissue to help increasing HGP through GNG. Our aims were 1) to determine if there is an active IL-6 signaling in the liver of animals exposed to one bout of prolonged intense exercise; 2) to determine if IL-6, can directly modulate gluconeogenic genes mRNA abundance, independently of exercise and glucoregulatory hormones. Therefore, we studied liver IL-6 signaling and gluconeogenic genes mRNAs at the end of one bout of exhaustive exercise in rats. To evaluate the ability of IL-6 to directly activate gluconeogenic genes independently of exercise, we studied gluconeogenic genes modulation in response to recombinant IL-6 injection in resting animals.

Research Design and Methods

Ethics

Experiments were carried out in accordance with the Helsinki Accords for Human Treatment of Animals during Experimentation, and received prior approval from the animal ethics committee for animal research of the Centre de Recherche du Service Santé des Armées (La Tronche, France).

Animals

Female Wistar rats (175-200 g) were purchased from Charles River (L'Arbresle, France).

They were housed at 22 ± 2°C, on a 12-12 h light-dark period, and provided with food and water ad libitum. For the exercise experiment they were randomly assigned to four experimental groups: resting rats (Rest, n=8) and 3 active groups, performing one bout of exercise and studied at the end of exercise (Ex, n=8), 2 hours (Ex+2h, n=8) or 6 hours (Ex+6h, n=8) later.

Exercise protocol

All animals were accustomed to running on a rodent treadmill for 15-20 min per day for 5 days (10-20 m.min -1 ; 0° grade). After 4 days at rest, active animals ran on the treadmill (22 m.min -1 , 5° grade) till exhaustion.

IL-6 injection

Animals were fasted overnight and injected intraperitonealy with 3 or 10 µg.kg body weight -1 human recombinant IL-6 (Biosource, Camarillo, CA, USA) and compared to control rats injected with endotoxin free PBS (Cont). Animals were sacrificed 1 hour after injection.

Tissue processing

Resting animals were sacrificed at different time corresponding to the other groups. After exercise, rats from Ex+2h and Ex+6h groups had free access to food and water. Animals were anaesthetized with pentobarbital (70 mg.kg body weight -1 ip). Liver was sampled and a part was frozen in liquid nitrogen and stored at -80°C for protein and glycogen quantification, another part was placed in 1ml RNAlater (Ambion, Austin, TX, USA), kept at 4°C for 24 h and then frozen at -20°C for mRNA study. Whole blood was sampled, centrifuged and plasma was frozen at -80°C. Animals were killed by removal of the heart.

RNA extraction

Hepatic tissue mRNA were isolated using the MagNA Pure LC instrument (Roche Applied Science, Mannheim, Germany). 150 to 200 mg of liver were disrupted in a volume corresponding to 6,66 µl MagNA Pure LC mRNA isolation kit II lysis buffer (Roche Applied Science, Mannheim, Germany) per mg of tissue, with a Polytron mixer (Kinematica, Switzerland). Lysate was centrifuged and 50µl was transferred in a fresh tube and completed to a 300-µl final volume with lysis buffer. Extraction was then performed following the manufacturer's protocol.

Reverse transcription

Reverse transcription was carried out using the Reverse Transcriptase Core Kit (Eurogentec, Seraing, Belgium). The reaction was performed following the manufacturer's instructions.

The reaction mix contained 1 µl buffer (10X), 2 µl dNTP (2.5 mM each dNTP), 0.5 µl oligo-dT primer (50 µM), 0.2 µl RNase inhibitor (20 units.µl -1 ), 2 µl MgCl 2 (25 mM), 0.25 µl Euroscript reverse transcriptase (50 U.µl -1 ), 1.05 µl RNase free water and 3 µl template mRNA.

Real-time PCR

The PCR reactions were carried out in a 20µl final volume with the LC Fast Start DNA Master SYBR Green kit (Roche Applied Science, Mannheim, Germany) using 2 µl of cDNA solution (1/20 dilution). Quantitative PCRs were performed using (Roche Applied Science, Mannheim, Germany) with 50 cycles of 95 °C for 20 s, annealing temperature (table 1) for 5 s and 72 °C for 8 s. Quantification was performed with the comparative threshold cycle method (28) using RelQuant software (Roche Diagnostics), with a pool of cDNA samples as calibrator. Five housekeeping genes were necessary to validate the normalization using genorm software [START_REF] Vandesompele | Accurate normalization of real-time quantitative RT-PCR data by geometric averaging of multiple internal control genes[END_REF]. These were Hypoxanthine-guanine phosphoribosyl transferase (HPRT), cyclophylin A (CycA), acidic ribosomal binding protein (ARBP), actin ß and glyceraldehydes-3-phosphate dehydrogenase (GAPDH). Primers characteristics are provided as supplemental data. The final quantification was performed using the geometric averaging of the five quantifications. In each experiment, all samples were treated at the same time for mRNA extraction, reverse transcription and PCR reaction.

Western blotting

Liver sections were homogenized in ice-cold lysis buffer (20 µl.mg -1 ) containing 20 mM HEPES, pH 7.4, 2 mM EGTA, 50 mM "-glycerophosphate, 1 mM dithiothreitol, 1 mM Na 3 VO 4 , 1% Triton X-100, 10% glycerol, 1 µl.ml -1 Protease inhibitor cocktail set III (Calbiochem, Fontenay-sous-bois, France). Samples were rotated for 1 h at 4°C and centrifuged (15,000 g, 15 min, 4°C). Protein content was determined using the BCA method (Roche/Hitachi 912 instrument, Roche Diagnostics, Mannheim, Germany).

Each sample (50 µg of total proteins) was separated by SDS-PAGE (10% resolving gel).

Proteins were transferred electrophoretically on nitrocellulose membranes (Hybond C-extra RPN 2020E, Amersham, Orsay, France). Membranes were blocked with non-fat milk in Trisbuffered-Saline solution added with 0.1% Tween 20 (TBST). Blots were incubated overnight at 4°C with anti-phospho-STAT-3 (Ser727) or anti-phospho-C/EBPβ (Ser105) antibody (Cell Signaling Technology, Beverly, MA, USA), rinsed with TBST and incubated with horse radish peroxidase-conjugated donkey anti-rabbit antibody (Santa Cruz Biotechnology, Santa Cruz, CA, USA) for 1h30. Blots were subjected to Enhanced cheluminescence reagent kit (Amersham, Orsay, France) and exposed to Hyperfilm ECL RPN 3103K (Amersham, Orsay, France). Membranes were then stripped in a solution containing 60 mM Tris-HCL, pH 7.4, 5% SDS and 0.07% "-mercapto ethanol, for 30 min at 50°C, washed with TBST, and incubated overnight with anti STAT-3 antibody or C/EBPβ (Cell Signaling Technology, Beverly, MA, USA) and treated as described above. The bands obtained were quantified using a densitometer (GS-700, Bio-Rad, Marnes-la-Coquette, France).

