come

Introduction

Emily Eells

Our speech, that is, all our utterances (including creative works), is filled with others' words, varying degrees of otherness or varying degrees of "our-own-ness", varying degrees of awareness and detachment. These words of others carry with them their own expression, their own evaluative tone, which we assimilate, rework, and reaccentuate. [START_REF] Bakhtin | Speech Genres and Other Late Essays[END_REF] Bakhtin 1 "Parasitic? Moi?" reads the caption to the picture attached to a recent article in The Economist. [START_REF]This reference is indebted to James McCabe's insightful and informative work on the use of French in The Economist[END_REF] Reports of the latest ornithological research are thus illustrated with a note of humour in the bird's complacent surprise that it could possibly be accused of behaving like a parasite. The shift from the English adjective "parasitic" to the French pronoun " moi" maps the question addressed by this collection of essays. What difference in meaning -if any -does it make to have the bird say "moi" rather than "me"? Does it denote an association of national characteristics such as French pride and selfimportance, or is it merely a modulation in tonality and accent? The responses presented here explore the implications of using foreign words which vary from the aesthetic to the sociological, from the political to the comic.

3

Using a French term instead of an English one is never a case of simple substitution, despite what the Queen of Hearts in Through the Looking-Glass implies to Alice when she advises her to: "Speak in French when you can't think of the English for a thing". [START_REF]Lewis Carroll Through the Looking-Glass, and What Alice Found There[END_REF] The foreign word brings something more to the text, adding value even if only in terms of playfulness and pleasure. As Beckett says of the central character in More Pricks than Kicks "he never used the English word when the foreign pleased him better". [START_REF] Beckett | More Pricks Than Kicks[END_REF] Theodor Adorno's two essays "On the Use of Foreign Words" and "Words from Abroad" 5 provide a theoretical framework for the following collection of articles. His statement: "The foreign word is better whenever its literal translation is not literal, for whatever reason" 6 defines its objective: to explore the myriad reasons -political, sociological, artistic or subjectivewhy, in what sense and in what ways, the foreign word is "better" than its literal translation.

4

This volume of articles opens with an extended inquiry into the significance of the word "decency" which Thierry Labica foregrounds as a foreign word twice removed. It derives from the Latin "decree", meaning to be fitting, and the French "décent"; Mallarmé cites it in his essay on "Les Mots anglais" as an example of a French word transformed into an English word by the simple addition of a suffix : "décence donne DECENCY." [START_REF] Mallarmé | Les mots anglais[END_REF] Paradoxically, this word of foreign origin is used to refer to a defining characteristic of Englishness. Thierry Labica reads Lord Chesterfield's letters to his son as a treatise on "decency": being English means belonging to a national community by embracing its qualities of decency which serve as a kind of passport to citizenship of a universal cosmopolitan community. External decency is a performance which creates another self, estranged and set apart from the inner self which has its own qualities of decency, namely such virtues as honesty, open-heartedness and authenticity. Labica uses neologism to represent that divide between inner and outer decency, referring to it with the verbal form "s'étranger" and the adjective "étrangé".

Orwell's notion of "common decency" is indebted to Lord Chesterfield whom the author of Why Orwell Matters seems to echo when he comments that the term of decency -"habitually applied to Orwell both as man and as writer" -is "yet another of the supposedly 'English virtues'". [START_REF] Hitchens | Why Orwell Matters[END_REF] Orwell's essay "Politics and the English language" -the subject of Pierre Guerlain's article -is a master class in the art of decent discourse, purged of political propaganda. It opens with Orwell's statement that English is "in a bad way" and develops into a critique of a slovenly use of the language which he believed resulted from oppressive ideology. He condemns pretentious use of foreign words in his list of the "swindles and perversions" characterizing a sloppy use of English. He argues that "a mass of Latin words blurs the outline and covers up the details" and does not mince his words when he offers his advice on how to improve written English: "Never use a foreign phrase […] if you can think of an everyday English equivalent." [START_REF] George | Politics and the English Language[END_REF] 6

The literary studies in this collection reflect on how some writers-in-the-making use foreign words to forge their signature style. Nathalie Saudo-Welby's analysis of Sarah Grand's The Beth Book traces the protagonist's development from a young girl at a loss for words to a mature orator, which involves learning the language of others. Beth experiences her mother tongue as a foreign, man-made language estranged from her own sensitivities, and turns to foreign languages to find her voice. Her status as a woman who uses the foreign to shape a language of her own struck one of the first reviewers of the novel who followed suit by borrowing French terms to define her as a "femme", a "fille, incomprise" [sic] and an "enfant terrible". [START_REF]See 'Sarah Grand's New Novel[END_REF] The use of foreign words grants Beth freedom of speech as a woman while at the same time anticipating the language revolution of the early twentieth century which was characterized by plurilingualism. Jean Rhys (herself a translator from French) locates the instability of her characters' states of mind in the interstices of language. Juliana Lopoukhine's analysis of how she makes the foreign language into the language of the affect in After Leaving Mr Mackenzie exemplifies Adorno's point: "Foreign words become the bearers of subjective contents: of the nuances." [START_REF] Adorno | On the Use of Foreign Words[END_REF] French words irrupt in the text and serve to trace the pangs of emotion experienced by the protagonist Julia in the wake of a romantic relationship. The shifts in typography convey the untranslatable and the inarticulable, beginning with the card advertising the Parisian hotel where Julia takes refuge. The information given in French is framed by stark black lines, like mourning stationery. The italicized foreign words destabilize the text by setting the letters at a slant, just as Julia has been thrown off-kilter by estrangement from her lover.

7

In his reflections entitled Extraterritorial, George Steiner examines "the emergence of linguistic pluralism or 'unhousedness'" [START_REF] Steiner | Extraterritorial : Papers on Literature and the Language Revolution[END_REF] which he identifies as the defining feature of modernism, studying in particular two of the authors under scrutiny in this volume: Vladimir Nabokov and Samuel Beckett. The latter's first novel Fair to Middling Women was written in English but punctuated with numerous foreign words, in a kind of hybrid language which Chiara Monti so aptly termed his "monolinguisme polyglotte". [START_REF] Montini | Genèse de la poétique bilingue de Samuel Beckett[END_REF] Pascale Sardin's reading of this novel highlights how the foreign word partakes in Beckett's aesthetic ambition to create what he calls the "non-mot". The foreign words in his novel make the first steps towards the "non word" as they seem to function as tokens devoid of any semantic significance.

