Alban R Defleur
email: alban.defleur1@gmail.com

Emmanuel Desclaux
email: emmanuel.desclaux@unice.fr

Impact of the last interglacial climate change on

Keywords: Last interglacial, MIS 5e, Climate change, ecosystems, Neanderthals, behavior, cannibalism

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

-Introduction

Throughout the past 400,000 years, the last interglacial period (Eemian, MIS 5e, LIG 128 to 114 ky) was the warmest (Working Group of PAGES, 2016), with a temperature about 2 degrees higher than the present, and a sea level rise between 6 -9 meters above the last century level [START_REF] Dutton | Sea-level rise due to polar ice-sheet mass loss during past warm periods[END_REF]. This warm climatic oscillation (Milankovitch, 1941) is documented in marine sediment cores [START_REF] Emiliani | Pleistocene temperatures[END_REF][START_REF] Shackleton | The last interglacial in the marine and terrestrial records[END_REF], ice cores [START_REF] Northgrip-Members | High-resolution record of Northern Hemisphere climate extending into the last interglacial period[END_REF], NEEM community members, 2013,) and terrestrial records from long sequences in middle-Europe (de Beaulieu andReille, 1989, Brewer et al., 2008). Due to the scarcity of Western European sites with precisely dated stratigraphic sequences bearing stone tools, faunal remains, and reliable ecological data, little is known of the biotopes and landscape changes associated with global warming during this period. Understanding how European Neanderthal hunter-gatherer communities, having adapted for tens of thousands of years to an earlier cold climate, were able to adjust to the major climatic change of the MIS 5e, has been elusive.

The distribution of Eemian Interglacial-aged sites is sporadic throughout

Western Europe. There are none in Britain [START_REF] Roebroeks | Dense forests, cold steppes, and the palaeolithic settlement of northern Europe [and comments and replies[END_REF][START_REF] Ashton | The absence of humans from Last Interglacial Britain[END_REF]. Some authors argue that their absence is due to inaccessibility resulting from rapid sea level rise rather than to the development, during MIS 5e, of habitats inhospitable to Neanderthals [START_REF] Roebroeks | Dense forests, cold steppes, and the palaeolithic settlement of northern Europe [and comments and replies[END_REF][START_REF] Roebroeks | Last interglacial (eemian) occupation of the North European plain and adjacent areas[END_REF]. However, we know that Britain was occupied during the earlier interglacial periods (Robert et al., 1995, Stringer and[START_REF] Stringer | New age estimates for the Swanscombe hominid, and their significance for human evolution[END_REF].

In Northern Germany several open-air sites (i.e. Gröbern, Rabütz, Grabschütz, Taubach, Neumark Nord and Lehringen), characterized by a temperate fauna and flora, are attributed to the optimum Eemian Interglacial. The site of Lehringen is famous for a well-preserved yew spear found with a Palaeoloxodon antiquus carcass [START_REF] Thieme | Neue Untersuchungen zum eemzeitlichen Elefanten-Jagdplatz Lehringen[END_REF]Veil, 1985, Wenzel, 2007). This discovery documents the recolonization of this area by temperate-adapted species following the strong climatic improvement that occurred during MIS 5e.

Among similar sites, Neumark-North has yielded the widest spectrum of temperate-adapted fauna (Gaudzinski-Windheuser andRoebroeks, 2011, Gaudzinski-Windheuser et al., 2014). Most of these sites are located in an area previously completely covered by the MIS 6 Scandinavian ice sheet, and these localities form a fairly dense group, in contrast to the general scarcity of Eemian sites in the rest of Western Europe.

In Southern Europe few Eemian Interglacial-aged (MIS 5e) sites have been found, including Krapina (Gorjanovic-Kramberger, 1906, Malez and[START_REF] Malez | The Upper Pleistocene fauna from the Neanderthal man site in Krapina (Croatia, Yougoslavia)[END_REF], in Croatia, and Bolomor cave [START_REF] Arsuaga | Fossil human remains from Bolomor cave (Valensia, Spain)[END_REF][START_REF] Sanudo | Site formation dynamics and human occupations at Bolomor Cave (Valencia, Spain); An archaeostratigraphic analysis of level I to XII (100-200 ka)[END_REF], in Spain. Saccopastore, in Italy, had previously been attributed to the Eemian but is now considered much older (Masa et al., 2015).

In France two Eemian sites are known. In the north is the site of Caours, [START_REF] Antoine | Le tuf de Caours (Somme, France) : mise en évidence d'une séquence éémienne et d'un site paléolithique associé[END_REF][START_REF] Sier | Evidence for the Blake Event recorded at the Eemian archaeological site of Caours, France[END_REF]. In the south, only Baume Moula-Guercy (BMG) is attributed with certainty to MIS 5e.

BMG is located near the village of Soyons in Ardèche (Southeastern France), between the Alps and the Massif Central, on the west bank of the Rhône River (Fig. 1,3). Stratigraphic and biochronologic analyses [START_REF] Saos | Stratigraphical, sedimentological and magnetic study of middle and upper pleistocene infilling of Baume Moula-Guercy (Soyons, Ardèche)[END_REF] allowed subdivision of the 8 meter thick deposits into 19 layers. These layers were then associated with three main climatic phases corresponding to MIS 6 to MIS 4. MIS 5 is represented by layers XV to XI and MIS 5e by layers XV and XIV.

Layer XV, the most important layer of this cave, comprises a silty-sand sediment about 40 cm thick that covers approximately 30-40 m 2 . The excellent state of bone and charcoal preservation in this layer has enabled reconstruction of the natural environment and landscapes that Neanderthals experienced during the Eemian period.

The quest to understand the causes of Paleolithic cannibalism is almost as old as prehistory itself [START_REF] Piette | Les grottes de Gourdan (Haute-Garonne)[END_REF], and continues to be a widely discussed topic in prehistory and paleoanthropology. In a recent study [START_REF] Cole | Assessing the calorific significance of episodes of human cannibalism in the Palaeolithic[END_REF] notes: In the majority of studies, the interpretation is that cannibalism was practiced for nutritional reasons, although there has never been a way to quantify how nutritional these episodes may be. Here we propose that the Neanderthal cannibalism evident at BMG cave layer XV is a nutritional cannibalism that emerged in response to the profound upheaval of their previously enduring environmental conditions. We also suggest that paleoanthropological sites of Eemian age are extremely rare due to a drastic decline in the European human population during this period.

-Methods

The initial discovery of human remains from Baume Moula-Guercy layer XV revealed characteristic traces suggesting an episode of Neanderthal cannibalism [START_REF] Defleur | Neanderthal Cannibalism at Moula-Guercy, Ardèche, France[END_REF]. Simultaneously, observation of the associated fauna from layer XV indicated that it corresponded to a temperate episode, in contrast to the cold-adapted fauna found in overlying layers [START_REF] Defleur | Première mise en évidence d'une séquence éémienne à restes humains dans le remplissage de la Baume Moula-Guercy (Soyons, Ardèche)[END_REF]. Using a paleoanthropological (interdisciplinary) approach to our investigation of BMG, the recovered samples and geological context were analyzed and compared with relevant publications. Topics assessed include; 1) delineating characteristics that distinguish cannibalism from other forms of bone damage -paleopathology (linear enamel hypoplasia: LEH), taphonomy, typology of lithic artefacts, and paleoethnography; 2) describing faunal and floral remains -vertebrate and invertebrate palaeontology (systematic, biostratigraphy and especially paleoecology including the cenogram method), palaeoanthropology, palaoebotany (anthracology), archaeozoology; 3) determining applicable dating methods -isotopic geochemistry (radiocarbon dating analysis: 40 Ar/ 39 Ar, Uranium-series and ESR, Strontium isotope ratios ( 87 Sr/ 86 Sr) applied on tooth enamel of tooth enamel); 4) establishing stratigraphic contextarchaeostratigraphy, quaternary geology (sedimentology and stratigraphy).