Liver and muscle glycogen content

Liver or plantaris muscle sections (30 mg or 20 mg respectively) were disrupted in NaOH (2 M). Samples were submitted to amyloglucosidase (Sigma-Aldrich, Steinheim, Germany) in acetate buffer (0.3 M) for 2 hours at 37°C to digest glycogen. Released glucose was quantified by spectrophometric measurement (340 nm) of NADH production in the presence of hexokinase and glucose-6-phosphate dehydrogenase (Sigma-Aldrich, Steinheim, Germany), as previously described (4).

Blood glucose

Blood glucose was measured according to the glucose oxidase methods with Glucose GOP-PAP kit on Roche/Hitachi 912 instrument (Roche Diagnostics, Mannheim, Germany).

IL-6 protein

rat IL-6 (R&D systems, Lille, France) in exercise experimentation and sandwich ELISA kit for human IL-6 (R&D systems, Lille, France) in recombinant IL-6 injected rats.

Hormones

Plasma hormones were measured using BioPlex Instrument (Biorad, France) with rat endo kit (Millipore, France) for glucagon and insulin, rat stress kit for corticosterone.

Statistical analysis

All data are presented as means ± S.E.M. A one factor ANOVA was used to evaluate the global effects of exercise and recovery, as well as recombinant IL-6 injection. When appropriate, the Newman-Keuls post hoc test was used for inter-group comparisons. Animals weight and running time values were compared using an unpaired Student t test. The significance level for all comparisons was set at P<0.05.

Results

Liver response to exercise

Endurance time: Mean endurance time was 123±10 min with no differences between groups (data not shown).

Carbohydrate metabolism: Plasma glucose was lower in Ex group than Rest and Ex+2h (8.25 ± 0.33; 6.71 ± 0.51 and 8.26 ± 0.34 mmol.l -1 respectively, data not shown) (P<0.05). Plantaris muscle glycogen content decreased immediately at the end of exercise (Rest: 24.11 ± 0.74; Ex: 3,37 ± 0.6 µmol glycosyl units.g -1 , datanot shown) (P<0.01). There was a profound decrease in hepatic glycogen in response to exercise (main effect P<0.001), with only 5% of the resting values at the end of exercise (P<0.001). Liver glycogen increased during recovery, but was still 35% lower in Ex+6h than in Rest group (P<0.001) (fig 1A). Plasma glucose and gluco-regulatory hormones: Recombinant IL-6 had no effect on plasma glucose (data not shown). There were no changes in plasma insulin/glucagon ratio (15.07 ± 3.42, 13.03 ± 2.94 and 14.03 ± 1.69 for Cont, 3 µg.kg -1 and 10 µg.kg -1 groups, respectively) and corticosterone was unchanged (252.27 ± 38.24 ng.ml -1 , 254.81 ± 35.12 ng.ml -1 and 247.48 ± 49.16 ng.ml -1 for Cont, 3 µg.kg -1 and 10 µg.kg -1 groups, respectively).

Correlation between PEPCK and IL-6-induced genes mRNAs:

Discussion

During prolonged intense exercise, a strong relationship between muscle IL-6 production and carbohydrate availability has been described [START_REF] Pedersen | Muscle as an endocrine organ: focus on musclederived interleukin-6[END_REF] and IL-6 is able to increase endogenous glucose production in exercising humans [START_REF] Febbraio | Interleukin-6 is a novel factor mediating glucose homeostasis during skeletal muscle contraction[END_REF]. However, the mechanisms involved in IL-6 action are unknown. In this work, we show that: 1) prolonged intense exercise is associated to an active IL-6 signaling in the liver, involving both STAT-3 and C/EBPß pathways; 2) Recombinant IL-6 can acutely modulate hepatic PEPCK mRNA levels independent from exercise and gluco-regulatory hormones.

We previously reported that exhaustive running exercise protocol in rats induces an increase in muscle IL-6 transcription and in plasma IL-6 concentration (3). This was confirmed in the present study where plasma IL-6 peaked at the end of exercise and decreased as soon as 2 hours recovery, as described in human [START_REF] Ostrowski | Evidence that interleukin-6 is produced in human skeletal muscle during prolonged running[END_REF]. However, whether such a transient increase in circulating IL-6 can induce an active IL-6 signaling in the liver has never been assessed. When IL-6 binds to its receptor, signal transduction can involve STAT-3 and/or C/EBPß transcription factors to promote gene transcription [START_REF] Akira | IL-6-regulated transcription factors[END_REF]. In the present study we show that in addition to increased hepatic C/EBPß phosphorylation, STAT-3 phosphorylation substantially raised at the end of exercise. Therefore, plasma IL-6 concentration and IL-6activated transcription factors phosphorylation followed the same time course. When phosphorylated, both transcription factors are translocated in the nucleus (1), and to determine if transcription factors activation was effective in altering transcription in our experiment, we measured hepatic mRNA levels for IL-6-induced genes. SOCS-3, a cytokine signaling suppressor molecule, is highly induced at the transcriptional level by IL-6 [START_REF] Fujimoto | Regulation of cytokine signaling by SOCS family molecules[END_REF]. Here, we found an important increase in liver SOCS-3 mRNA in response to exercise, still significant after 2 hours recovery. Although commonly used to assess IL-6 activity, SOCS-3 is not fully specific of IL-6 and we studied IL-6R and C/EBPß, two IL-6-inducible genes in the liver in vivo [START_REF] Akira | A nuclear factor for IL-6 expression (NF-IL6) is a member of a C/EBP family[END_REF][START_REF] Saito | Molecular cloning of a murine IL-6 receptor-associated signal transducer, gp130, and its regulated expression in vivo[END_REF]. We found that mRNA levels for both genes were transiently activated at the end of exercise. Taken together, concomitant elevation in plasma IL-6, the increase in its transcription factors phosphorylation and IL-6-responsive genes activation in the liver strongly suggests that prolonged intense exercise can acutely induce an active IL-6 signaling in the liver. This result is consistent with the hepato-splanchnic removal of IL-6 previously described in exercising human [START_REF] Febbraio | Hepatosplanchnic clearance of interleukin-6 in humans during exercise[END_REF]. IL-6 increases endogenous glucose production during exercise by an unknown, hormone-independent mechanism [START_REF] Febbraio | Interleukin-6 is a novel factor mediating glucose homeostasis during skeletal muscle contraction[END_REF] and because we found an active signaling in the liver, we hypothesized that IL-6 could act directly on hepatic tissue to modulate HGP during prolonged and intense exercise.

As expected in an exhaustive running exercise, plasma glucose was slightly lower at the end of exercise and gluco-regulatory hormones measurement showed a drop in plasma insulin/glucagon ratio and an increase in plasma corticosterone at the end of exercise.