8

The aesthetic notion of the "non-mot" finds a political and sociological counterpart in Guillaume Leblanc's essay on how the foreigner is colonized by the language of the society he inhabits to such an extent that his own language is affected. In his monograph Dedans, dehors he defines his concept of the « no-langue », [START_REF] Leblanc | dehors : la condition d'étranger[END_REF] by which he means a language belonging to no-one -neither the foreigner, nor his host. Before expounding on the resonance of Le Blanc's philosophy, Jean-Jacques Lecercle returns to Adorno's incisive statement: "German words of foreign derivation are the Jews of language". [START_REF] Theodor | Minima Moralia: Reflections from Damaged Life[END_REF] He argues that one of Adorno's unacknowledged reasons for returning to his native Germany after self-imposed exile during the Second World War was the fact that one's mother tongue expresses the affect. Lecercle's study of the displaced writer takes Nabokov's Pnin as a literary example of a work authored by a cosmopolitan polyglot, portraying a Russian academic in exile in America and comically at odds with the English language.

9 Pnin, as well as Nabokov's first novel in English -The Real Life of Sebastian Knight -are the subject of Julie Loison-Charles' article in which she shows how the distance created by the use of foreign words allows for comic effect. Pnin is a linguistic clown -a "funny foreigner" to use Michael Cronin's term -who "bears the taint of ridicule and grapples with the intricacies of syntax and the treacheries of faux-amis." The momentary discomfort experienced by English-speaking readers when faced with the transliterated Russian words quickly resolves into a smile when they decipher the Russian words for Shakespeare and Hamlet. The chiasmus in Russian at the beginning of the passage sets the poetic tone, which uses foreignized words to evoke the cherished memory of the character's first reading of Hamlet. The poetic is debunked by Pnin's literary appreciation as he ranks Shakespeare in translation above the original version.

11 Exile, immigration and being forced to learn what has been called a "stepmother tongue" take on political implications in the post-colonial world. New literatures in English have enhanced the language by introducing indigenous terms and expressions. Bernard Cros reads the Dictionary of South African English on Historical Principles as a record of the development of a distinctive, territorialized brand of English. He reminds us that South Africa counts eleven official languages and that English is spoken by less than ten percent of the population. It nevertheless has tremendous political importance as it was the language of freedom used to ring the changes and to toll the end of the regime of apartheid imposed by speakers of Afrikaans. South African English is undergoing constant mutation and its vocabulary has been augmented with new words related to current events and developments in society, such as the example Cros gives of the expression "black diamond" used to refer to a black person from the emerging middle classes.

12 Cécile Birks makes Coetzee's autobiographical novel Summertime the object of her study. This narrative begins with extracts from the writer's notebooks, which immediately plunge the reader into a linguistic maelstrom. Commenting on an incident of racial violence reported in the press, the author enigmatically notes "Agenbite of inwit". The reader probably has no clue what that phrase means, and might even wonder what language is being used. Unless he or she remembers -or does the necessary research to discover -that Joyce uses the term in Ulysses to signify "remorse of conscience". Coetzee uses that strange expression to throw his readers off-balance and to make them experience the instability of living in South African culture.

13 Alice Braun concentrates on another autobiographical novel: the bone people by New Zealand born Keri Hulme. She stresses that the author feels a particular affinity with the Maoris and uses their words and expressions in her text, to which she appends a glossary. The novel focuses on a trio of isolated characters -Kerewen, a recluse and a painter; Joe, a lonely widower who has been forced to espouse European culture; and, thirdly, his adopted son Simon who has lost the ability to speak in a shipwreck. The interweaving of different languages reflects the complexity of the relationship between the characters and their cultural environment. That complexity is conveyed by the typography, starting with the author's choice to portray the bone people as ordinary people without capitalizing their identity in the title. In this novel, the foreign becomes a non-word in a no-language as the Maoris express themselves with body language, for example when they greet each other with a "hongi" in which they draw close, face to face, pressing their noses together and breathing the same air. The novel concludes with a line in the Maori's language, set in block capitals. The glossary spells out its meaning: "It is the end -or the beginning", thus serving as a powerful example of "How the Empire Writes Back". [START_REF] Bill Ashcroft | The Empire Writes Back: Theory and Practice in Post-Colonial Literatures[END_REF] 14 This volume issues a salutary reminder. Although English has become a global language as a consequence of linguistic and cultural imperialism, it is constantly being enriched by the words of others. They have a visual presence in the text, forming a pattern on the page in the way they are set off by quotation marks, italicized or cast in a different font. They are adopted and adapted into the language through typography, which marks their degree of assimilation. Foreign words make the text into a musical score denoting different accents, tones of voice and modes of pronunciation. They are set in-between words of another language, where they are caught in the interstices of language. This process marks what Deleuze would call the "devenir langue". Foreign words are encircled by the words of the host text, imprisoned in the chain of words in the other language. To echo the French expression Jean Rhys translated at the end of the novel studied in this collection of articles, they are "entre chien et loup": they no longer have the wildness of the wolf, but do not yet have the tameness of the dog.

NOTES

  [START_REF] Cronin | Across the Lines: Travel, Language, Translation[END_REF] Julie Loison-Charles illustrates how the ludic verve of Nabokov's prose is sparked off by the friction of different languages. The following passage from Pnin gently mocks the eponymous antihero's nostalgia for his family and homeland in the way it weaves together Russian words and allusions to Shakespeare: And here […] there is, we recollect, also a willow and also wreaths. But where to check properly? Alas, 'Gamlet', Vil'yama Shekspira […] was not represented in Waindell College Library, and whenever you were reduced to look up something in the English version, you never found this or that beautiful, noble, sonorous line that you remembered all your life from Kroneberg's text in Vengerov's splendid edition. Sad![START_REF] Nabokov | [END_REF] 10

… plïla i pela, pela i plïla … … she floated and she sang, she sang and she floated … Of course! Ophelia's death! Hamlet! In good old Andrey Kroneberg's Russian translation, 1844 -the joy of Pnin's youth, and of his father's and grandfather's young days!