-Results

-The Layer XV: a deep reconfiguration of ecosystems

During the Pleistocene, climate change resulted in a general reconfiguration of ecosystems and redistribution of all living species on the European continent [START_REF] Pecl | Biodiversity redistribution under climate change: Impacts on ecosystems and human well-being[END_REF] The cenogram method (Fig. 2) confirms that layer XV is consistent with a temperate climate with a predominance of mixed forest biotopes under wellmarked warm conditions. The Hermann tortoise (Testudo hermanni) was also recovered from Layer XV. This tortoise is currently limited to the Mediterranean coastline and has not been reported to be in the Baume Moula-Guercy area in historic or prehistoric times. Currently, the northern-most range of the Hermann tortoise derives from a Holocene locality that is d more than 100 km south [START_REF] Forest | Le lièvre, la tortue et l'aurochs[END_REF] of BMG. Two other reptile species, the Montpellier snake (Malpolon monspessulanus) and ladder snake (Rhinechis scalaris) are no longer dispersed in the study area and have a more southerly distribution [START_REF] Manzano | Les amphibiens et les reptiles des sites du Pléistocène moyen et supérieur du pourtour méditérranéen (Caune de l'Arago, grotte du Lazaret[END_REF]. The presence and current ecological requirements of these three species of reptiles indicates that the climate of this region was warmer and probably drier than at present.

A climatic shift to warmer and drier conditions, inferred from the reptiles recovered at BMG correlates with the study of chironomids assemblages (aquatic insects families which respond rapidly to environmental changes).

These insect remains were recovered from the pollen sequence at the Les Echets mire (130 km to the north of BMG) [START_REF] Gandouin | Past environment and climate changes at the last Interglacial/Glacial transition (Les Échets, France) inferred from subfossil chironomids (Insecta) C[END_REF].

Charcoal recovered from a cave in the vicinity of BMG (MIS 4) only represents the remains of a single species of pine (Pinus type silvestris) [START_REF] Defleur | Industries, biostratigraphie, restes humains et datation du gisement moustérien de la Baume Néron (Soyons, Ardèche)[END_REF]. Flora remains recovered around hearths (Layer XV) belong to several species. These include: deciduous oak (Quercus sp.), beech (Fagus sylvatica), These findings allow us to attribute the chronological position of layer XV to the initial phase of MIS 5e. Recent U-series and ESR dating has provided a minimum date of 119 ± 13 ky for Layer XV. The probable age range is approximately 120-130 ky [START_REF] Willmes | A comprehensive chronology of the Neanderthal site Moula-Guercy, Ardèche, France[END_REF]. This result is in accordance with palaeoecological and biochronological data provided by both the large and small vertebrate fauna.

-Rare cases of Neanderthal cannibalism

It is commonly accepted that inferences of prehistoric human cannibalism are best based upon similarity of butchery techniques found on human and ungulate remains, (skinning, defleshing, disarticulation), the same mode of breakage of long bones and skulls for the extraction of marrow and brains, and the same discard pattern and context. When numerous human remains are discovered on an undisturbed living floor, with similar patterns of damage, mixed with animal remains, stone tools, and fireplaces they can legitimately be interpreted as evidence of cannibalism. [START_REF] Villa | Cannibalism in the Neolithic[END_REF][START_REF] White | Prehistoric cannibalism at Mancos 5MTUMR-2346[END_REF][START_REF] Defleur | Neanderthal Cannibalism at Moula-Guercy, Ardèche, France[END_REF][START_REF] Rosas | Paleobiology and comparative morphology of a late Neandertal sample from El Sidrón, Asturias, Spain[END_REF]. The archaeological material ideally must have remained in situ since its abandonment and must have been recovered in a precise stratigraphic framework with meticulous excavation and recovery methods.

An inventory carried out in 2008 shows that about 220 European sites have yielded human remains belonging to the Neanderthal lineage [START_REF] Mann | Les Néandertaliens européens[END_REF]. The large majority of these vestiges are isolated human remains (teeth, fragments of skulls or long bones).

The issue of cannibalism currently arises for four Neanderthal sites, all located in Western Europe: Krapina in Croatia (Gorjanovic-Kramberger, 1906) BMG in France ( [START_REF] Defleur | Neanderthal Cannibalism at Moula-Guercy, Ardèche, France[END_REF], El Sidron in Spain [START_REF] Rosas | Paleobiology and comparative morphology of a late Neandertal sample from El Sidrón, Asturias, Spain[END_REF] and La

Troisième caverne de Goyet in Belgium [START_REF] Rougier | Neanderthal cannibalism and Neanderthal bones used as tools in Northern Europe[END_REF] (Fig. 4).

However, the ideal criteria outlined above are not all present in each case. For example, at El Sidron, all the human remains are in a secondary position and the faunal remains are scarce, but the evidence of cannibal behavior is not in dispute.

At La Troisième caverne and Krapina, excavations were carried out between 1865-1868 and 1899-1905 and the contextual association between the human bones and their environmental contexts are not equivalent to those of sites excavated recently.

Other sites, also few in number, show traces of specific activities involving the deceased. Most of these cases have cut marks on cranial and sometimes postcranial remains: Vindija [START_REF] Malez | Neuere paläo-anthropologische Untersuchungen am material aus der HoleVindija (Kroatien)[END_REF], Combe-Grenal [START_REF] Garralda | Neanderthal cut marks: Combe-Grenal and Marillac (France). A SEM analysis[END_REF], Zafarraya (Finlayson et al., 2006).

However, if cut marks on human remains are a necessary criteria, they are not sufficient to support the hypothesis of cannibalism. Indeed, these traces can also result from peri or post mortem manipulation of corpses associated with secondary funerary practices. Such funeral practices can include the preservation of skulls or other bones of the deceased for symbolic or other purposes as it is probably the case in Kebara [START_REF] Bar-Yosef | Le squelette moustérien de Kébara 2[END_REF].

At the site of Marillac/Les Pradelles in northwestern France a sample of 95 human remains have been recovered [START_REF] Garralda | Neanderthal cut marks: Combe-Grenal and Marillac (France). A SEM analysis[END_REF]. Review of the published data for this site shows that the human remains come from five different stratigraphic layers and the ages obtained for these layers cover a significant span of time (50-70 ky). Tooth marks and digestive traces (acid etching) associated with large carnivores were recognized on some bones. We note also the absence of trunk elements and pelvic and scapular girdles. While some of the bones possess cut marks from stone tools, percussion marks have not been observed [START_REF] Garralda | Neanderthal cut marks: Combe-Grenal and Marillac (France). A SEM analysis[END_REF]. In a taphonomic study of these human remains [START_REF] Mussini | Les restes humains moustériens des Pradelles (Marillac-le-Franc, Charente, France): étude morphométrique et réflexions sur un aspect comportemental des Néandertaliens[END_REF] considers this site as "a probable case of cannibalism".

-Evidence for a nutritional cannibalism at BMG layer XV

In BMG layer XV, the 120 human remains from 6 individuals (2 adults, 2 the skulls, all the long bones and other bones of adults and children. Some anvil striae, adhering flakes, internal vault release, crushing, and peeling [START_REF] White | Prehistoric cannibalism at Mancos 5MTUMR-2346[END_REF] and probable Neanderthal chewing on two hand phalanges were also observed. The human remains were recovered in a mixed context with faunal remains on a few square meters near the east wall of the cave [START_REF] Defleur | Neanderthal Cannibalism at Moula-Guercy, Ardèche, France[END_REF]. The numerous refitting pieces and projections indicate that there was probably only a single episode of cannibalism in this layer. The fact that skeletal elements from all regions of the body suggests that they were intact prior to the human butchery event. Further, none of the remains is in anatomical relationship to one another, indicating the body were completely dismembered.

The low number of unretouched flint flakes [START_REF] Defleur | Mousterian lithic assemblages from Moula-Guercy cave (Soyons, Ardèche). Excavations 1993-1999[END_REF] and large mammals, in addition to other zooarchaeological data suggest a succession of temporary hunting camps with occupation occurring during the summer or autumn [START_REF] Valensi | Archaeozoological data from the Mousterian level from Moula-Guercy (Ardèche, France) bearing cannibalized Néanderthals remains[END_REF].

For over a century, different hypotheses have been suggested to explain

Neanderthal cannibalism: consumption of individuals from the group (endocannibalism) or outside the group (exo-cannibalism), and survival cannibalism.

In the case of the BMG layer XV several arguments accredit the survival necessity cannibalism thesis:

BMG Cannibalism Not Mortuary Practice -Neanderthal burials, a hallmark of Mousterian culture, come from 17 sites (34 individuals) and cover almost the entire Neanderthal dissemination area from the Atlantic Ocean to the Middle East and Central Asia [START_REF] Defleur | Les sépultures moustériennes[END_REF]. All were discovered at cave sites. However, many areas (Italy, Spain, Balkan countries and Central Europe) have yet to produce evidence of Neanderthal burials, possibly reflecting different funerary behaviors such as burial outside of caves. In contrast, Neanderthal cannibalism is limited to four localities in Western Europe (Fig. 4). Thus, cannibalism, unlike burial, was probably not a usual mortuary practice of the Neanderthals.