There was a 86% decrease in plantaris muscle glycogen levels. We found both a profound decrease in hepatic glycogen content (95%) and a substantial increase in liver key gluconeogenic enzymes mRNA, PEPCK and G6Pase. PEPCK is regulated at the transcriptional level and increased mRNA levels during exercise are associated to increased enzyme activity [START_REF] Friedman | Role of glucocorticoids in activation of hepatic PEPCK gene transcription during exercise[END_REF][START_REF] Wasserman | Hepatic fuel metabolism during muscular work: role and regulation[END_REF]. Interestingly, we found a concomitant and substantial increase in PGC-1α mRNA, a key regulator of fasting-induced hepatic GNG [START_REF] Yoon | Control of hepatic gluconeogenesis through the transcriptional coactivator PGC-1[END_REF]. Taken together, these data confirm that our model of long lasting exercise is a challenge for glucose homeostasis and that HGP is very dependent on GNG at the end of exercise. Muscle IL-6 gene transcription starts early during exercise, but IL-6 release in the plasma is delayed [START_REF] Helge | The effect of graded exercise on IL-6 release and glucose uptake in human skeletal muscle[END_REF][START_REF] Keller | Transcriptional activation of the IL-6 gene in human contracting skeletal muscle: influence of muscle glycogen content[END_REF].

Because circulating IL-6 concentration peaks at the end of exercise when GNG becomes crucial to maintain plasma glucose levels, and because this pathway is regulated at the transcriptional level, we hypothesized that IL-6 could favor HGP by modulating gluconeogenic genes mRNA levels, particularly PEPCK. PEPCK control during exercise is complex and in order to understand if IL-6 is able to acutely modulate gluconeogenic genes independent of exercise-related factors, we injected 2 different doses of recombinant IL-6 to inactive rats and explored hepatic PEPCK and G6Pase mRNA levels one hour after injection. Injected doses were calculated to mimic exerciseinduced plasma concentration. To avoid post-absorptive GNG blockade by insulin, animals were fasted overnight. Injections resulted in a rise in circulating IL-6, with final plasma concentrations flanking concentrations observed at the end of exhaustive running exercise. As expected, we found a dose dependent increase in hepatic SOCS-3 and IL-6R mRNA levels reflecting IL-6 signaling activation in the liver. This stimulation induced a dose dependent increase in PEPCK mRNA levels, consistent with in vitro results [START_REF] Blumberg | Interleukin-6 stimulates gluconeogenesis in primary cultures of rat hepatocytes[END_REF]. During food deprivation, increased hepatic PEPCK mRNA levels result from both increased gene transcription and mRNA stabilization through AMPc [START_REF] Hanson | Regulation of phosphoenolpyruvate carboxykinase (GTP) gene expression[END_REF]. In the present work, we did not measure gene transcription, however the important and rapid increase in mRNA levels suggests that PEPCK transcripts are stabilized in response exercise and IL-6 injection. No changes in PGC-1α and G6Pase mRNA abundance were found. Since G6Pase catalyses the last step of glucose production, allowing glucose to leave the hepatocyte, this enzyme is common to both GNG and glycogenolysis. In our experiment, rats were fasted overnight (approximatively 14 hours). In this situation, gluconeogenesis accounts for only 40-50% of hepatic glucose output [START_REF] Katz | Gluconeogenesis and the Cori cycle in 12-, 20-, and 40-hfasted humans[END_REF][START_REF] Landau | Contributions of gluconeogenesis to glucose production in the fasted state[END_REF], whereas glycogenolysis is fully induced after 4 to 6 hours food deprivation [START_REF] Cahill | Fuel Metabolism in Starvation[END_REF]. Therefore, a full activation of PEPCK gene is not expected, whereas G6Pase was probably already activated, and no additive effect of IL-6 injection was found.

An increase in liver PGC-1α mRNA levels at the end of 1 hour exercise as been previously reported [START_REF] Hoene | Acute regulation of metabolic genes and insulin receptor substrates in the liver of mice by one single bout of treadmill exercise[END_REF] and we show here that this is a transient rise. Whether newly transcribed PGC-1α is involved in PEPCK transcription during exercise is not known and need to be explored, however our results show that IL-6 effect on PEPCK gene is independent of any effect on PGC-1α mRNA levels. Plasma insulin/glucagon ratio and corticosterone were unchanged in IL-6 injected rats, suggesting that gluco-regulatory hormones do not mediate IL-6 effect. We found that PEPCK mRNA levels correlated with SOCS-3 and IL-6R mRNA levels in control and exercised rats and this correlation was stronger in IL-6 injected rats, further suggesting a co-regulation by IL-6 (Fig 8A and8B). Differences observed between the two situations may be due to the fact that hormonal factors are the main regulators of hepatic PEPCK mRNA levels during exercise.

Independently of the debate concerning IL-6 and insulin sensitivity, the effect of IL-6 on HGP is still controversial. This cytokine has been described in vivo to whether activate [START_REF] Stith | Endocrine and carbohydrate responses to interleukin-6 in vivo[END_REF][START_REF] Tsigos | Dose-dependent effects of recombinant human interleukin-6 on glucose regulation[END_REF] or inhibit HGP [START_REF] Christ | Mechanism of the impairment of the glucagon-stimulated phosphoenolpyruvate carboxykinase gene expression by interleukin-6 in rat hepatocytes: inhibition of the increase in cyclic 3',5' adenosine monophosphate and the downstream cyclic 3',5' adenosine monophosphate action[END_REF][START_REF] Inoue | Role of hepatic STAT3 in brain-insulin action on hepatic glucose production[END_REF]. These differences could be explained by very different environment in term of gluco-regulatory hormones, cytokines, gluconeogenic substrates, but also by very different stimulation times (ranging from hours to days) and injected doses. Two studies have investigated the effect of one ip injection of IL-6 in overnight fasted rodents, and found no effect on PEPCK gene transcription [START_REF] Hill | Altered transcriptional regulation of phosphoenolpyruvate carboxykinase in rats following endotoxin treatment[END_REF][START_REF] Ramadoss | STAT3 targets the regulatory regions of gluconeogenic genes in vivo[END_REF]. The discrepancy between these results and the increase in PEPCK mRNA we report could be explained by a substantially higher hepatic IL-6 signaling activation in previous studies. Indeed, in the first study, authors injected high IL-6 doses to mimic the same acute phase response as an acute inflammatory agent (turpentine) and stimulation time was longer [START_REF] Hill | Altered transcriptional regulation of phosphoenolpyruvate carboxykinase in rats following endotoxin treatment[END_REF]. In the second study [START_REF] Ramadoss | STAT3 targets the regulatory regions of gluconeogenic genes in vivo[END_REF], despite similar fasting time, doses and stimulation time, the authors report a 30 fold increase in hepatic SOCS-3 mRNA whereas we found a 12 or 4.2 fold increase in exercise and injection studies respectively. Thus, it appears that acute effects of IL-6 on gluconeogenic genes are very dependent on circulating concentrations. In our experiment, animals were fasted to mimic exercise-induced variations in hormones and IL-6 was injected alone. Furthermore, animals were sacrificed only 1h after injection to mimic the very transient IL-6 stimulation we observed during exercise and plasma concentrations were similar to concentrations observed in exhaustive exercise. Therefore, although we did not directly measure HGP or gluconeogenic flux, our results are consistent with the positive effect of IL-6 on HGP during prolonged exercise in human [START_REF] Febbraio | Interleukin-6 is a novel factor mediating glucose homeostasis during skeletal muscle contraction[END_REF], and strongly suggest that IL-6 is able to directly modulate liver PEPCK mRNA to increase HGP through GNG.