Angeline Rouers

Raphaël Jeger-Madiot

Arnaud Moris

Stéphanie Graff-Dubois

Lymphocytes T folliculaires helper et VIH Unis pour le meilleur et pour le pire

> Les lymphocytes T folliculaires helper (Tfh) ont été découverts dans les années 2000 et sont, depuis, le sujet d'intenses recherches visant à comprendre leur(s) origine(s), leur(s) différenciation(s) et leurs fonctions. Les Tfh interagissent avec les lymphocytes B (LB) dans les organes lymphoïdes secondaires. Ces interactions permettent la différenciation des LB en cellules produisant des anticorps de haute affinité. Les Tfh sont particulièrement étudiés dans des contextes pathologiques, comme les maladies auto-immunes et l'infection par le virus de l'immunodéficience humaine (VIH). Dans le cas de l'infection par le VIH, des anticorps hautement neutralisants ont été identifiés chez quelques patients. La production de ces anticorps nécessite une maturation longue et complexe des LB. L'étude de la fonction des Tfh, en lien avec la qualité de ces anticorps, pourrait aider à développer de nouvelles immunothérapies et stratégies vaccinales visant à induire des anticorps neutralisants. < de la réponse immune) et les LTh régulateurs (qui contrôlent cette réponse). Les LTh effecteurs regroupent, en plus des classiques Th1 et Th2, les lymphocytes Th17, Th9, Th22 [2] et les Tfh, pour lymphocytes T folliculaires helper [3]. Les LTh régulateurs comprennent les Treg naturels (nTreg), les Treg induits (iTreg) et les Tfh régulateurs (Tfr) [3, 4]. Les Tfh, interagissent avec les lymphocytes B (LB) dans les organes lymphoïdes secondaires 1 (OLII). Ces interactions permettent aux LB de se différencier en cellules produisant des anticorps de haute affinité qui constituent une barrière efficace et durable contre différents pathogènes. Les LB se différencient ensuite en cellules mémoires ou en plasmocytes, cellules sécrétrices d'anticorps. On distingue les plasmocytes à courte durée de vie (appelés plasmablastes) qui sécrètent d'importantes quantités d'anticorps dans les OLII et dans le sang, et les plasmocytes à longue durée de vie qui se différencient dans la moelle osseuse [5]. Les LB peuvent également se différencier en LB mémoires, une population de cellules quiescentes à longue durée de vie rapidement mobilisables lors d'une seconde rencontre avec l'antigène pour lequel ils ont été sélectionnés, ou d'une réinfection par un pathogène par exemple [6]. Il existe des sous-populations distinctes de LB mémoires qui, lors de leur remobilisation, forment des centres germinatifs 2 (CG) secondaires ou se différencient en plasmablastes [7, 8].

Sorbonne universités, UPMC Univ Paris 06, Inserm U1135, CNRS ERL 8255, Centre d'immunologie et des maladies infectieuses, CIMI-Paris, 91, boulevard de l'Hôpital, 75013 Paris, France. angeline.rouers@gmail.com Les lymphocytes T exprimant CD4 (LT CD4) jouent un rôle central dans la réponse immunitaire. On les nomme classiquement LT « helper » (LTh) ou LT auxiliaires car ils potentialisent des cellules du système immunitaire présentant diverses fonctions. Pendant longtemps, la réponse lymphocytaire T CD4 auxiliaire a été définie de façon restrictive. Il était en effet considéré qu'un LT CD4 naïf se différenciait à la suite de son interaction avec une cellule dendritique (DC, pour dendritic cells), soit en LTh1 (lymphocyte T helper de type 1), pouvant soutenir la réponse cellulaire, soit en LTh2, impliqué dans la réponse humorale [1]. Outre leurs fonctions auxiliaires, les LT CD4 peuvent aussi avoir des fonctions cytotoxiques [START_REF] Marshall | Cytotoxic CD4 T cells in antiviral immunity[END_REF], souvent oubliées et généralement attribuées aux LT CD8. Aujourd'hui, nous avons une vision plus complète de la diversité des LTh. On distingue les LTh effecteurs (acteurs Vignette (Photo © Delphine Sauce).

La différenciation en Tfh est multifactorielle. Elle se déroule dans les OLII en plusieurs étapes successives aussi bien chez la souris que chez l'homme [START_REF] Crotty | T follicular helper cell differentiation, function, and roles in disease[END_REF]. Les LT CD4 naïfs interagissent d'abord avec des cellules dendritiques -cellules présentatrices d'antigènes (CPA) dites professionnelles -par l'intermédiaire de leur TCR (T-cell receptor). Le TCR reconnaît à la surface de la cellule dendritique un peptide, issu de la phagocytose et de la dégradation de l'antigène, présenté par les molécules du complexe majeur d'histocompatibilité de classe II (CMH-II). L'affinité de liaison du TCR au complexe CMH-II/peptide joue un rôle déterminant dans la différenciation des LT CD4 [START_REF] Fazilleau | The function of follicular helper T cells is regulated by the strength of T cell antigen receptor binding[END_REF]. Une forte affinité du TCR combinée à l'interaction des molécules de co-stimulation, comme B7-H2 (ou ICOS-L), le ligand d'ICOS (inducible costimulator) exprimé par les DC, induisent une signalisation permettant la différenciation du lymphocyte en lymphocyte T précurseur de Tfh (ou pré-Tfh) [START_REF] Fazilleau | The function of follicular helper T cells is regulated by the strength of T cell antigen receptor binding[END_REF][START_REF] Choi | ICOS receptor instructs T follicular helper cell versus effector cell differentiation via induction of the transcriptional repressor Bcl6[END_REF]. La différenciation en pré-Tfh s'accompagne également de l'expression par les cellules d'un facteur de relocalisation vers la zone B des OLII (ou cortex) : le récepteur CXCR5 (chemokine C-X-C motif receptor 5) qui lie la chimiokine CXCL13 produite essentiellement par les cellules dendritiques folliculaires localisées au niveau de la zone B. La zone B présente donc de fortes concentrations de CXCL13, attirant ainsi les pré-Tfh situés initialement dans la zone T des OLII (ou paracortex) [START_REF] Nurieva | Generation of T follicular helper cells is mediated by interleukin-21 but independent of T helper 1, 2, or 17 cell lineages[END_REF] (Figure 2). À noter que les Tfh produisent également la chimiokine CXCL13 qui présente alors une activité autocrine [START_REF] Crotty | Follicular helper CD4 T cells (TFH)[END_REF].