Cannibalism Due to Lifestyle Adjustments -Two of the four examples of Neanderthal cannibalism, Baume Moula-Guercy and Krapina [START_REF] Rink | ESR ages for Krapina hominids[END_REF], come from the MIS 5e, a short but significant period of climatic and ecological change representing only about 10% of Neanderthal existence during the Upper Pleistocene, as compared to 90% lived in cold or very cold phases.

-Changes in ecosystems reported above affect the lifestyles of human communities. As today, Pleistocene interglacial deciduous forests exhibited a high primary biomass predominantly represented by trees, vegetation, invertebrates and non-mammalian vertebrates (Ramade, 1984; [START_REF] Gamble | The Paleolithic settlement of Europe[END_REF][START_REF] Kelly | The foraging spectrum. Diversity in hunter-gatherer lifeways[END_REF]. The medium-sized ungulate biomass was -BMG layers above and below layer XV include open steppe and wooded prairie biotopes where small, medium-sized and large species were available (cf. supra) as Neanderthal prey.

In contrast, the BMG Layer XV was located between a largely dominant interglacial forest environment providing a small number of medium sized mammal species difficult to kill, and rare open landscapes with powerful and dangerous animals. This change in availability would have disrupted the traditional way of life and behavior of Neanderthal communities hitherto accustomed to abundant large prey.

Few Occupation Sites Due to Population Stress

-The small number of sites attributable to MIS 5 in general and to the 5e in particular on the European continent cannot be explained by taphonomic causes (as suggested by Surovell andBrantigham, 2007, Surovell et al., 2009). Rather it is probably due to a sharp demographic contraction (i.e. Richter, 2006;Speller, 2000;[START_REF] Gaudzinski-Windheuser | On Neanderthal Subsistence in Last Interglacial Forested Environments in Western Europe[END_REF][START_REF] Williams | The use of summed radiocarbon probability distributions in archaeology: a review of methods[END_REF], Rolland, 2015[START_REF] Benito | The ecological niche and distribution of Neanderthals during the Last Interglacial[END_REF].

Some authors [START_REF] Roebroeks | Dense forests, cold steppes, and the palaeolithic settlement of northern Europe [and comments and replies[END_REF][START_REF] Roebroeks | Last interglacial (eemian) occupation of the North European plain and adjacent areas[END_REF] hypothesis that the rarity of Neanderthal traces during MIS 5e results from various factors, including its short duration, and a combination of geological and historical research factors.

Archaeological research conducted over more than a century in southern France -which is probably one of the regions of Europe where

Neanderthal occupation was the most intense -shows that the hypothesis of an historically based lack of research does not correspond to the archaeological reality. In Southern France, the deposits are numerous to deliver occupation of MIS 6 the 4 and 3, exceptionally of MIS 5d,c,b,a, and never of MIS 5e. Hitherto, only layer XV of BMG is confidently and unambiguously attributed to MIS 5e in this region.

Neanderthals had likely abandoned most parts of Europe because of lack of natural resources compared to the previous and subsequent periods, and because the low population density of the groups was insufficient to allow adequate human reproduction. However, some regions such as the southeast of France may have remained permanently inhabited (Fig. 5).

Dependence on Protein from Meat

-Isotopic analyses show that Neanderthals were at the top of the food chain; meat and fat were their main food sources [START_REF] Richards | Neanderthal diet at Vindija and Neanderthal predation: The evidence from stable isotopes[END_REF][START_REF] Wissing | Isotopic evidence for dietary ecology of late Neandertals in North-Western Europe[END_REF], thus their population density had to be small. They had a robust skeleton and their average muscular mass was higher than that observed in modern people, implying a high-energy cost for travel and activities. Their DEE (daily energy expenditure) has been estimated [START_REF] Churchill | Bioenergetic perspectives on Neanderthals thermoregulatory and activity budgets[END_REF] between 3,500 and 5,000 kcal per day for men, a much higher expenditure than the average DEE of recent hunter-gatherer populations. This resulted in a greater dependence on the flesh and fat of large herbivores, with very little exploitation of small game. Populations would have been more vulnerable during periods when animal protein became scarce. Although protein was a dominant component of their diet, it may have taken exceptional circumstances for human flesh to become meat. Hunger likely induced this behavior. "People usually eat because they are hungry, and most prehistoric cannibals were therefore probably hungry." [START_REF] White | Once were Cannibals[END_REF], and even more so during abrupt climate change.

-In layer XV, potential secondary preys are only represented by porcupine, Hermann tortoise and fishes. While fish remains have been recovered in only a small number of Mousterian sites, the study of fish bones from layer XV suggests that this group fish from the Rhône River. The rabbit (Oryctolagus cuniculus), which is widely represented in the contemporary layer IV at Bolomor cave in Spain [START_REF] Blasco | Small and large game: human use of diverse faunal ressources at level IV of Bolomor cave (Valencia, Spain)[END_REF] is not represented in layer XV.

-In layer XV, the percentage of cut marks on human bones (>50%) is much higher than on the remains of deer (14%), which is the most abundant and largest mammal recovered [START_REF] Valensi | Archaeozoological data from the Mousterian level from Moula-Guercy (Ardèche, France) bearing cannibalized Néanderthals remains[END_REF]. These findings indicate that human carcasses were exploited more than those of large mammals, suggesting an exacerbated need for protein. Cut marks on metapodials and phalanges confirm this observation.

-Faunal remains are scarce. Only 290 fragments of ungulates have been taxonomically identified and 193 are deer (66.5 %) [START_REF] Valensi | Archaeozoological data from the Mousterian level from Moula-Guercy (Ardèche, France) bearing cannibalized Néanderthals remains[END_REF].

The extent of the identified fauna could be misleading. This is because it corresponds to both a palimpsest of short-lived human occupations and, in small part, to the faunal remains left by carnivores during periods of Neanderthal absence. For example, we have no proof that the rare remains of elephant and rhinoceros correspond to prey hunted by humans.

The presence in this layer of the fallow and roe deer are possibly also linked to carnivore activities (Valensi et al., 2013). Thus, contrary to assumption made by [START_REF] Yustos | Cannibalism in the Neanderthal world: An Extensive Revision[END_REF], the faunal list cannot be used as an argument to think that Neanderthals have never missed animal proteins at BMG layer XV (Fig. 6).

-The layer XV episode was of short duration. The total consumable soft tissue of the six individuals recovered does not exceed in caloric value that of a deer and an ibex [START_REF] Cole | Assessing the calorific significance of episodes of human cannibalism in the Palaeolithic[END_REF]. It is suggested by be about the equivalent of two days of meat consumption for a group of 15 to 25 people [START_REF] Hayden | Neandertal Social Structure?[END_REF], or 4 days by rationing.

Endured Episodes of Famine

-Study of the teeth of several individuals from layer XV shows defects in enamel development called "linear enamel hypoplasia" (LEH) [START_REF] Hlusko | Neanderthal teeth from Moula-Guercy, Ardèche, France[END_REF] at a frequency comparable to that observed on the cannibalized individuals of Krapina (Hutchinson et al. 1999). LEH can result from any number of physiological stresses, such as severe illness or malnutrition, signs of frequent nutritional stress (Hlusko et al., op. cit.). A recent study of LEH in the Hadza in Tanzania shows that LEH occurrence is most often associated with malnutrition [START_REF] Ungar | Toddlers in Transition: Linear Enamel Hypoplasias in the Hadza of Tanzania[END_REF]. Although observed in other sites [START_REF] Ogilvie | Incidence and patterning of dental enamel hypoplasia among the Neanderthals[END_REF], LEHs are expressed at Baume Moula-Guercy on molars, a finding that could indicate long periods of physiological stress and support the hypothesis of severe famines. These signs of developmental arrest or insufficient development are also present at El Sidron cave "almost all individuals in El Sidrón associated with dental remains have at least one episode of physiological stress stigmatized by tooth enamel hypoplasia (HED) that affects two or three teeth" [START_REF] Rosas | The Neandertals from El Sidrón (Asturias, Spain).Updating of a new sample[END_REF]. Of the four cases of cannibalism found in Neanderthals, at least three show signs of hypoplasia attesting that these populations had to endure periods of food stress. No equivalent study has been published for the fourth site.