Plasma IL-6 concentrations obtained in our IL-6-injected animals reach the highest levels described in human for intense and prolonged exercise [START_REF] Ostrowski | Evidence that interleukin-6 is produced in human skeletal muscle during prolonged running[END_REF]. However, rats injected with 3 or 10µg.kg -1 recombinant IL-6 had only a 2 and 2.7 fold increase in hepatic PEPCK mRNA (versus 5.5 at the end of prolonged intense exercise). This further confirms that muscle-derived IL-6 cannot be considered a major regulator of glucose homeostasis during exercise but rather a positive modulator of HGP, acutely supporting hormonal control in very demanding exercises [START_REF] Febbraio | Interleukin-6 is a novel factor mediating glucose homeostasis during skeletal muscle contraction[END_REF].

Conclusion

Prolonged intense exercise is a very challenging physiological situation for wholebody glucose homeostasis. IL-6 has been shown to increase endogenous glucose production during exercise in human through an unknown mechanism. Here we show that exhaustive running exercise in rats induces an increase in plasma IL-6 and an active IL-6 signaling in the liver. Because HGP was very dependent on GNG at the end of exercise, and GNG is regulated at the transcriptional level, we tested IL-6 ability to modulate gluconeogenic genes mRNA abundance in resting rats. We found a dose dependent increase in PEPCK mRNA levels in response to IL-6 injection, with no changes in glucoregulatory hormones. Taken together, our results suggest that muscle-derived IL-6 could contribute increasing HGP by directly upregulating PEPCK during intense and prolonged exercise.

Legends

  Gluconeogenic genes: Prolonged running exercise induced an increase in both PEPCK and G6Pase mRNA levels (main effects P<0.001) with a maximum at the end of exercise (P<0.001). Mean mRNA values decreased to values measured in Rest group as soon as 2h after exercise (fig 1B and 1C). Changes in PGC-1α mRNA levels exhibited a similar evolution (P <0.0001) (fig 1D). Glucoregulatory hormones: Exercise resulted in a prolonged decrease in plasma insulin/glucagon ratio (main effect P<0.01), values being lower than control in Ex and Ex+2h groups (P<0.05) (fig 2A). Plasma corticosterone concentrations increased in response to running (main effect P<0.05), peaking at the end of exercise (fig 2B). Concentrations increased 6h after exercise (P<0.05) (fig 2B). Plasma IL-6: Exercise induced an increase in plasma IL-6 protein (main effect P<0.05) peaking at the end of exercise (P<0.01). Plasma IL-6 concentrations decreased to values measured in resting animals as soon as 2h after exercise (fig 3A). IL-6 signaling in liver tissue: Liver STAT-3 and C/EBPß phosphorylation ratio significantly increased in response to exercise (global effects P<0.05) (fig 3B and C), with a maximum at the end of exercise (P<0.05 and P<0.01 respectively). increased (main effects P<0.001) peaking at the end of exercise (P<0.001) (fig 4A and B). SOCS-3 mRNA levels were still slightly elevated at Ex+2h compared to resting values (P<0.05). Interestingly, we found a 3.5 fold increase in IL-6R mRNA at the end of exercise (P<0.05) (fig 4C). Correlation between PEPCK and IL-6-induced genes mRNAs in Rest an Exercise groups: We found significant correlations between PEPCK mRNA levels and two IL-6-induced genes mRNA levels, SOCS-3 (R 2 = 0.5546 ; P<0.05) (fig 7A) and IL-6R (R 2 = 0.7707 ; P<0.01) (fig 7B). Liver response to recombinant IL-6 IL-6 responsive genes in liver tissue: Recombinant IL-6 injection resulted in a dose dependent peak of recombinant IL-6 plasma levels (fig 5A) (P<0.01) and induced an increase in both SOCS-3 and IL-6R mRNA levels (P<0.01 and P<0.05 respectively) (fig 5B and C).

  Gluconeogenic genes: Recombinant IL-6 injection induced a dose dependent increase in hepatic PEPCK mRNA levels (main effect P<0.01) (fig 6A). No effects of IL-6 injection was found on G6Pase and PGC-1α mRNA levels (fig 6B and C).

  We found significant correlations between PEPCK mRNA levels and two IL-6-induced genes mRNA levels, SOCS-3 (R 2 = 0. 5764 ; P<0.01) (fig 7C) and IL-6R (R 2 = 0. 6094 ; P<0.01) (fig 7D).

Figure 1 :

 1 Figure 1: Liver carbohydrate metabolism in response to exhaustive exercise Liver glycogen content (A), PEPCK (B), G6P (C) and PGC-1α (D) mRNA levels were measured in resting rats (Rest), at the end of exercise (Ex), 2 hours (2h) and 6 hours (6h) after the end of exercise. * : significantly different from the other groups, P<0.001.

Figure 2 :

 2 Figure 2: Glucoregulatory hormones in response to exhaustive exercise Plasma insulin and glucagons ratio (A) and corticosterone levels (B) were measured in resting rats (Rest), at the end of exercise (Ex), 2 hours (2h) and 6 hours (6h) after the end of exercise. # : significantly different from Rest group, P<0.05 ; ## : significantly different from Rest group, P<0.01.

Figure 3 :

 3 Figure 3: Liver IL-6 signaling in response to exhaustive exercise Plasma IL-6 protein level (A), liver total and phospho STAT-3 (B) and C/EBPβ (C) were measured and phospho/total ratio was calculated in resting rats (Rest), at the end of exercise (Ex), 2 hours (2h) and 6 hours (6h) after the end of exercise. * : significantly different from the other groups, P<0.05 ; * * : significantly different from the other groups, P<0.01.

Figure 4 :

 4 Figure 4: Liver IL-6-inductible genes mRNA levels in response to exhaustive exercise Liver SOCS-3 (A), C/EBP-β (B) and IL-6R (C) mRNA levels were measured in resting rats (Rest), at the end of exercise (Ex), 2 hours (2h) and 6 hours (6h) after the end of exercise. * : significantly different from the other groups, P<0.001; # : significantly different from Rest group, P<0.05.

Figure 5 :

 5 Figure 5: Plasma rh IL-6 protein levels, liver SOCS-3 and IL-6R mRNA levels in response to recombinant IL-6 injection. Plasma recombinant human IL-6 protein levels (A) Liver SOCS-3 (B) and IL-6R (C) mRNA levels were measured in resting fasted rats 1 hour after recombinant IL-6 injection (3 or 10µg.kg bodyweight -1 ) or an equal volume of PBS buffer (Cont). * : significantly different from Cont group, P<0.05 ; # : significantly different from 3µg.kg -1 group, P<0.05.

Figure 6 :

 6 Figure 6: Liver gluconeogenic genes mRNA levels in response to recombinant IL-6 injection. Liver PEPCK (A), G6P (B), PGC-1α (C) mRNA levels were measured in resting fasted rats 1 hour after recombinant IL-6 injection (3 or 10µg.kg bodyweight -1 ) or an equal volume of PBS buffer (Cont). * : significantly different from Cont group, P<0.05; * * : significantly different from Cont group, P<0.01.

Figure 7 :

 7 Figure 7: Association between IL-6 responsive genes and PEPCK mRNA in liver in response to exercise and to recombinant IL-6 injection.