L'infection par le virus de l'immunodéficience humaine (VIH) est associée à un dysfonctionnement des LB mémoires et une altération de la production d'anticorps [START_REF] Moir | Insights into B cells and HIV-specific B-cell responses in HIV-infected individuals[END_REF]. Sachant le rôle déterminant que jouent les Tfh dans la différenciation et la maturation des LB, comprendre l'origine et la dynamique des populations de Tfh, ainsi que la nature des interactions entre ces cellules et les LB, en particulier dans le contexte de l'infection par le VIH, est donc d'une importance primordiale. Des cellules présentant des caractéristiques communes à celles des Tfh ont été observées dans le sang. Ces Tfh sanguins correspondraient au versant mémoire des Tfh des ganglions. Ils pourraient être le reflet des événements qui se produisent dans les OLII. Un intérêt particulier est donc donné aujourd'hui à ces populations de Tfh sanguines dont l'étude est plus aisée que celle des cellules des organes lymphoïdes chez l'homme.

Les lymphocytes T folliculaires helper : origine(s), différenciation(s) et fonctions

La différenciation d'un LT CD4 en LTh effecteur est conditionnée par un environnement cytokinique propre à chaque filiation. La sécrétion d'interleukine (IL-) 6 et d'IL-21 par les DC favorise la différenciation des lymphocytes en Tfh chez la souris [START_REF] Karnowski A | B and T cells collaborate in antiviral responses via IL-6, IL-21, and transcriptional activator and coactivator, Oct2 and OBF-1[END_REF]. Chez l'homme, c'est l'IL-12 qui joue un rôle majeur [START_REF] Schmitt | Human dendritic cells induce the differentiation of interleukin-21producing T follicular helper-like cells through interleukin-12[END_REF]. La différenciation vers l'un ou l'autre des soustypes de LTh est associée à l'expression de facteurs de transcription spécifiques de la voie de différenciation (Figure 1). Au cours de l'initiation de la différenciation des lymphocytes en Tfh, le répresseur transcriptionnel Bcl-6 (B-cell lymphoma 6), dont l'expression est nécessaire et suffisante à la différenciation des Tfh, est exprimé précocement [START_REF] Baumjohann | Cutting edge: distinct waves of BCL6 expression during T follicular helper cell development[END_REF]. semaines après l'exposition. La charge virale est ensuite progressivement contrôlée par l'activité antivirale des LT CD8 + . Le nombre de LT CD4 se stabilise alors, marquant l'entrée dans la phase chronique qui peut parfois durer plus de 10 ans. Cette phase peut aussi être appelée « latente », bien que le virus continue de se répliquer. Lors de la phase chronique, en l'absence de traitement, la destruction des LT CD4 qui touche particulièrement les cellules des ganglions et de la rate, est progressive [START_REF] Pantaleo | HIV infection is active and progressive in lymphoid tissue during the clinically latent stage of disease[END_REF]. Une chute des LT CD4 en dessous de 200 cellules/μL de sang entraîne l'apparition de maladies opportunistes et caractérise l'entrée dans la phase Sida.

Les lymphocytes T folliculaires régulateurs

Les Tfh au cours de l'infection

La proportion de Tfh dans le contexte de l'infection par le VIH ou le SIV dépend du stade de l'infection (primoinfection ou chronique), de la population de Tfh étudiée (définie par l'expression d'une combinaison de marqueurs), ou de la vitesse de progression vers le stade Sida (patients progresseurs ou contrôleurs5 ) (Tableau I) Dans les ganglions, les Tfh constituent des cibles privilégiées de l'infection. Ils représentent d'ailleurs un compartiment majeur pour la réplication et la production de particules virales [START_REF] Perreau | Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production[END_REF]. In vitro, les CG-Tfh issus des ganglions humains non infectés sont hautement susceptibles à l'infection par le VIH [START_REF] Kohler | Germinal center T follicular helper cells are highly permissive to HIV-1 and alter their phenotype during virus replication[END_REF]. Les Tfh sont également des cibles pour le SIV chez les macaques [START_REF] Fukazawa | B cell follicle sanctuary permits persistent productive simian immunodeficiency virus infection in elite controllers[END_REF].

Les virus VIH ou SIV présentent la particularité de former des réservoirs : intégré dans le génome de la cellule hôte qu'il infecte, le virus peut persister à l'état latent. Les traitements antirétroviraux contrôlent la réplication du virus et la charge virale, mais ne permettent pas l'élimination du virus de l'organisme. Chez les patients, une fraction importante de Tfh porte dans leur génome des copies d'ADN intégrées du VIH [START_REF] Perreau | Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production[END_REF][START_REF] Colineau | HIV-infected spleens present altered follicular helper T Cell (Tfh) subsets and skewed B cell maturation[END_REF]. Il est maintenant clairement établi que le virus persiste dans les cellules mémoires, particulièrement dans les Tfh [START_REF] Boritz | Multiple origins of virus persistence during natural control of HIV infection[END_REF]. De façon intéressante, quelques rares patients contrôlent l'infection en l'absence de traitement sans pour autant éliminer le virus (il s'agit des patients contrôleurs naturels). Malgré ce contrôle naturel, le VIH persiste chez ces individus en particulier dans les Tfh des ganglions [START_REF] Boritz | Multiple origins of virus persistence during natural control of HIV infection[END_REF]. Paradoxalement, alors que l'infection des LT CD4 par le VIH conduit à la mort de la cellule en moins de 48h, de nombreuses études ont mis en évidence une augmen-cas des Tfh. L'expression de NFAT2 par les Tfr est essentielle pour amorcer l'expression de CXCR5 et donc leur migration dans les follicules. Ces cellules régulatrices expriment les marqueurs classiques des Tfh ainsi que CTLA-4 (cytotoxic T-lymphocyte-associated protein 4) et Foxp3 (forkhead box P3), marqueurs caractéristiques des lymphocytes Treg [4]. Selon une étude récente, les Tfr pourraient également se différencier à partir de LT CD4 naïfs en contexte inflammatoire [START_REF] Aloulou | Follicular regulatory T cells can be specific for the immunizing antigen and derive from naive T cells[END_REF].

Tfh et infection par le VIH

Dans des pathologies humaines aussi diverses que les maladies autoimmunes, les cas d'immunodéficiences primaires et acquises, ou les cancers, la réponse humorale est altérée. Ces pathologies sont associées à un dysfonctionnement des Tfh à l'origine d'une perturbation de la différenciation des LB et donc à une dérégulation de la production d'anticorps [START_REF] Tangye | The good, the bad and the ugly -TFH cells in human health and disease[END_REF]. HD domain-containing protein 1), qui limite la réplication virale, est moins exprimé dans les Tfh que dans les LT CD4, rendant ainsi cette population particulièrement susceptible à l'infection [START_REF] Ruffin | Low SAMHD1 expression following T-cell activation and proliferation renders CD4 + T cells susceptible to HIV-1[END_REF]. L'infection par le VIH pourrait aussi altérer les fonctions des Tfh, notamment leur capacité à interagir avec les LB. Ainsi, in vitro, les Tfh des ganglions ou des amygdales de patients infectés par le VIH ont perdu leur capacité à aider les LB pour la production d'IgG [START_REF] Cubas R A | Inadequate T follicular cell help impairs B cell immunity during HIV infection[END_REF].