-Data from study of the microwear on molar teeth suggests that BMG layer XV individuals likely had a diverse diet. While meat was included in their diet they may have had a comparatively richer plant resource base [START_REF] Hlusko | Neanderthal teeth from Moula-Guercy, Ardèche, France[END_REF]. These observations suggest that this population could have been faced with episodes of dietary stress and that they regularly consumed plant resources provided by their environment.

Consumption of Deceased

-Unlike Krapina and other sites, such as the Sima de los Huesos (Bocquet-Appel and [START_REF] Bocquet-Appel | Age Distribution of Hominid Samples at Atapuerca (SH) and Krapina, could indicate Accumulation by Catastrophe[END_REF], at BMG layer XV the individuals were not selected from any age group. Younger people aged under 15-16 years represent 2/3 of cannibalized individuals, along with an elderly robust individual and another gracile young-adult possibly female [START_REF] Defleur | Neanderthal Cannibalism at Moula-Guercy, Ardèche, France[END_REF][START_REF] Hlusko | Neanderthal teeth from Moula-Guercy, Ardèche, France[END_REF], Mersey et al., 2013a, Mersey et al., 2013b).

All age classes are close to natural mortality cases observed in recent hunter-gatherers [START_REF] Gurven | Longevity among Hunter-Gatherers: A Cross-Cultural Examination[END_REF]. In this case, cannibalism could have been an opportunistic consumption of the deceased of a human group, or part of it.

-In all known societies, cannibalism is accompanied by practices designed to dispel guilt. No evidence of social gestures of this nature have been recovered from the XV layer, though searches in this layer are as yet incomplete.

-Discussion

Analysis of the data found in BMG layer XV provides a benchmark for establishing the temperate conditions of the last interglacial, which is previously not documented for the European continent. The consistency of these data fills a gap for terrestrial climate change of the last interglacial. Our findings relate to reconstructing paleoenvironments and landscapes and human behavior from this period.

Firstly, we provide an additional terrestrial argument in favor of higher temperatures for the Eemian period than during peak Holocene. [START_REF] Brewer | The climate in Europe during the. Eemian: a multi-method approach using pollen data[END_REF].

Secondly, our study fills a void in defining landscape features during MIS 5e. At this time, the cave surroundings corresponded to a mosaic landscape inhabited by a fauna that is now either gone extinct, has disappeared from the region, or is similar to that existing today. The vast deciduous forest sheltered a fauna still present today, apart from the fallow deer which disappeared from the region at the end of MIS 5. Relict groups such as the giant deer, straight-tusked elephant and narrow-nosed rhinoceros inhabited the wooded prairie. Open sunlit areas of bushy vegetation were inhabited by Mediterranean reptiles as well as the porcupine (Hystrix cf. vinogradovi). These species are now extinct or located several tens of kilometers to the south. Stratigraphic analysis suggests that the transition between the cold, arid climate of MIS 6 and the temperate, humid climate of the MIS 5e occurred rapidly, as has been observed elsewhere [START_REF] Velichko | Comparative analysis of vegetation and climate changes during the Eemian interglacial in Central and Eastern Europe In: The Climate of Past Interglacials[END_REF]. The arctic lemming (Dicrostonyx torquatus), adapted to a circumpolar climate, disappears with layer XVI, whereas the Hermann tortoise, whose distribution is strictly Mediterranean, appears with layer XV.

These paleoenvironmental data potentially sheds light on changes that our current ecosystems might undergo if our planet experiences global warming in the future.

Thirdly, as some authors have pointed out [START_REF] Kelly | Hunter-gatherer mobility strategies[END_REF], Bailley et al., 1989[START_REF] Kelly | The foraging spectrum. Diversity in hunter-gatherer lifeways[END_REF], Bindford, 2001), tropical or temperate forests are hostile living environments for all hunter-gatherer groups today, as in the past, because protein from medium-sized ungulates is much less than from prey found in an open Mammoth steppe formation. Based on study of the human teeth [START_REF] Hlusko | Neanderthal teeth from Moula-Guercy, Ardèche, France[END_REF], this population was subject to frequent nutritional stresses and the presence of cannibalistic behaviour which, unlike ritual burial practices, is the exception for European Neanderthals. Climate-environmental episodes encountered by this human population during its history on the European continent have certainly influenced its periodic demographic collapses, as successive bottleneck events lead to genetic weaknesses, rapid evolution and low variability in Neanderthals (e.g. [START_REF] Briggs | Targeted retrieval and analysis of multiple Neandertal genomes[END_REF][START_REF] Premo | Culture, population structure, and low genetic diversity in Pleistocene hominins[END_REF], Bocquet-Appel and Degiovani, 2013[START_REF] Rogers | history of Neanderthals and Denisovans[END_REF].

Using a model from sub-contemporary hunter-gatherer populations, Neanderthals were probably divided into autonomous groups of 15 to 25 individuals related to 10 or 20 other allied groups [START_REF] Hayden | Neandertal Social Structure?[END_REF], Lalueza et al., 2011[START_REF] Rosas | The Neandertals from El Sidrón (Asturias, Spain).Updating of a new sample[END_REF].

During the Eemian period, networks of exchange and alliances, established for millennia among these human groups, may have been disrupted by new geographical barriers, as melting ice increased river volume and forests dominated the landscape. The scarcity of large prey species may have resulted in increased residential mobility and possibly disrupted old social connections, as their high trophic level and high energy requirements forced expanded hunting forays in search of widely dispersed medium-sized mammals [START_REF] Verpoorte | Neanderthal energetics and spatial behavior[END_REF].

At Baume Moula-Guercy, situated along the Rhône River, fishery resources may have mitigated these phenomena (Guillaud in progress, Kelly, 1995, Roselló-Izquierdo and[START_REF] Roselló-Izquierdo | Ictiofaunas Musterienses de la peninsula Iberica: ¿Evidencias de pesca Neandertal?[END_REF] Many recent works have changed the image of Neanderthals by showing their aptitude for language [START_REF] D'anastasio | Micro-Biomechanics of the Kebara 2 Hyoid and Its Implications for Speech in Neanderthals[END_REF][START_REF] Krause | The Derived FOXP2 Variant of Modern Humans Was Shared with Neanderthals[END_REF], the complexity of their lithic technology, shared with anatomically modern humans, the first known processing technology with the complex manufacturing of birch pitch, [START_REF] Koller | High-Tech in the Middle Palaeolithic: Neandertal-Manufactured Pitch Identified[END_REF], the use of natural pigments [START_REF] Peter | Selection and Use of Manganese Dioxide by Neanderthals[END_REF], the use of ornamental objects (Radovcic et al., 2015), their assistance of weak individuals [START_REF] Schultz | Results of the anatomical-palaeopathological investigations on the Neanderthal skeleton from Kleine Feldhofer Grotte (1856) including the new discovery from 1997/2000[END_REF], and the practice of ritual burial [START_REF] Defleur | Les sépultures moustériennes[END_REF]. The available data show that from a technical, cognitive, symbolic, emotional, and analytical (ability to abstract, anticipate and reflect on complex situations) perspective, the Neanderthals were likely similar to modern humans, who replaced them on the European continent around 40-35 ky ago. Their remarkable adaptability allowed them to live in very different environments and climates, from the Atlantic to the Altai Mountains and even more to the east [START_REF] Li | The easternmost Middle Paleolithic (Mousterian) from Jinsitai Cave, North China[END_REF], for more than 300,000 years. Fig. 2: Cenograms of the levels XV to IV of Baume Moula-Guercy, showing a gradual opening of the environment. This method corresponds to the study of the weight distribution of species in a mammalian community, at the exception of bats and carnivores. A cenogram is built by putting the logarithm of the mean body weight of each species of the community on the Y-axis; and the species classified in decreasing order of size on the X-axis. Four main categories of cenograms have been distinguished end it is possible to characterize the type of environments (open or closed) and the climatic conditions (arid or humid). Thus, a continuous distribution of weight is characteristic of a closed habitat. On the other hand, the lack of medium-weight species (between 500 g and 8 kg) is seen in faunas of open landscapes. Otherwise, an abundance of large-size species (over 8 kg) characterizes humid conditions, whereas a steep slope is associated with arid conditions. The slope for the small species reflects the temperature. A large number of small species (a gentle slope) indicates a rather warm climate. Chapelle-aux-Saints, 5=Le Moustier, 6=Roc-de-Marsal, 7=La Quina, 8=Spy, 9=Feldhofer, 10=Kilk-Koba, 11=Mezmalskaya, 12=Dederyeh, 13=Shanidar, 14=Amud, 15=Kebara, 16=Tabun and 17=Teshik Tash.