Gene

Table 1


Pascal Koiran December
email: pascal.koiran@ens-lyon.fr

A Hitting Set Construction, with Applications to Arithmetic Circuit Lower Bounds

A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form P t j=0 cj X α j (a + bX) β j . From our algorithm we derive an exponential lower bound for representations of polynomials such as Q 2 n i=1 (X i -1) under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the "hardness from derandomization" approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers.

Introduction

The large body of work on hardness versus randomness tradeoffs shows that the two tasks of proving lower bounds and derandomizing algorithms are roughly equivalent. This equivalence holds both in the boolean and arithmetic world. We focus here on the arithmetic world [START_REF] Kabanets | Derandomizing polynomial identity test means proving circuit lower bounds[END_REF]. The equivalence between lower bounds and derandomization suggests a new approach to lower bounds (see e.g. [START_REF] Kabanets | Derandomizing polynomial identity test means proving circuit lower bounds[END_REF][START_REF]Proving lower bounds via pseudo-random generators[END_REF]): let us derandomize algorithms first, and much-coveted lower bounds will follow. This "hardness from derandomization" approach is very appealing, but apparently has not yet led to many new lower bound results. There have been some recent advances in derandomization, however, especially for identity testing of small-depth arithmetic circuits, e.g. [START_REF] Saxena | An almost optimal rank bound for depth-3 identities[END_REF][START_REF] Karnin | Black box polynomial identity testing of depth-3 arithmetic circuits with bounded top fan-in[END_REF] and for the more difficult problem of blackbox circuit reconstruction [START_REF] Karnin | Reconstruction of generalized depth-3 arithmetic circuits with bounded top fan-in[END_REF]. Also techniques have been developed for obtaining simultaneously lower bounds and identity tests [START_REF] Saxena | Diagonal circuit identity testing and lower bounds[END_REF], thereby reinforcing the intuition that these two problems are intimately connected.

In this paper we use the "hardness from derandomization" approach to obtain lower bounds for a certain class of arithmetic circuits. More precisely, we prove lower bounds for representations of univariate polynomials under the form t j=0 c j X α j (a + bX)

β j , (1)
where the c j , a and b are rational numbers. Polynomials of this form were first considered in [START_REF] Kaltofen | On the complexity of factoring bivariate supersparse (lacunary) polynomials[END_REF] due to their role in the factorization of sparse bivariate polynomials. Indeed, such an expression vanishes identically if and only if Y -a -bX is a linear factor of the bivariate polynomial t j=0 c j X α j Y β j . Obviously, any univariate polynomial can be expressed under form (1) by expanding it as a sum of monomials (the resulting β j are all 0). Representation [START_REF] Koiran December | A possible pseudorandom generator against arithmetic circuits[END_REF] can potentially be much more compact than the "sum of monomials" representation, however, due to the presence of the possibly large exponents α j and β j (note that X α j can be computed in about log α j multiplications by repeated squaring; the same trick applies of course to (a+bX) β j ). The presence of possibly large exponents makes lower bounds and deterministic identity testing nontrivial.

Lower Bound Statement

A simple version of our lower bound result is as follows.

Theorem 1. Consider a family of polynomials (P n ) of the form

P n = Nn i=1 (X i -1). (2)
Assume that P n can be expressed under form (1) with t polynomially bounded in n and the bit sizes of the c j , α j and β j polynomially bounded in n. Then N n must be polynomially bounded in n as well.

We define the bit size of c j as the sum of the bit sizes of its numerator and denominator. Note that there is no restriction on the size of the coefficients a and b in this theorem (they may grow arbitrarily fast as a function of n). Here we have expressed our result as a function of a single parameter n for the sake of clarity. We give in Theorem 4 a more precise (and slightly more general) lower bound where the dependency on each parameter is worked out carefully. In particular, we work with the projective height H(c) of the tuple c = (c j ). This is a more appropriate notion of "size" of c than the naive bit size used in Theorem 1. The projective height is defined in Section 2.2.

The "obvious" arguments such as degree comparison between (1) and ( 2) only show that N n must be exponentially bounded in n. Theorem 1 should therefore be viewed as an exponential lower bound. One can also see the exponential nature of our lower bound by considering the polynomials 2 n i=1 (X i -1): it follows from Theorem 4 that for some constant ǫ > 0, these polynomials cannot be expressed under form (1) if t and the bit sizes of the c j , α j and β j are bounded by 2 ǫn .

We note that the polynomials P n were suggested by Agrawal as good candidates for proving lower bounds. As observed by Agrawal [START_REF] Koiran December | A possible pseudorandom generator against arithmetic circuits[END_REF], if it could be shown that P n is hard to compute by general arithmetic circuits, it would follow that the permanent is hard to compute by arithmetic circuits. This also follows from a general result (Theorem 5 of [START_REF] Koiran | Interpolation in Valiant's theory[END_REF], see also [START_REF] Bürgisser | On defining integers in the counting hierarchy and proving lower bounds in algebraic complexity[END_REF]) which roughly speaking shows the following: if the permanent has polynomial size arithmetic circuits then exponential-size products of easy-to-compute polynomials are themselves easy to compute.

Note also that there is a formal similarity between (2) and the wellknown Pochhammer-Wilkinson polynomial n i=1 (X -i) where roots of unity are replaced by integers. The Pochhammer-Wilkinson polynomial is widely conjectured to be hard to compute [START_REF] Bürgisser | On implications between P-NP hypotheses: decision versus computation in algebraic complexity[END_REF][START_REF] Bürgisser | On defining integers in the counting hierarchy and proving lower bounds in algebraic complexity[END_REF][START_REF] Lipton | Straight-line complexity and integer factorization[END_REF][START_REF] Shub | On the intractability of Hilbert's Nullstellensatz and an algebraic version of "P=NP[END_REF]. As explained in Section 6, it is possible to obtain a good lower bound for representations of this polynomial under form (1).

Main Ideas and Connections to Previous Work

Our lower bound is based on the construction of hitting sets for polynomials of the form [START_REF] Koiran December | A possible pseudorandom generator against arithmetic circuits[END_REF]. Recall that a hitting set H for a set F of polynomials is a (finite) set of points such that there exists for any non-identically zero polynomial f ∈ F at least one point a ∈ H such that f (a) = 0. Hitting sets are sometimes called "correct test sequences" [START_REF] Heintz | Testing polynomials which are easy to compute[END_REF]. By a natural abuse of notation, we will sometimes say that H is a hitting set for a polynomial f if it a hitting set for the singleton {f }.

The existence of polynomial size hitting sets for general arithmetic circuits follows from standard probabilistic arguments. A much more difficult problem is to give explicit (deterministic) constructions of small hitting sets. It is easy to see that this problem is equivalent to blackbox deterministic identity testing: any hitting set for H yields an obvious black-box identity testing algorithm (declare that f ≡ 0 iff f evaluates to 0 on all the points of H); conversely, assuming that F contains the identically zero polynomial, the set of points queried by a black box algorithm on the input f ≡ 0 must be a hitting set for F.