L'infection par le VIH

Ce défaut pourrait être lié à une surexpression de PD-L1 à la surface des LB qui inhiberait la sécrétion d'IL-21 par les Tfh via la signalisation PD-1 [START_REF] Cubas R A | Inadequate T follicular cell help impairs B cell immunity during HIV infection[END_REF]. Plus récemment, nous avons démontré, par une analyse du profil transcriptionnel, que les Tfh issus de la rate de patients chroniquement infectés ont un défaut d'expression de gènes codant les molécules de co-stimulation OX40, CD40L et ICOS nécessaires aux interactions Tfh/LB et à la maturation des LB [START_REF] Colineau | HIV-infected spleens present altered follicular helper T Cell (Tfh) subsets and skewed B cell maturation[END_REF].

Des Tfh dans le sang

Des LT exprimant la molécule CD4 et le récepteur CXCR5 (CD4 + CXCR5 + ) ont été identifiés dans le sang [START_REF] Schaerli | CXC chemokine receptor 5 expression defines follicular homing T cells with B cell helper function[END_REF]. Ils expriment également la L-selectine (CD62L) et le récep-tation de la proportion de Tfh dans les ganglions des patients traités au stade chronique de l'infection [START_REF] Perreau | Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production[END_REF][START_REF] Lindqvist | Expansion of HIV-specific T follicular helper cells in chronic HIV infection[END_REF]. L'accumulation des Tfh est plus marquée lorsque les patients ne reçoivent pas de traitements antirétroviraux, suggérant un lien avec la persistance du virus. La fréquence des Tfh est en effet corrélée positivement avec la charge virale mesurée dans le plasma [START_REF] Perreau | Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production[END_REF][START_REF] Lindqvist | Expansion of HIV-specific T follicular helper cells in chronic HIV infection[END_REF]. Les OLII concentrent d'importantes quantités de virus [START_REF] Embretson | Massive covert infection of helper T lymphocytes and macrophages by HIV during the incubation period of AIDS[END_REF], notamment dans les cellules dendritiques folliculaires, ce qui pourrait expliquer l'accumulation des Tfh. Cette accumulation de Tfh lors de la phase chronique de l'infection a également été observée dans les ganglions et la rate de macaques infectés par le SIV [START_REF] Xu | Persistent SIV infection drives differentiation, aberrant accumulation, and latent infection of germinal center follicular T helper cells[END_REF][START_REF] Petrovas | CD4 T follicular helper cell dynamics during SIV infection[END_REF][START_REF] Hong | Spatial alterations between CD4+ T follicular helper, B, and CD8 + T cells during simian immunodeficiency virus infection: T/B cell homeostasis, activation, and potential mechanism for viral escape[END_REF]. Cependant, lors de la primo-infection, la proportion de Tfh diffère selon la vitesse à laquelle les macaques progresseront plus tard vers le stade Sida [START_REF] Moukambi | Early loss of splenic Tfh cells in SIV-Infected rhesus macaques[END_REF][START_REF] Hong | Early lymphoid responses and germinal center formation correlate with lower viral load set points and better prognosis of simian immunodeficiency virus infection[END_REF]. Ainsi, l'établissement précoce des CG lors de l'infection est un marqueur de bon pronostic chez le macaque [START_REF] Hong | Early lymphoid responses and germinal center formation correlate with lower viral load set points and better prognosis of simian immunodeficiency virus infection[END_REF].

Tableau II. Éléments de la littérature pouvant expliquer le paradoxe entre accumulation et infection préférentielle des Tfh lors de l'infection par le VIH.

Bcl2 : B-cell lymphoma 2 ; BST2 : bone marrow stromal cell antigen 2 ; CTLA-4 : cytotoxic T-lymphocyte-associated protein 4 ; CXCL13 : chemokine (C-X-C motif) ligand 13 ; IL : interleukin ; LEDGF/p75 : lens epithelium-derived growth factor ; PD-L2 : Programmed cell-death ligand 2 ; SAMHD1 : SAM domain and HD domain-containing protein 1 ; Tfh : T follicular helper cell ; Tfr : T follicular regulatory cell ; TRIM5 : tripartite motifcontaining 5.

Selon l'agent infectieux rencontré, les molécules inflammatoires produites par les cellules immunitaires sont différentes. Le contexte inflammatoire ainsi engendré influence l'apparition dans le sang de pTfh aux profils fonctionnels variés. L'étude du phénotype, de la fréquence et des fonctions des pTfh dans le sang permettrait donc d'apprécier la qualité de l'établissement de la réponse humorale [START_REF] Morita | Human blood CXCR5 + CD4 + T cells are counterparts of T follicular cells and contain specific subsets that differentially support antibody secretion[END_REF]. Les pTfh semblent en effet être de bons marqueurs de l'efficacité vaccinale [START_REF] Schultz | Circulating HIV-specific interleukin-21 + CD4 + T cells represent peripheral Tfh cells with antigen-dependent helper functions[END_REF]. En ce qui concerne l'infection par le VIH, un seul essai vaccinal (RV144) 6 a conduit, à ce jour, à une protection, bien que limitée, contre l'infection par le virus avec une réduction de 31,2 % du nombre de personnes infectées chez les individus vaccinés par rapport à ceux ayant reçu un placebo [START_REF] Rerks-Ngarm | Vaccination with ALVAC and AIDSVAX to prevent HIV-1 infection in Thailand[END_REF]. Dans cet essai, les personnes exposées mais non-infectées présentaient une proportion plus élevée de pTfh spécifiques du VIH, produisant de l'IL-21, que des individus participant à d'autres protocoles de vaccination qui n'ont pas permis d'obtenir de protection [START_REF] Schultz | Circulating HIV-specific interleukin-21 + CD4 + T cells represent peripheral Tfh cells with antigen-dependent helper functions[END_REF]. Chez les individus vaccinés lors de l'essai, la protection a pu être corrélée à la présence d'anticorps pouvant induire la lyse des cellules infectées par le virus par la réaction d'ADCC (antibody-dependent cellular cytotoxicity)7 [START_REF] Bonsignori | Antibody-dependent cellular cytotoxicity-mediating antibodies from an HIV-1 vaccine efficacy trial target multiple epitopes and preferentially use the VH1 gene family[END_REF]. Ces observations suggèrent un rôle important de la réponse anticorps, et donc des Tfh, dans l'établissement de la protection induite par la vaccination. Des résultats similaires ont été obtenus au cours d'un essai vaccinal dans le modèle d'infection par le SIV de primates non humains [START_REF] Vargas-Inchaustegui | Vaccine induction of lymph node: resident simian immunodeficiency virus Env-specific T follicular helper clls in Rhesus macaques[END_REF].