  yew (Taxus baccata), elm (Ulmus sp.), dogwood (Cornus mas), holly (Ilex aquifolium), scots pine (Pinus type silvestris), common juniper (Juniperus communis), wired broadleaf (Phillyrea latifolia), buckthorn (Ramnus saxatilis), blackthorn (Prunus spinosa), ivy (Hedera helix), and common wild rose (Rosa sp.)[START_REF] Defleur | Présentation paléo-environnementale du remplissage de la Baume Moula-Guercy à Soyons (Ardèche) : implications paléoclimatiques et chronologiques[END_REF]. All of these species indicate a deciduous forest environment. They are characteristic of a temperate climate similar to that of the present, although wired broadleaf is a typically Mediterranean species. The rapid warming of the climate at the end of MIS 6 led to the development of a thick deciduous forest throughout the continent, including the northern areas previously occupied by glaciers and permafrost[START_REF] Gaudzinski-Windheuser | The Eemian Interglacial lake-landscape at Neumark-Nord (Germany) and its potential for our knowledge of hominin subsistence strategies[END_REF]. The present local climate is significantly affected by the Rhône Valley corridor as it has resulted in Mediterranean climatic conditions extending to within 50 km to the south of the cave. The presence of Hermann tortoise in this part of France in the Eemian period indicates that the hot and dry Mediterranean climate must have reached much farther north than at present (Fig.3).

  adolescents (≈ 12/15 years) and 2 children (≈ 4/5 and ≈ 7/8 years) exhibit all the criteria characteristic of Neanderthal cannibalism: in situ material on an undisturbed living floor, with fireplaces, fauna and a lithic industry, precise stratigraphic framework and modern excavations. Preservation of the osteological surfaces is excellent. The bone remains show neither the action of carnivores nor weathering alteration. Cut marks are spread over 50% of the human remains and distributed over the entire skeleton from the cranium and mandible to the metapodials and phalanges. Percussion marks are visible on all

  much lower and more dispersed in contrast to open steppe formation where a rich pasturage favored abundant species, Mammoth being a typical representative. Throughout Neanderthal history, the open steppe biome was largely dominant.

Fourthly, our inventory

  of Mousterian sites attributed to MIS 5e in the southern half of France and elsewhere in Europe shows a collapse of the human population during the Eemian period of global warming. According to Prüfer et al. (2017), the Neanderthal population has been low throughout the life of this species, on the order of tens of thousands of individuals, even at its peak and contrary to what one might suppose by referring to the many artifacts collected on the Eurasian continent.

Fig. 1 :

 1 Fig. 1: Vertebrate faunas of the MIS 5e in Western Europe.

Fig. 3 :

 3 Fig. 3: Distribution over time of three thermophilous reptiles (Testudo hermanni, Rhinechis scalaris and Malpolon monspessulanus) identified in level XV of Baume Moula-Guercy.

Fig. 4 :

 4 Fig.4: Sites with Neanderhal cannibalism (letters) and burials (numbers). With : A= Baume Moula-Guercy, B=El Sidron, C=La Troisième caverne de Goyet, D=Krapina, 1=Roche à Pierrot, 2=La Ferrassie, 3=Le Regourdou, 4=La Chapelle-aux-Saints, 5=Le Moustier, 6=Roc- 7=La Quina, 8=Spy, 9=Feldhofer, 11=Mezmalskaya, 12=Dederyeh, 13=Shanidar, 14=Amud, 15=Kebara, 16=Tabun and 17=Teshik Tash.

Fig. 5 :

 5 Fig.5: Prehistoric sites (stratified cave sites) from MIS 6 to MIS 3 in southern France (south of the 46 th parallel). This region is one of the richest in Middle Palaeolithic sites of Europe. The presence of sites from all periods from the end of MIS 6 to the end of MIS 3 suggests a clear decline or collapse in human demography during MIS 5e.

Fig. 6 :

 6 Fig. 6: Quantification of the large mammal species of the level XV of Baume Moula-Guercy, in terms of NISP (number of identified specimens) and MNI (minimal number of individuals). With: A = relative proportions of ungulates, carnivores and Homo neanderthalensis according to NISP; B = relative proportions of species consumed by humans according to MNI and C = comparison between MNI and NISP of species consumed by humans.

  The three species of reptile found in layer XV; Testudo hermanni, Malpolon monspessulanus, and Rhinechis scalaris, are specifically Mediterranean and are no longer living in the Soyons region. These bioclimatic markers show that the climate around the cave during the MIS 5e period was warmer and probably drier than at present, and at least comparable to that of the current Mediterranean coast. The Mediterranean climate extended more widely to the north than in the present day, following the Rhône Valley. Layer XIV also contains remains of Testudo hermanni, suggesting a relative stability of climatic conditions in this area throughout the duration of the Eemian, as shown by pollen data in the south of the European continent

  . A few regions in Europe, such as the southeast of France, remained inhabited and could represent one of the population pools from which the Neanderthal groups of Western Europe reconstituted. The first 87 Sr/ 86 Sr isotopic analysis carried out on two Neanderthal teeth from layer XV provides direct evidence of a childhood food source from within the Rhône Valley[START_REF] Willmes | Strontium isotope tracing of prehistoric human mobility in France[END_REF][START_REF] Willmes | A comprehensive chronology of the Neanderthal site Moula-Guercy, Ardèche, France[END_REF] and a local origin for this population.

Acknowledgements

We are grateful to Leslea Hlusklo, Tim White and Marc Cheylan for useful discussions. We thank Bernard Magnaldi for his help in illustration work, Rebecca Jabbour, Dorothy Dechant, Gary Richards and Lois Woodthorpe for the correction and improvement of English text and the many researchers that stand behind the results presented: especially Evelyne Crégut-Bonnoure, Emilie Guillaud, Patricia Valensi, Véronique Michel, Salvador Bailon and Michel Thinon. The authors thank the editor, as well as both anonymous reviewers and the associate editor for providing contribution to greatly improving the manuscript.

highlighted at Baume Moula-Guercy is not a mark of bestiality or sub-humanity.

The synthesis of the data makes it possible to interpret this occurrence as a short and single episode of survival endo-cannibalism in response to nutritional stress induced by rapid and radical environmental changes.

Following this episode, the same human cultural group continued to include this site in its seasonal cycle throughout the Eemian period [START_REF] Defleur | Mousterian lithic assemblages from Moula-Guercy cave (Soyons, Ardèche). Excavations 1993-1999[END_REF]. This site demonstrates the practice of cannibalism by Neanderthals [START_REF] Defleur | Neanderthal Cannibalism at Moula-Guercy, Ardèche, France[END_REF] and provides a credible explanation for this behaviour.

Competing interests

The authors have no competing interests


Jean-François Coulombel
email: jfcoulom@math.univ-lille1.fr

The hyperbolic region for hyperbolic boundary value problems

Keywords: AMS subject classification: 35L50, 35L35 Hyperbolic systems, boundary value problems, hyperbolic region

The well-posedness of hyperbolic initial boundary value problems is linked to the occurrence of zeros of the so-called Lopatinskii determinant. For an important class of problems, the Lopatinskii determinant vanishes in the hyperbolic region of the frequency domain and nowhere else. In this paper, we give a criterion that ensures that the hyperbolic region coincides with the projection of the forward cone. We give some examples of strictly hyperbolic operators that show that our criterion is sharp.

Introduction

In this paper, we consider initial boundary value problems for hyperbolic systems. Such problems read:

     L u := ∂ t u + d j=1 A j ∂ x j u = F (t, x) , (t, x) ∈ R + × R d + , B u(t, y, 0) = g(t, y) , (t, y) ∈ R + × R d-1 , u(0, x) = f (x) , x ∈ R d + , (1)
where the spatial domain is the half-space R d + = {x ∈ R d , x d > 0} and the notation x = (y, x d ) is used. The A j 's are N × N real matrices, and B is a p × N real matrix. We always assume d ≥ 2 in what follows.