There is a general connection between lower bounds and derandomization of polynomial identity testing [START_REF] Kabanets | Derandomizing polynomial identity test means proving circuit lower bounds[END_REF]. This connection is especially apparent in the case of black-box derandomization. Namely, let H be a hitting set for F. The polynomial P = a∈H (X -a) cannot belong to F since it is nonzero and vanishes on H. The same remark applies to all nonzero multiples of P . If F is viewed as some kind of "complexity class", we have therefore obtained a lower bound against F by exhibiting a polynomial P which does not belong to F. This connection between hitting sets and arithmetic lower bounds has been known for at least 30 years [START_REF] Heintz | Testing polynomials which are easy to compute[END_REF], but has led to suprisingly few lower bound results. 1 To the best of our knowledge, only one lower bound of this type is known: Agrawal [2, Corollary 65] has shown that certain multilinear polynomials cannot be computed by circuits with unbounded fanin addition gates of size n 2-ǫ and depth (2 -ǫ) log n. The lower bound applies to polynomials with coefficients computable in PSPACE (this complexity class was independently defined in [START_REF] Koiran | VPSACE and a transfer theorem over the reals[END_REF], where it is called VPSPACE; further results on this class and other space-bounded classes in Valiant's model can be found in [START_REF] Koiran | VPSPACE and a transfer theorem over the complex field[END_REF][START_REF] Mahajan | Small-space analogues of Valiant's classes[END_REF][START_REF] Poizat | À la recherche de la définition de la complexité d'espace pour le calcul des polynômes à la manière de Valiant[END_REF]).

We have pointed out in Section 1.1 that a lower bound for P n against general arithmetic circuits would imply a lower bound for the permanent. For the same reason (Theorem 5 of [START_REF] Koiran | Interpolation in Valiant's theory[END_REF]), a hitting set construction against general arithmetic circuits would imply a lower bound for the permanent.

Our hitting set construction builds on work from [START_REF] Kaltofen | On the complexity of factoring bivariate supersparse (lacunary) polynomials[END_REF][START_REF] Kaltofen | Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields[END_REF]. In [START_REF] Kaltofen | On the complexity of factoring bivariate supersparse (lacunary) polynomials[END_REF] we designed a deterministic identity testing algorithm for expressions of the form (1) as an intermediate step toward an algorithm for the factorization of "supersparse" bivariate polynomials. Our identity testing algorithm was not black-box. Rather, it was based on a structure theorem (a so-called "gap theorem") which makes it possible to recognize easily identically zero expressions. Here we build on this work to construct hitting sets. These sets turn out to be made of roots of unity, explaining why we obtain a lower bound for polynomials of the form [START_REF]Proving lower bounds via pseudo-random generators[END_REF].

In terms of the class of arithmetic circuits studied, the work which seems closest to ours is by Saxena [START_REF] Saxena | Diagonal circuit identity testing and lower bounds[END_REF]. He gives lower bounds and identity testing algorithms for "diagonal circuits", i.e., sums of powers of (multivariate) linear functions, and more generally for sums of products of a small number of powers of linear functions. Our circuits fall in this category since they compute sums of products of two powers of linear functions. Our results and methods are quite different, however. He uses non-black-box methods, whereas we use black-box methods. Moreover, his lower bounds break down for powers of high degree whereas we can handle high degree powers (indeed, for univariate polynomials the only challenge is to prove lower bounds for polynomials of high degree since any low degree polynomial can be represented efficiently as a sum of monomials, assuming that field constants are given for free).

Organization of the paper

As in [START_REF] Kaltofen | On the complexity of factoring bivariate supersparse (lacunary) polynomials[END_REF][START_REF] Kaltofen | Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields[END_REF] we use number-theoretic techniques and in particular properties of the height of algebraic numbers. Some background on the height function is provided in Section 2. Section 3 is technical: we obtain a height lower bound which we use in Section 4 to construct our hitting sets. From there, the lower bound theorem of Section 5 follows easily from the approach outlined in Section 1.2. Finally, we suggest some possible extensions of our results in Section 6.

Number Theory Background

In this section we provide some background on the height function, first for algebraic numbers and then more generally for points in projective space.

Heights of Algebraic Numbers

For any prime number p, the p-adic absolute value on Q is characterized by the following properties: |p| p = 1/p, and |q| p = 1 if q is a prime number different from p. For any x ∈ Q \ {0}, |x| p can be computed as follows: write x = p α y where p is relatively prime to the numerator and denominator of y, and α ∈ Z. Then |x| p = 1/p α (and of course |0| p = 0). We denote by M Q the union of the set of p-adic absolute values and of the usual (archimedean) absolute value on Q.

Let d, e ∈ Z be two non-zero relatively prime integers. By definition, the height of the rational number d/e is max(|d|, |e|). There is an equivalent definition in terms of absolute values:

for x ∈ Q, H(x) = ν∈M Q max(1, |x| ν ). Note in particular that H(0) = 1.
More generally, let K be a number field (an extension of Q of finite degree). The set M K of normalized absolute values is the set of absolute values on K which extend an absolute value of M Q . For ν ∈ M K , we write ν|∞ if ν extends the usual absolute value, and ν|p if ν extends the p-adic absolute value. One defines a "relative height" H K on K by the formula

H K (x) = ν∈M K max(1, |x| ν ) dν . (3)
Here d ν is the so-called "local degree". For every p (either prime or infinite),

ν|p d ν = [K : Q]. The absolute height H(x) of x is H K (x) 1/n , where n = [K : Q].
It is independent of the choice of K. The above material is standard in algebraic number theory. More details can be found for instance in [START_REF] Lang | Algebra[END_REF] or [START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF]. We will also need a special case of a result due Amoroso and Zannier and already used in [START_REF] Kaltofen | Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields[END_REF].

Lemma 1. Let θ be a root of unity and a, b ∈ Q such that α = a + bθ is not a root of unity. If α = 0 we have H(α) ≥ C where C > 1 is an absolute constant.

Proof. This follows from Theorem 1.1 of [START_REF] Amoroso | A relative Dobrowolski lower bound over Abelian varieties[END_REF] since the cyclotomic extension Q(θ) is Abelian over Q (see for instance [START_REF] Van Der Waerden | Moderne Algebra[END_REF], Section 8.4).

Projective Height

One can define a notion of (relative) height for a point c = (c 0 , . . . , c t ) in K t+1 by the formula

H K (c) = ν∈M K |c| dν ν ,
where |c| ν = max 0≤j≤t |c j | ν . This is the classical notion of height for a point in projective space ( [START_REF] Hindry | Diophantine Geometry: an Introduction[END_REF], section B.2). As a projective notion, H K (c) should be invariant by scalar multiplication. Indeed, for λ ∈ K \ {0} we have H K (λc) = H K (c). This follows from the product formula:

ν∈M K |λ| dν ν = 1
for any ν ∈ K \ {0}. Note also that the (relative) height of an algebraic number x ∈ K is equal to the projective height of the point (1,

x) ∈ K 2 .
As in the previous section, we can define an absolute height by the formula

H(c) = H K (c) 1/n where n = [K : Q] and K is chosen so that c ∈ K t+1 .
In our main lower bound theorem (Theorem 4) we measure the size of the rational tuple c = (c j ) in (1) by its projective height instead of the naive bit size used in Theorem 1. To compute the height of a rational tuple, we first note that H(c) = max j |c j | if the c j are relatively prime integers. The general case c j ∈ Q is therefore quite easy: reduce to the same denominator to obtain integer coefficients, divide by their gcd and take the maximum of the absolute values of the resulting integers (so in particular H(c) ∈ N for any c in Q t+1 ).