Ces différentes études suggèrent que les pTfh pourraient participer à la mise en place d'une réponse anticorps efficace et/ou protectrice contre le virus. L'expansion des pTfh pourrait également être utilisée comme un marqueur de l'efficacité vaccinale.

Tfh et induction d'anticorps neutralisants : vers de nouvelles stratégies vaccinales ?

La découverte récente d'anticorps capables de neutraliser un large spectre d'isolats du VIH (les bnAb pour broadly neutralizing antibodies) qui sont produits par une poignée de patients infectés par le VIH, a relancé l'espoir de pouvoir développer un vaccin efficace. La plupart de ces anticorps présentent un taux important de modifications résultant de mutations somatiques dans les gènes codant les teur CCR7. Ils correspondraient à une sous-population de LT « central mémoire » (les Tcm). Sur le plan fonctionnel, la co-culture de LB isolés du sang avec des LT CD4 CXCR5 + conduit à une plus forte sécrétion d'anticorps par les LB que la co-culture avec des LT n'exprimant pas CXCR5 [START_REF] Morita | Human blood CXCR5 + CD4 + T cells are counterparts of T follicular cells and contain specific subsets that differentially support antibody secretion[END_REF]. Ainsi, Les LT CD4 CXCR5 + circulants correspondraient au versant mémoire périphérique des Tfh, les pTfh [START_REF] Chevalier | CXCR5 expressing human central memory CD4 T cells and their relevance for humoral immune responses[END_REF]. Comparée à celle des Tfh, l'expression de CXCR5 et d'ICOS à la surface des pTfh est moins intense et celle de Bcl-6 est perdue [START_REF] Schaerli | CXC chemokine receptor 5 expression defines follicular homing T cells with B cell helper function[END_REF][START_REF] Morita | Human blood CXCR5 + CD4 + T cells are counterparts of T follicular cells and contain specific subsets that differentially support antibody secretion[END_REF]. Les pTfh semblent également sécréter moins de cytokines que les Tfh des OLII [START_REF] Schaerli | CXC chemokine receptor 5 expression defines follicular homing T cells with B cell helper function[END_REF] [START_REF] Morita | Human blood CXCR5 + CD4 + T cells are counterparts of T follicular cells and contain specific subsets that differentially support antibody secretion[END_REF]. Enfin, les pTfh sécrétant de l'IL-21 semblent les plus proches des Tfh des OLII, que ce soit phénotypiquement (expression de Maf et de CXCR5) ou sur le plan fonctionnel [START_REF] Schultz | Circulating HIV-specific interleukin-21 + CD4 + T cells represent peripheral Tfh cells with antigen-dependent helper functions[END_REF]. L'expression des marqueurs ICOS, CCR7 et PD-1 témoigne de l'état activé (ICOS + PD-1 + CCR7 -) ou quiescent (ICOS -PD-1 +/-CCR7 + ) des populations de pTfh. Une sous-population de pTfh CCR7 -PD-1 hi CXCR5 + CD4 + semble provenir de LT n'étant jamais entrés dans les CG [START_REF] He | Circulating precursor CCR7loPD-1hi CXCR5+ CD4+ T cells indicate Tfh cell activity and promote antibody responses upon antigen reexposure[END_REF]. Ces pTfh peuvent induire une réponse anticorps rapide lors d'une réexposition à l'antigène, suggérant l'existence d'une réponse mémoire établie à partir des Tfh [START_REF] He | Circulating precursor CCR7loPD-1hi CXCR5+ CD4+ T cells indicate Tfh cell activity and promote antibody responses upon antigen reexposure[END_REF].

pTfh et infection par le VIH

Les pTfh, comme les Tfh des OLII, sont préférentiellement infectés par le VIH. Ils constituent un réservoir viral dans le sang [START_REF] Pallikkuth | Peripheral T follicular helper cells are the major HIV reservoir within central memory CD4 T cells in peripheral blood from chronic HIV infected individuals on cART[END_REF]. Récemment, l'expression de CD32a, un récepteur pour la région Fc des immunoglobulines (Fc gamma RIIA), a été identifiée comme un marqueur des LT CD4 porteurs de virus intégré latent, c'est-à-dire un marqueur du réservoir [START_REF] Descours | CD32a is a marker of a CD4 T-cell HIV reservoir harbouring replication-competent proviruses[END_REF]. domaines variables des immunoglobulines. Ces modifications sont certainement apportées par les LB lors de leur maturation, grâce à leurs interactions répétées avec les Tfh. Ce taux important de mutations somatiques des nAbs suggère que les LB à l'origine de leur production ont subi plusieurs cycles de maturation. L'hypothèse selon laquelle les Tfh sont importants pour le développement de ces bnAb a donc rapidement été posée. La capacité de neutralisation du VIH par les anticorps serait en fait corrélée à la proportion de pTfh [START_REF] Locci | Human circulating PD-1+CXCR3-CXCR5+ memory Tfh cells are highly functional and correlate with broadly neutralizing HIV antibody responses[END_REF]. Une étude plus récente a confirmé ce lien chez des patients contrôleurs naturels, c'est à dire contrôlant naturellement l'infection. Chez ces patients, les DC polarisent la différenciation des LT CD4 en Tfh sécréteurs d'IL-21 qui expriment fortement ICOS et sont capables d'induire la production d'importantes quantités d'IgG par les LB [START_REF] Martin-Gayo | Circulating CXCR5+CXCR3+PD-1lo Tfh-like cells in HIV-1 controllers with neutralizing antibody breadth[END_REF]. Ces résultats soulignent donc l'importance du lien existant entre les premières étapes de différenciation en Tfh et la production, plus tardive, d'anticorps à forte capacité de neutralisation. Les réponses au sein des centres germinatifs des OLII sont également importantes à considérer. Il a en effet été montré, chez le singe, que le développement d'anticorps neutralisants dépendait de la proportion des CG-Tfh [START_REF] Havenar-Daughton | Direct probing of germinal center responses reveals immunological features and bottlenecks for neutralizing antibody responses to HIV Env trimer[END_REF]. La quantité, mais aussi la qualité, des Tfh induits semblent donc déterminer l'efficacité de la réponse anticorps dirigée contre le virus. Seul l'essai vaccinal RV144 a donné lieu à une réduction modérée du taux d'infection par le VIH [START_REF] Rerks-Ngarm | Vaccination with ALVAC and AIDSVAX to prevent HIV-1 infection in Thailand[END_REF]. La conception d'un vaccin fondé sur la réponse anticorps est complexe à mettre en place en raison des capacités exceptionnelles du virus à s'adapter, par mutation, à la réponse immune de l'hôte. Durant la dernière décennie, des vaccins permettant d'induire une réponse immunitaire cytotoxique dépendant des LT CD8 spécifiques du VIH ont été développés sans succès [START_REF] Koup | Vaccine design for CD8 T lymphocyte responses[END_REF]. Désormais, la découverte des bnAb a changé le paradigme de la vaccination anti-VIH. Un vaccin universel permettant la production de bnAb, reposera nécessairement sur sa capacité à induire la différenciation de Tfh qui soient spécifiques du virus. La preuve de concept de la possibilité d'induire des bnAb contre le VIH a été réalisée chez la souris. Une stratégie d'immunisation en « primeboost » (c'est à dire avec plusieurs stimulations) utilisant différents antigènes produits à partir de la protéine d'enveloppe du virus a en effet permis de générer des bnAb spécifiques [START_REF] Mouquet | Tailored immunogens for rationally designed antibody-based HIV-1 vaccines[END_REF].