The well-posedness of (1) can be characterized with the help of a complex valued function ∆, that is known as the Lopatinskii determinant and that depends on the variables (z, η), z ∈ C with Im z ≤ 0 and η ∈ R d-1 . We refer to the original articles [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF][START_REF] Sakamoto | Mixed problems for hyperbolic equations. I. Energy inequalities[END_REF]10] as well as to the recent book [2, chapter 4] for a detailed description of the theory. The function ∆ can be chosen to be positively homogeneous of degree 0 with respect to the variables (z, η). If ∆ does not vanish on the closed half-sphere {Im z ≤ 0, |z| 2 + |η| 2 = 1}, then (1) is strongly well-posed, meaning that source terms in L 2 give rise to a unique solution u in L 2 that depends continuously on the data. When ∆ vanishes in the open half-sphere {Im z < 0, |z| 2 + |η| 2 = 1}, (1) is ill-posed.

In [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF], an open class of weakly well-posed problems has been exhibited. This so-called WR class is made of problems for which ∆ does not vanish in the open half-sphere but vanishes at first order in the hyperbolic region of {Im z = 0, |z| 2 + |η| 2 = 1}. Problems in the WR class arise 1 naturally in shock wave theory in fluid dynamics, see e.g. [2, chapter 15], and in other various physical contexts. They give rise to amplification and instability phenomena for geometric optics expansions, see e.g. the review [START_REF] Majda | Nonlinear geometric optics for hyperbolic mixed problems[END_REF] and the references therein, and to an increase in the speed of propagation, see e.g. [START_REF] Ikawa | Mixed problem for the wave equation with an oblique derivative boundary condition[END_REF]. For an arbitrary system, it may be difficult to locate the hyperbolic region and to determine whether a given problem of the form (1) belongs to the WR class. In this paper, we are interested in finding a criterion that allows to compute easily the hyperbolic region in terms of the so-called forward cone. Our criterion involves the decomposition of the characteristic polynomial as a product of irreducible factors. When the irreducible factor associated with the extreme eigenvalues has degree 2, the hyperbolic region coincides with the projected forward cone. This criterion covers some well-known cases such as the linearized Euler equations, the wave equation and the elasticity system. We give some counter-examples with irreducible hyperbolic polynomials of degree 3 or 4 for which the hyperbolic region does not coincide with the projected forward cone. This shows that our criterion is sharp.

Notations In all this article, ξ denotes a frequency vector in R d that is decomposed as ξ = (η, ξ) with η ∈ R d-1 , ξ ∈ R. For instance, (0, 1) denotes the last vector of the canonical basis of

R d . If (τ, ξ) ∈ R × R d , we define π(τ, ξ) = (τ, η) ∈ R × R d-1
the vector obtained by deleting the last coordinate of ξ.

Main results

Let us consider the operator L in (1) and introduce the symbol:

A(ξ) := d j=1 ξ j A j , ξ ∈ R d . (2)
The characteristic polynomial of L is:

P (τ, ξ) := det(τ I + A(ξ)) , (τ, ξ) ∈ R × R d . (3)
We make the following assumption of hyperbolicity with constant multiplicity:

Assumption 1. There exist some real valued analytic functions λ 1 , . . . , λ q on R d \{0} and some integers α 1 , . . . , α q such that the characteristic polynomial P defined by (3) satisfies:

∀ (τ, ξ) ∈ R × (R d \ {0}) , P (τ, ξ) = q j=1 (τ + λ j (ξ)) α j , λ 1 (ξ) < • • • < λ q (ξ) .
Moreover, the λ j (ξ)'s are semi-simple eigenvalues of the matrix A(ξ) (their geometric and algebraic multiplicty are equal).

For simplicity, we also assume that the boundary {x d = 0} is non-characteristic, that is:

Assumption 2. The matrix A d is invertible.
The two main objects used in this paper are the so-called forward cone and hyperbolic region, and are defined as follows (see e.g. [START_REF] Benzoni-Gavage | Multidimensional hyperbolic partial differential equations[END_REF]):

Definition 1.
• The characteristic variety of the operator L is Char L := {(τ, ξ) ∈ R × R d /P (τ, ξ) = 0}. The forward cone Γ is the connected component of (1, 0) in the complementary set of Char L.

• The hyperbolic region H is the set of all (τ, η) ∈ R × R d-1 \ {(0, 0)} such that the matrix:

A (τ, η) := -A -1 d (τ I + A(η, 0)) , (4)
is diagonalizable with real eigenvalues.

The nonzero elements of πΓ always belong to the hyperbolic region H , see [2, chapter 8]. Recall that π(τ, ξ) = (τ, η). Thanks to the fact that H is a symmetric cone:

(τ, η) ∈ H , s ∈ R \ {0} =⇒ (s τ, s η) ∈ H , we thus have: (πΓ ∪ -πΓ) \ {(0, 0)} ⊂ H . (5)
In this paper, we are interested in characterizing the operators L for which the opposite inclusion in (5) holds, meaning that the hyperbolic region H coincides with the projected forward cone πΓ and its symmetric set -πΓ.

Let us first begin with the special (easy) case when the spectrum of A d is either positive or negative: Proposition 1. Let assumptions 1 and 2 hold. Then we have either λ 1 (0, 1) > 0 or λ q (0, 1) < 0 if and only if πΓ = R × R d-1 . In that case, we have H = R × R d-1 \ {(0, 0)}, and H is connected.

We now consider the general case where A d has positive and negative eigenvalues. Our first main result is the following: Theorem 1. Let assumptions 1 and 2 hold, and assume that the inequalities λ 1 (0, 1) < 0 < λ q (0, 1) hold. Then πΓ and -πΓ are two disjoint connected components of H . Consequently, we have H = πΓ ∪ -πΓ if and only if H has exactly two connected components.

Our purpose is now to give necessary or sufficient conditions on the operator L that ensure that H has two connected components. Our criterion below involves the decomposition of P as a product of irreducible factors. We therefore recall the following result:

Proposition 2 ([2]).
Let assumption 1 hold. Then the characteristic polynomial P splits as:

P (τ, ξ) = J j=1 P j (τ, ξ) β j , (6)
where the polynomials P j are normalized by P j (1, 0) = 1 and satisfy the following properties:

• each P j is a homogeneous polynomial of (τ, ξ),

• the P j 's are irreducible in R[τ, ξ] and pairwise distinct,

• for ξ ∈ R d \ {0}, the roots of each P j (•, ξ) are real and simple,

• for ξ ∈ R d \ {0}, the roots of P j (•, ξ) and P k (•, ξ) are pairwise distinct if j = k.

Up to reordering the P j 's, we can always assume that -λ 1 (ξ) is a root of P 1 (•, ξ) for all ξ. This convention is used from now on. Our criterion is the following: Theorem 2. Let assumptions 1 and 2 hold, and assume that the inequalities λ 1 (0, 1) < 0 < λ q (0, 1) hold. Assume furthermore that in the decomposition (6), P 1 has degree 2. Then H = πΓ ∪ -πΓ.

As we shall see with some explicit examples below, the criterion of Theorem 2 is "optimal". More precisely, there are examples of strictly hyperbolic operators L with P 1 of degree 3 or 4 such that the corresponding hyperbolic region H has more than two connected components. In [1, page 1080], the authors expected1 that for hyperbolic operators with constant multiplicity, H would coincide with πΓ ∪ -πΓ. Our examples show that this is unfortunately not true. As a matter of fact, we believe that in space dimension 2, as soon as the degree of P 1 is greater than 3, it may happen that H has more than two connected components.

We shall also give an example where P 1 has degree 3 and where we still have H = πΓ∪-πΓ. Therefore the criterion of Theorem 2 is not a necessary and sufficient condition. However Theorem 2 predicts that the only general case where the hyperbolic region is easily computable corresponds to a polynomial P 1 of degree 2. This situation occurs for the linearized Euler equations, as well as for the wave equation or the elasticity system (these were the examples treated in [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF]). When the degree of P 1 is greater than 3, the hyperbolic region can have many connected components, and it may become difficult to check the WR condition for the system (1) since the hyperbolic region must then be determined by computing the spectrum of A , which may be difficult, especially when the size N of the system is large.

Proof of the main results

We recall that the forward cone Γ coincides with the set

{(τ, ξ) ∈ R × R d /τ + λ 1 (ξ) > 0}, see e.g. [2, chapter 1]. In a similar way, -Γ = {(τ, ξ) ∈ R × R d /τ + λ q (ξ) < 0}.
We also recall that Γ and -Γ are open and convex.