A Height Lower Bound

The goal of this section is to establish the following lower bound. Proposition 1. Let (a, b) be a pair of rational numbers different from the five "excluded pairs" (0, 0), (±1, 0) and (0, ±1).

There is a universal constant C > 1 such that the inequality

H(a + bθ) ≥ C (4)
holds for any root of unity θ which is not a 6th root of unity.

The inequality H(a + bθ) ≥ C implies in particular that a + bθ is not a root of unity, since roots of unity are of height 1.

The main tool in the proof of Proposition 1 is the height lower bound of Lemma 1. In light of this lemma, to complete the proof of Proposition 1 we just need to understand when a + bθ can be a root of unity. Lemma 2. Let θ be a root of unity and (a, b) a pair of rational numbers different from the five excluded pairs (0, 0), (±1, 0) and (0, ±1). If θ is not a 6th root of unity then α = a + bθ is nonzero, and is not a root of unity.

Proof. We will need some properties of cyclotomic polynomials. Recall that if θ is a root of unity of order n, its minimal polynomial is the cyclotomic polynomials ψ n . By definition, the conjugates of θ are the other roots of its minimal polynomial. The roots of ψ n are exactly the roots of unity of order n. There are φ(n) such roots, where φ(n) is Euler's totient function. It is known that φ(n) ≥ √ n for n > 6. We therefore have φ(n) ≥ 3 for n > 6. From this it follows that φ(n) ≥ 3 except for n = 1, 2, 3, 4 or 6.

The conclusion of the lemma clearly holds true in the case b = 0. We therefore assume in the remainder of the proof that b = 0.

The only rational roots of unity are +1 and -1, which are 6th roots of unity, hence α = 0. If both θ and a + bθ happen to be roots of unity then θ lies at the intersection of the unit circle of the complex plane, and of the circle defined by the condition |a + bz| = 1. By excluding the 5 excluded pairs, we have made sure that these two circles are distinct. They have therefore at most 2 intersection points. If θ ′ is a conjugate of θ, the point a + bθ ′ is also a root of unity and must therefore lie at the intersection of the two circles. Since there are at most two intersection points, θ has at most one conjugate. This happens only when θ is a root of a cyclotomic polynomial ψ n of degree φ(n) ≤ 2, and we have seen that there are only 5 possible values for n. The two roots of order 4, ±i, can be ruled out since a ± bi is a root of unity only when (a, b) is equal to one of the two excluded pairs (0, ±1). We are left with the roots of unity of order 1, 2, 3 or 6, that is, with the 6th roots of unity.

⊓ ⊔ Remark 1. If θ 6 = 1, a + bθ can be a root of unity for appropriate values of a and b. For instance, if θ = e iπ/3 then 1 -θ = e -iπ/3 . If θ = e 2iπ/3 then 1 + θ = e iπ/3 .

Remark 2. In the remainder of this paper we will apply (4) only to p-th roots of unity where p is prime.

Hitting Set Construction

It is well known that roots of unity yield hitting sets for sparse polynomials.

Lemma 3. Let K be a field of characteristic 0 and f ∈ K[X] a nonzero univariate polynomial of degree at most d with at most m nonzero monomials. Then there are less than m log d prime numbers p for which f (X) is identically zero modulo X p -1.

Here we restrict to fields of characteristic 0 but this lemma is stated in [START_REF] Bläser | Deterministically testing sparse polynomial identities of unbounded degree[END_REF] for arbitrary integral domains. A multivariate version can be found in Lemma 5 of [START_REF] Kaltofen | Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields[END_REF]. Lemma 3 can be immediately restated in the language of hitting sets: Lemma 4. Let K be a field of characteristic 0, P a set of at least m log d prime numbers and H the set of all p-th roots of unity (in the algebraic closure of K) for all p ∈ P.

Then H is a hitting set for the set of all polynomials f ∈ K[X] of degree at most d with at most m nonzero monomials.

In the next proposition and theorems, the projective height comes into play. Recall that this notion is defined in Section 2; in particular, we explain at the end of that section how to compute H(c) when the c j are rational (which is the case in Theorem 2). For a rational tuple, the logarithm of the projective height gives a more appropriate notion of "size" than the naive bit size. In the next proposition, we use the projective height for tuples of algebraic numbers. Namely, following Lenstra [START_REF] Lenstra | Finding small degree factors of lacunary polynomials[END_REF] we define the height H(p) of a polynomial p = t j=0 c j X j ∈ Q[X] as the projective height H(c). Proposition 2. Let p ∈ Q[X] be a polynomial with at most t + 1 nonzero terms. Assume that p can be written as the sum of two polynomials q and r where each monomial of q has degree at most β and each monomial of r has degree at least γ. Let x ∈ Q * be a root of p that is not a root of unity. If γ -β > log(t H(p))/ log H(x) then x is a common root of q and r.

The proof of Proposition 2 can be found in [START_REF] Kaltofen | Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields[END_REF]. It is essentially the same as the proof of Proposition 2.3 of [START_REF] Lenstra | Finding small degree factors of lacunary polynomials[END_REF].

Theorem 2 (Gap Theorem for Hitting Sets). Let f ∈ Q[X] be a polynomial of the form (1), with (a, b) different from the five excluded pairs of Proposition 1. Assume without loss of generality that the sequence (β j ) is nondecreasing, and assume also there exists l such that

β l+1 -β l > log(t(t + 1)H(c))/ log C (5)
where C is the constant of Proposition 1, and H(c) is the projective height of the tuple c = (c j ). Let H be a set of roots of unity with θ 6 = 1 for all θ ∈ H. Let g = l j=0 c j X α j (a + bX) β j and h = t j=l+1 c j X α j (a + bX) β j . If H is a hitting set for g and h, H is also a hitting set for f = g + h.

Proof. We need to show that f (θ) = 0 for all θ ∈ H implies f = 0. If θ ∈ H is a root of f then a + bθ is a root of the univariate polynomial p(X) = t j=0 c j θ α j X β j . The height of p satisfies the inequality H(p) ≤ (t + 1)H(c). The factor t + 1 is due to the fact that each monomial of p "comes" from at most t + 1 terms of (1); see [START_REF] Kaltofen | Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields[END_REF], Lemma 3 for a proof. Since θ 6 = 1 we have H(a + bθ) ≥ C > 1 by Proposition 1. We can therefore apply Proposition 2, and it follows that x = a + bθ is a common root of the two univariate polynomials q = l j=0 c j θ α j X β j and r = t j=l+1 c j θ α j X β j . This means exactly that g(θ) = h(θ) = 0. If these two equalities apply to every θ ∈ H we have g = h = 0 since H is supposed to be a hitting set for both g and h. Hence f = g + h = 0.