Conclusion

Figure 1 .

 1 Figure 1. Différenciation d'un lymphocyte CD4 + naïf en lymphocyte T « helper ». Selon l'environnement cytokinique imposé par la cellule dendritique (DC), le LT CD4 se différencie en un LTh particulier. Chaque LTh est associé à l'expression d'un facteur de transcription « signature ». Les lymphocytes T régulateurs naturels ne sont pas représentés dans cette figure car ils sont initiés dès leur développement dans le thymus. BCL6 : B-cell lymphoma 6 ; BNC2 : Basonuclin 2 ; DC : cellule dendritique ; FOXO4 : Forkhead Box O4 ; Foxp3 : forkhead box P3 ; GATA3 : GATA binding protein 3 ; IFN : interferon ; IL : interleukin ; iTreg : induced T regulatory cell ; LT : lymphocyte T ; Tbet : T-box transcription factor ; Tfh : T follicular helper cell ; TGF : transforming growth factor ; Th : T helper ; TNF : tumor necrosis factor ; ROR : RAR-related orphan receptors.

Figure 2 .

 2 Figure 2. Différenciation des Tfh et réaction du centre germinatif. Une première interaction entre un LT CD4 et une DC dans la zone T permet d'initier la différenciation vers un profil Tfh. En bordure de la zone B, les pré-Tfh interagissent avec des LB pré-activés. Les LB retournent dans la zone B accompagnés des Tfh pour former un CG. Des contacts répétés entre les LB et les Tfh dans le centre germinatif permettent la maturation des anticorps grâce à l'hypermutation somatique et la commutation de classe isotypique. Les anticorps produits ont une haute affinité pour l'antigène. Suite à leurs interactions avec les Tfh, les LB se différencient en LB mémoires ou en plasmablastes. BCR : B cell receptor ; CG : centre germinatif ; CMH : complexe majeur d'histocompatibilité ; DC : cellule dendritique ; IL : interleukin ; LB : lymphocyte B ; TCR : T cell receptor, Tfh : T follicular helper cell.

Les

  Tfh sont indispensables pour la différenciation des LB et la production d'anticorps. Leur différenciation à partir de LT CD4 se déroule en trois étapes successives dans les OLII : (1) un premier contact avec une DC dans la zone T ; (2) une interaction entre un pré-Tfh et un LB à la bordure de la zone T et de la zone B ; (3) une liaison soutenue entre Tfh et LB dans le CG permettant la maturation complète des LB et la production d'anticorps de haute affinité. Une population mémoire des Tfh (pTfh), dont l'origine reste à confirmer, est détectable dans le sang. Ces cellules semblent être de bons marqueurs de l'efficacité vaccinale.

  La découverte de la coopération entre les LT et les LB conduisant à la production d'anticorps remonte aux années 1960. Parmi les travaux princeps, une étude a montré que la production d'anticorps dirigés contre des globules rouges de mouton était diminuée chez des souris dépourvues de thymus, mettant en évidence le rôle des LT dans la production d'anticorps. Ce rôle des LT dans la production d'anticorps a été confirmé quelques années plus tard par des expériences d'interactions spécifiques réalisées in vitro. Cependant, à cette époque, il était suggéré que les antigènes s'accumulaient à la surface du LT pour être présentés au LB. Au début des années 1970, avec la découverte du rôle des molécules du complexe majeur d'histocompatibilite de classe II (CMH-II) dans la présentation des antigènes, les mécanismes d'interactions entre LT et LB ont été élucidés. Depuis, il est clairement établi que les antigènes sont reconnus de façon spécifique par le récepteur des LT CD4 (le TCR) sous la forme de peptides qui sont associés aux molécules du CMH-II et présentés à la surface du LB. Cette interaction conduit à la maturation et à la différenciation des LB.

	Dans le cas de l'infection par le VIH, les Tfh ont été étudiés chez les
	personnes infectées, sous traitement ou non, et chez des patients
	infectés mais contrôlant naturellement l'infection [5]. Le modèle
	d'infection expérimental des macaques par le SIV (virus de l'immuno-
	déficience simienne) permet aussi d'étudier la dynamique des Tfh dans
	le contexte infectieux.