Proof of Proposition 1

Let us first assume that λ 1 (0, 1) > 0, or in other words that A d has only positive eigenvalues. Let (τ 0 , η 0 ) ∈ R × R d-1 . We introduce the function:

g : ξ ∈ R -→ τ 0 + λ 1 (η 0 , ξ) .
The homogeneity and continuity properties of λ 1 show that g(ξ) ∼ ξ λ 1 (0, 1) as ξ tends to +∞. Consequently there exists a real number ξ 0 such that g(ξ 0 ) > 0, and we get (τ 0 , η 0 ) ∈ πΓ. A similar proof with ξ tending to -∞ works if we consider the case λ q (0, 1) < 0.

Let us now assume that πΓ = R × R d-1 . Since (0, 0) ∈ πΓ, there exists ξ 0 ∈ R \ {0} such that λ 1 (0, ξ 0 ) > 0. We have:

λ 1 (0, ξ) = ξ λ 1 (0, 1) , if ξ ≥ 0, -|ξ| λ q (0, 1) , if ξ ≤ 0.
This shows that we have either λ 1 (0, 1) > 0 (when ξ 0 > 0) or λ q (0, 1) < 0 (when ξ 0 < 0). The proof of Proposition 1 is complete. In this case, the hyperbolic region H is connected and πΓ = -πΓ (even though Γ and -Γ are disjoint).

Proof of Theorem 1

We now assume that the inequalities λ 1 (0, 1) < 0 < λ q (0, 1) hold. Then the proof of Proposition 1 above shows that (0, 0) ∈ πΓ ∪ -πΓ. More precisely, πΓ does not contain any element of the form (τ, 0) with τ ≤ 0, and -πΓ does not contain any element of the form (τ, 0) with τ ≥ 0.

Let us first begin with the following result: In particular, we necessarily have η = 0 (otherwise τ > 0 and τ < 0). Moreover there exist some real numbers ξ 1 , ξ q such that:

τ + λ 1 (η, ξ 1 ) > 0 , τ + λ q (η, ξ q ) < 0 .
For all j = 1, . . . , q, we introduce the real analytic function:

f j : ξ ∈ R -→ τ + λ j (η, ξ) .
Following an argument used in the proof of Proposition 1, we know that f 1 (ξ) tends to -∞ as ξ tends to ±∞. In the same way, f q (ξ) tends to +∞ as ξ tends to ±∞. The intermediate value Theorem shows that f 1 and f q vanish both at least twice: there exist ξ 1 < ξ 1 and ξ q < ξ q such that f 1 (ξ 1 ) = f 1 (ξ 1 ) = f q (ξ q ) = f q (ξ q ) = 0. If 2 ≤ j ≤ q -1, we have:

f j (ξ 1 ) = τ + λ j (η, ξ 1 ) > τ + λ 1 (η, ξ 1 ) > 0 , f j (ξ q ) < 0 .
This shows that there exists a real number ξ j such that f j (ξ j ) = 0.

It is not difficult to check that the q + 2 real numbers ξ 1 , . . . , ξ q , ξ 1 , ξ q are pairwise distinct.

The polynomial P (τ , η,

•) = f α 1 1 . . . f αq q
has degree N , and we have shown that ξ 1 , . . . , ξ q are roots of multiplicity at least equal to α 1 , . . . , α q and ξ 1 , ξ q are roots of multiplicity at least equal to α 1 , α q . We thus obtain:

N ≥ 2 α 1 + α 2 + • • • + α q-1 + 2 α q = N + α 1 + α q ,
which is a contradiction. The proof of Lemma 1 is complete.

Our goal now is to show that πΓ is the connected component of (1, 0) in H , and similarly that -πΓ is the connected component of (-1, 0) in H .

We first use the fact that H is open. This is indeed a consequence of the block structure condition proved in [START_REF] Métivier | The block structure condition for symmetric hyperbolic systems[END_REF] for the matrix A defined in (4). We thus know that the connected components of H are also open. Let Ω denote the connected component of (1, 0) in H and let us assume that πΓ = Ω, in other words πΓ is strictly included in Ω. We thus consider an element (τ 0 , η 0 ) ∈ Ω \ πΓ. The set Ω is open and connected. It is therefore pathwise connected. Let us consider a continuous path {(τ s , η s ), s ∈ [0, 1]} in Ω that joins (τ 0 , η 0 ) and (1, 0). Consider now s := inf{s ∈ [0, 1] , (τ s , η s ) ∈ πΓ}. It is standard to show that (τ s , η s ) belongs to the boundary of πΓ.

Up to now, we have constructed an element (τ , η) ∈ H ∩ ∂(πΓ). In particular, we have η = 0. We are going to show that (τ , η) is a glancing mode for which the matrix A (τ , η) is not diagonalizable. We know that (τ , η) is the limit of a sequence (τ n , η n ) n∈N of elements of πΓ. For all integer n there exists a real number ξ n such that τ n + λ 1 (η n , ξ n ) > 0. The sequence (τ n , η n ) n∈N is bounded because it converges so the sequence (ξ n ) n∈N is necessarily bounded. Up to extracting a subsequence, it converges towards a real number ξ, and we obtain:

τ + λ 1 (η, ξ) ≥ 0 .
The quantity τ + λ 1 (η, ξ) cannot be positive because (τ , η) belongs to the boundary of πΓ and not to πΓ. We thus have τ + λ 1 (η, ξ) = 0. Moreover, we have ∂ ξ λ 1 (η, ξ) = 0 for otherwise we could find a real number ξ close to ξ such that τ + λ 1 (η, ξ) > 0. We have thus obtained:

τ + λ 1 (η, ξ) = ∂ ξ λ 1 (η, ξ) = 0 ,
which means that (τ , η) is a so-called glancing mode. For such frequencies, the matrix A is not diagonalizable, see [START_REF] Métivier | The block structure condition for symmetric hyperbolic systems[END_REF]. For the sake of completeness, we briefly recall the proof of this claim. First of all, we have Ker (A (τ , η) -ξ I) = Ker (τ I + A(η, ξ)) so the geometric multiplicity of the eigenvalue ξ equals α 1 . Moreover, we can write:

τ + λ 1 (η, ξ) = (ξ -ξ) 2 ϑ 1 (ξ) ,
where ϑ 1 is smooth in a neighborhood of ξ. A simple calculation then gives:

det(A (τ , η) -ξ I) = (ξ -ξ) 2 α 1 ϑ 2 (ξ) ,
where ϑ 2 is smooth in a neighborhood of ξ. This shows that the algebraic multiplicity of the eigenvalue ξ equals at least 2 α 1 , so A (τ , η) is not diagonalizable.

We have therefore proved that πΓ is the connected component of (1, 0) in H , and in a similar way -πΓ is the connected component of (-1, 0) in H . In particular, H has at least two connected components, and the proof of Theorem 1 follows.

Proof of Theorem 2

Let us first observe that P 1 can not have degree 1. Otherwise, λ 1 would be a linear function of ξ. In particular, we would have λ 1 (-ξ) = -λ 1 (ξ) for all ξ ∈ R d . But we also have λ 1 (-ξ) = -λ q (ξ) so q would equal 1, and this is incompatible with the assumption that A d has two distinct eigenvalues.

Let us assume from now on that P 1 has degree 2. Because P 1 is homogeneous of degree 2, we have P 1 (λ 1 (ξ), -ξ) = 0 for all ξ. Applying to -ξ, we get P 1 (-λ q (ξ), ξ) = 0 for all ξ, so we obtain:

P 1 (τ, ξ) = (τ + λ 1 (ξ)) (τ + λ q (ξ)) .
Consider now (τ, η) ∈ H . In particular, the characteristic polynomial of the matrix A (τ, η) has real roots. We compute: det(ξ I -A (τ, η)) = (det A -1 d ) P (τ, η, ξ) , so P (τ, η, •) has only real roots. Using (6), P 1 (τ, η, •) has only real roots. Let ξ 1 ∈ R satisfy P 1 (τ, η, ξ 1 ) = 0. We have either τ + λ 1 (η, ξ 1 ) = 0 or τ + λ q (η, ξ 1 ) = 0. In the first case, we necessarily have ∂ ξ λ 1 (η, ξ 1 ) = 0, otherwise (τ, η) would be a glancing mode and, as we have seen at the end of the proof of Theorem 1, A (τ, η) would not be diagonalizable. Consequently, there exists ξ close to ξ 1 such that τ + λ 1 (η, ξ) > 0, and (τ, η) ∈ πΓ. In the second case, similar arguments lead to (τ, η) ∈ -πΓ. We have thus obtained H ⊂ πΓ ∪ -πΓ. Together with (5), the proof of Theorem 2 is complete.