We are now ready to state our main hitting set theorem. The bound will depend on 3 parameters:

Notice that this is essentially the gap bound in [START_REF] Bläser | Deterministically testing sparse polynomial identities of unbounded degree[END_REF].

Theorem 3 (Hitting Set Construction). Let P be a set of at least (t + 1)(δt + 1) log(d + tδ) prime numbers, with δ as in ( 6) and p ≥ 5 for all p ∈ P.

Let H be the set of all p-th roots of unity for all p ∈ P. Then H is a hitting set for the set of polynomials that can be represented under form (1) with α j ≤ d for all j, the rational tuple c of projective height H(c) ≤ M , and (a, b) different from the two pairs (0, ±1).

Proof. We proceed by reduction to Lemma 4. As in Theorem 2, we will assume without loss of generality that the sequence (β j ) is nondecreasing. We can of course assume that (a, b) = (0, 0) since the corresponding polynomial in (1) would be identically zero. We will also assume that (a, b) = (±1, 0). In that case, f can be written as a sum of t+1 monomials of degree at most d and we can apply Lemma 4: H is a hitting set for f since |P| ≥ (t + 1) log d (the same argument could of course be applied to any pair (a, b) with b = 0).

The remainder of the proof is divided in two cases. We first consider the case where there is no gap in f in the sense of Theorem 2, that is, β l+1 -β l ≤ δ for all l. In this case, factoring out the polynomial (a+bX) β 0 if necessary, we assume without loss of generality that β 0 = 0. This is legitimate since the nonzero polynomial (a+bX) β 0 does not vanish at any point of H (recall that the elements of H are irrational numbers). From the relations β 0 = 0 and β l+1 -β l ≤ δ we find that β t = max l β l ≤ δt. Expanding each factor (a + bX) β j in (1) as a sum of monomials, we see that f can be written as a sum of at most (t + 1)(δt + 1) monomials, each of degree at most d + tδ. Lemma 4 therefore implies that H is a hitting set for f . We finally consider the case where there are gaps in f . By "breaking f at the gaps", we write f = s i=1 f i where each f i is a sum of consecutive terms c j X α j (a + bX) β j from (1). More precisely, we make sure that there is no gap inside each f i in the sense that the difference between two consecutive exponents β j in f i is bounded by δ, and there is a gap between f i and f i+1 in the sense that the difference between the smallest exponent β j in f i+1 and the biggest one in f i is greater than δ.

We have seen that H is a hitting set for each of the f i . Applying Theorem 4 repeatedly (s -1 times), we see that H is a hitting set for f as well.

Remark 3. The pair (a, b) = (0, ±1) is excluded from Theorem 3. This case can easily be handled with Lemma 4: f is a sum of t + 1 monomials of degree at most d + d ′ , where d ′ = max j β j . We can therefore replace the set P in Theorem 3 by a set of prime numbers of cardinality at least (t + 1) log(d + d ′ ). By contrast, the bound in Theorem 3 does not depend on d ′ . Also, we can construct a single hitting set which covers uniformly the two cases (a, b) = (0, ±1) and (a, b) = (0, ±1) by replacing the bound (t + 1)(δt + 1) log(d + tδ) in Theorem 3 by the maximum of this bound and (t + 1) log(d + d ′ ).

Lower Bound Theorem

As explained in Section 1.2, it is straightforward to obtain a lower from our hitting set construction.

Theorem 4 (Main Lower Bound). Let P be a set of prime numbers with p ≥ 5 for all p ∈ P, |P| ≥ (t + 1) max(log(d + d ′ ), (δt + 1) log(d + tδ))

and δ as in [START_REF] Bürgisser | On implications between P-NP hypotheses: decision versus computation in algebraic complexity[END_REF]. The polynomial

P = i∈P (X p i -1)
cannot be expressed under form (1) if α j ≤ d and β j ≤ d ′ for all j, and if the rational tuple c is of projective height H(c) ≤ M . The same lower bound applies to all nonzero multiples of P .

Proof. Let f be a polynomial which can be expressed under form (1) with α j ≤ d and β j ≤ d ′ for all j, and H(c) ≤ M . Let Q be a multiple of P . By Theorem 3 and the remark following it, the set of roots of Q is a hitting set for f . Hence we cannot have f = Q, unless Q = 0.

Theorem 1 follows from Theorem 4 since there are Ω(N/ log N ) prime numbers in the interval [2, N ].

One can try to extend our results in various ways. One possible direction is prove lower bounds for other polynomials than polynomials of the form i (X i -1). It was recently shown in [START_REF] Avendano | The number of roots of a lacunary bivariate polynomial on a line[END_REF] that nonzero polynomials represented under form (1) have at most 6t -4 real roots. As a result, any set of 6t -3 real numbers is a hitting set and we have lower bounds for polynomials with many real roots such as 2 n i=1 (X -i). Perhaps more importantly, one can look for lower bounds under more general representations than [START_REF] Koiran December | A possible pseudorandom generator against arithmetic circuits[END_REF]. We make two suggestions below.

1. Consider expressions of the form t j=0 c j (a + bX) α j (c + dX) β j .

(

) 8
Assuming that b = 0, the change of variable Y = a+bX brings us back to (1) and we can use the black-box algorithm of the present paper or the non-black-box algorithm of [START_REF] Kaltofen | On the complexity of factoring bivariate supersparse (lacunary) polynomials[END_REF] to perform deterministic identity testing. Unfortunately, the change of variable Y = a+bX is non-blackbox and as a result we do not have a lower bound for polynomials of the form i (X i -1). Nevertheless, the set of real numbers is invariant under this change of variable. As a result, it follows again from [START_REF] Avendano | The number of roots of a lacunary bivariate polynomial on a line[END_REF] that any set of 6t -3 real numbers is a hitting set for (8) and we still have exponential lower bounds for polynomials such as 2 n i=1 (X -i). The case b = 0 is even simpler: now we have polynomials of the form t j=0 c ′ j (c + dX) β j , where c ′ j = c j a α j . The change of variable Y = c + dX shows that by Descarte's rule of signs, such a polynomial can have at most 2t+1 real roots if it is nonzero. We can therefore construct a hitting set (any set of 2t + 2 real numbers will do) and derive good lower bounds. 2. Consider now expressions of the form t j=0 c j X α j (a j + b j X) β j .

In [START_REF] Koiran December | A possible pseudorandom generator against arithmetic circuits[END_REF] we have a j = a and b j = b for all j. Is deterministic identity testing feasible, either in a black-box or non-black-box way ? Is it possible to derive lower bounds for this form of polynomial representation ?

  (i) the parameter t in (1). (ii) d, the maximal value of the α j . iii) an upper bound M on the projective height H(c) of the tuple c. Given t, d and M we define δ = log(t(t + 1)M )/ log C.

As already observed in[START_REF] Heintz | Testing polynomials which are easy to compute[END_REF], hitting sets may be difficult to construct precisely because they yield lower bounds.

Acknowledgments

This work was to a large extent triggered by a question of Erich Kaltofen: can the polynomial (X n -1)/(X -1) be represented efficiently under form (1) ?