Some examples

The linearized isentropic Euler equations We consider the isentropic Euler equations and linearize them around a state that corresponds to a positive density ρ > 0, and a velocity u = u e d , with e d the last vector of the canonical basis of R d . We assume that the fluid is incoming and subsonic, that is: 0 < u < c , with c the sound speed corresponding to the density ρ. In space dimension d, the characteristic polynomial of the corresponding linear operator L splits as follows:

P (τ, η, ξ) = (τ + u ξ) 2 -c 2 (|η| 2 + ξ 2 ) (τ + u ξ) d-1 .
The matrix A d has one negative eigenvalue and d positive eigenvalues (counted with their multiplicity). The hyperbolic region H is given by:

H = (τ, η) ∈ R × R d-1 /τ 2 > (c 2 -u 2 ) |η| 2 ,
and has two connected components, which is consistent with the result of Theorem 2.

The wave equation Even though the analysis above is done for first-order systems, we feel free, as in [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF], to apply the results in the case of higher order hyperbolic equations or systems. For instance, in the case of the wave equation, the characteristic polynomial is:

P (τ, ξ) = τ 2 -|ξ| 2 .
The hyperbolic region H is given by:

H = (τ, η) ∈ R × R d-1 /τ 2 > |η| 2 ,
and has two connected components, which is consistent with the result of Theorem 2 because P has degree 2.

The elasticity system The linear elasticity system reads:

∂ 2 tt z -div α (∇z + ∇z T ) + (β -α) (divz) I = 0 , z ∈ R 3 ,
with α > 0, β > 0 the Lamé coefficients. The characteristic polynomial splits as:

P (τ, ξ) = (τ 2 -c 2 s |ξ| 2 ) 2 (τ 2 -c 2 p |ξ| 2 )
, with c 2 s = α and c 2 p = α + β. As expected from Theorem 2, the hyperbolic region has two connected components, and it is given (see [1, page 1091]) by:

H = (τ, η) ∈ R × R d-1 /τ 2 > c 2 p |η| 2 .
We refer to [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF] for examples of WR problems in the case of the wave equation or the elasticity system.

From now on, we only consider the case d = 2, so η ∈ R. In that case, we recall the result of [START_REF] Helton | Linear matrix inequality representation of sets[END_REF][START_REF] Lewis | The Lax conjecture is true[END_REF] that any homogeneous hyperbolic polynomial may be represented under the form (3) with suitable real symmetric matrices A 1 , . . . , A d . (This is Lax' conjecture.) We therefore work directly with the characteristic polynomial P and forget about the matrices A j 's.

Examples with an irreducible polynomial of degree 3 The following example is taken from [11, page 426]:

P (τ, η, ξ) = τ 3 -3 η 2 + ξ 2 τ + ξ 3 . (7)
Using Cardano's rule (Lemma 12.1 in [11]), it is straightforward to check that for (η, ξ) = 0, P (•, η, ξ) has three simple real roots. It is also straightforward to check that P is irreducible in R[τ, η, ξ] so the only factor in ( 6) is P itself with multiplicity 1. The hyperbolic region corresponds to the (τ, η) ∈ R 2 \ {(0, 0)} such that P (τ, η, •) has three simple roots. Cardano's rule shows that H = {(τ, η) ∈ R 2 /τ 2 > 3 η 3 }, so H has two connected components. This is a case where the irreducible factor associated with extreme eigenvalues has degree 3 but H still has two connected component, so H = πΓ ∪ -πΓ. The example of the polynomial [START_REF] Majda | Nonlinear geometric optics for hyperbolic mixed problems[END_REF] shows that the condition in Theorem 2 is only sufficient and is not necessary. However, it does not seem possible to improve the criterion in Theorem 2 as shown with the two examples below.

Let us now introduce the following polynomial:

P (τ, η, ξ) = τ 3 -3 η 2 + ξ 2 τ + η 3 + ξ 3 . (8)
Again, it is straightforward to check that P is irreducible in R[τ, η, ξ]. Cardano's rule shows that for (η, ξ) = 0, P (•, η, ξ) has three simple real roots. The hyperbolic region H is the set of (τ, η) such that P (τ, η, •) has three simple real roots. This is equivalent to asking that the polynomial P (τ, η, • + τ ) has three simple real roots. We compute:

P (τ, η, Ξ + τ ) = Ξ 3 -3 τ 2 Ξ + (η 3 -3 τ η 2 -τ 3 ) .
Applying Cardano's rule, we get:

H = {(τ, η) ∈ R 2 /4 τ 6 > (η 3 -3 τ η 2 -τ 3 ) 2 } .
The region H is depicted in red in figure 1, where we can see that H has four connected components. We now give a quick argument that shows that H has at least four connected components (this is sufficient to prove that H is not equal to πΓ ∪ -πΓ). We introduce the homogeneous polynomial:

Q(τ, η) := 4 τ 6 -(η 3 -3 τ η 2 -τ 3 ) 2 .
Standard trigonometric relations yield: The following inequalities are obtained from straightforward computations:

f (θ) := Q(cos θ, sin θ) = - 5
f (0) > 0 , f ( π 4 ) < 0 , f ( 2 π 5 ) > 0 , f ( π 2 ) < 0 , f (π) > 0 . (9)
The only possibly difficult evaluation is when θ = 2 π/5, for which we use the relations cos(π/5) = ϕ/2 and cos(2 π/5) = 1/(2 ϕ) where ϕ = (1+ √ 5)/2 is the golden ratio. The inequalities [START_REF] Sakamoto | Mixed problems for hyperbolic equations. I. Energy inequalities[END_REF] show that the set {θ ∈

[0, π]/f (θ) > 0} contains at least three disjoint intervals [0, θ 1 [, ]θ 2 , θ 3 [, ]θ 4 , π].
Since f is π-periodic, the set {θ ∈ R/(2 π Z)/f (θ) > 0} contains at least four connected components. As a matter of fact, we can see on figure 1 that there are exactly four connected components. The example [START_REF] Métivier | The block structure condition for symmetric hyperbolic systems[END_REF] shows that when the irreducible factor associated with extreme eigenvalues has degree 3, the equality H = πΓ ∪ -πΓ may not hold anymore. We give another example of this fact below with an irreducible factor of degree 4 for which the computations are much easier.

An example with an irreducible polynomial of degree 4 The following result is elementary:

Lemma 2. Let a, b ∈ R. Then the polynomial X 4 -2 a X 2 + b has four simple real roots if and only if a > 0, b > 0 and a 2 > b.

Let us then define the polynomial: P (τ, η, ξ) = τ 4 -4 η 2 + ξ 2 τ 2 + η 4 + ξ 4 .

(10)

Lemma 2 shows that for (η, ξ) = 0, P (•, η, ξ) has four simple real roots. We now consider P as a polynomial in ξ. We can apply Lemma 2 again: P (τ, η, •) has four simple real roots if and only if the following inequalities hold:

τ 2 > 0 , τ 4 -4 τ 2 η 2 + η 4 > 0 , 3 τ 4 + 4 τ 2 η 2 -η 4 > 0 .
We thus get:

H = (τ, η) ∈ R 2 / √ 7 -2 3 η 2 < τ 2 < (2 - √ 3) η 2 or (2 + √ 3) η 2 < τ 2 .
The hyperbolic region H is depicted in red in figure 2. It has six connected components, so H = πΓ ∪ -πΓ. [10] R. Sakamoto. Mixed problems for hyperbolic equations. II. Existence theorems with zero initial datas and energy inequalities with initial datas. J. Math. Kyoto Univ., 10:403-417, 1970.

[11] M. Williams. Boundary layers and glancing blow-up in nonlinear geometric optics. Ann. Sci. École Norm. Sup. (4), 33(3):383-432, 2000.

Lemma 1 .

 1 The sets πΓ and -πΓ are open, convex, and their intersection is empty. Proof. It is clear that π is linear and surjective, so π maps open sets onto open sets. Since Γ and -Γ are convex, and π is linear, πΓ and -πΓ are open and convex. It remains to show that their intersection is empty. Let us assume that (τ , η) ∈ πΓ ∩ -πΓ.

Figure 1 :

 1 Figure 1: The hyperbolic region (in red) with four connected components for the third degree polynomial P in (8).

Figure 2 :

 2 Figure 2: The hyperbolic region (in red) with six connected components for the fourth degree polynomial P in (10).

And so did the author of this article until he studied the examples given at the end of this article !