L'infection par le VIH se décline en trois phases : la primo-infection, la phase chronique et la phase Sida (syndrome de l'immunodéficience acquise). Lors de la phase de primo-infection (entre 2 et 4 semaines), le nombre de LT CD4 (cibles principales du virus) chute et la charge virale augmente de façon exponentielle pour atteindre un pic 3 à 4 Un peu d'histoire… (d'après N.A. Mitchison

[START_REF] Mitchison | T-cell-B-cell cooperation[END_REF]

)

Expression des facteurs de restriction et co-facteurs

	Virus	Stade	Progression	Organe (a) Prolifération/Survie Phénotype		Dynamique	Réf.
	VIH VIH SIV SIV Sous-expression du Chronique Chronique Observation Précoce Précoce marqueur de prolifération Ki-67	N.D. N.D. Population cellulaire VIH + Ganglions Ganglions Progresseurs Rate Progresseurs lents Ganglions Tfh spléniques Pas de meilleure prolifération CXCR5 + PD-1 + Bcl-6 + Interprétation CXCR5 + PD-1 + Bcl-6 + (CCR7 -CD45RA -) CXCR5 + PD-1 + CXCR5 + PD-1 + des Tfh VIH +	Accumulation En faveur de l'expansion ? Accumulation Perte Accumulation Non	En faveur de l'infection ? Non	[21] Réf. [26] [31] [32] [24]
	Surexpression du
	SIV apoptotique Bcl2 Précoce marqueur anti-	Progresseurs rapides Lymphocytes CXCR5 +		Ganglions Meilleure survie des Tfh VIH + CXCR5 + PD-1 +	Oui	Pas d'accumulation Non	[32] [53]
	SIV SIV Surexpression de PD-L2 Chronique Chronique	Progresseurs lents Progresseurs lents CG-Tfh	Ganglions Rate Ganglions Tfh VIH + plus sensible à CXCR5 + PD-1 + l'apoptose CD45RA-CD62L + CXCR5 + PD-1 + (b) Différenciation en Tfh	Non	Accumulation Accumulation	Non	[30] [52] [31]
	SIV	Chronique Observation	Progresseurs rapides Population cellulaire VIH + Ganglions	CXCR5 + PD-1 + Interprétation	Perte En faveur de l'expansion ?	En faveur de l'infection ?	[31] Réf.
	SIV Surexpression des gènes Sida	Progresseurs		Ganglions	CXCR5 + PD-1 +		Perte	[52]
	SIV IL-21 codant pour CXCL13 et Chronique	Contrôleurs Tfh spléniques		Ganglions Différenciation poussée en Tfh CXCR5 + PD-1 +	Oui	Préservation	Non	[52] [24]
	SIV SIV Surexpression du gène Chronique Chronique codant pour le récepteur de l'IL-6	Progresseurs ND LT CD4 VIH + plus sensibles à une différenciation en Tfh LT CD4 naïfs spléniques	Ganglions Ganglions	CXCR5 + PD-1 + CD28 fort CD95 fort CCR7 faible PD-1 fort	Oui	Accumulation Accumulation	Non	[28] [29] Non publié
					(c) Contrôle du CG
		Observation	Population cellulaire VIH +	Interprétation	En faveur de l'expansion ?	En faveur de l'infection ?	Réf.
	Diminution du ratio Tfr/ Tfh	Ratio Tfh/Tfr de macaques SIV +	Nombre insuffisant de Tfr pour contrôler les Tfh	Oui		Non	[54]
	Sous-expression des
	gènes codant pour CTLA4 et IL-10 et diminution de	Tfh VIH + spléniques		Suggère un défaut fonctionnel des Tfr VIH + (parmi les Tfh)	Oui		Non	[24]
	la sécrétion d'IL-10
	Sous-expression du gène
	codant pour le récepteur	Défaut fonctionnel des Tfr SIV + Tfr SIV +		Oui		Non	[54]
	de l'IL-2
		(d) Observation Population cellulaire VIH +	Interprétation	En faveur de l'expansion ?	En faveur de l'infection ?	Réf.
	Expression préférentielle du co-facteur LEDGF/ p75 et des facteurs de et TRIM5 restriction BST2/tetherin	Tfh et CG-Tfh		Meilleure survie des Tfh VIH + cellules couplée à une limitation de la réplication du virus dans ces	Oui		Oui	Non publié
	Plusieurs hypothèses pourraient expliquer ce paradoxe entre une pro-portion de Tfh plus élevée et leur infection préférentielle par le virus (Tableau II) : (a) les Tfh pourraient avoir une capacité de survie ou de Sous-expression du facteur de restriction SAMHD1 Tfh Plus forte susceptibilité à l'infection par le VIH	Non		Oui	[33]

prolifération plus importante que les LT CD4 ; (b) l'infection pourrait favoriser la différenciation des LT CD4 en Tfh ; (c) les mécanismes de régulation par les Tfr de l'interaction Tfh/LB pourraient être altérés ; (d) la balance d'expression de facteurs pro et anti-viraux dans les Tfh favoriserait l'intégration du virus et une faible réplication virale. De façon intéressante, le facteur de restriction SAMHD1 (SAM domain and Tableau I. Dynamique des Tfh observée dans différentes études. Bcl-6 : B-cell lymphoma 6 ; CCR : chemokine (C-C motif) receptor ; CXCR : chemokine (C-X-C motif) receptor ; PD-1 : programmed cell death 1 ; ND : non déterminé. « + » : le marqueur est exprimé ; « -» : le marqueur n'est pas exprimé.

  Cette découverte ouvre des perspectives fascinantes quant à la possibilité d'éliminer par immunothérapie le réservoir du virus chez les personnes traitées. À l'heure actuelle, la population de LT CD4 exprimant CD32a n'a pas été précisément caractérisée mais il est fort à parier que les Tfh infectés par le VIH expriment également ce marqueur.

	Aujourd'hui, nous avons une meilleure compréhension
	de la biologie des Tfh : leur processus de différen-
	ciation, leurs mécanismes d'interaction avec les LB
	et leur rôle dans la réaction du CG. Cependant, dans
	des pathologies comme l'infection par le VIH, certains
	aspects restent mal compris. Plusieurs hypothèses ont
	été émises concernant l'augmentation des Tfh chez les
	patients infectés par le VIH mais il s'agit vraisembla-
	blement d'un phénomène multifactoriel. Nous n'avons,
	à ce jour, aucune donnée concernant l'impact possible
	de l'infection par le VIH sur les fonctions des Tfh. Il reste
	donc à déterminer si les perturbations des fonctions des
	Tfh sont directement dues à leur infection par le VIH, ou
	si elles sont le résultat de modifications du microenvi-
	ronnement, ou d'autres acteurs cellulaires, impliqués
	dans la réaction du CG. ‡
	SUMMARY
	Follicular helper T cells and HIV -United for better
	and worse
	Follicular helper T cells (Tfh) have been discovered in
	lymph nodes and, since then, are the focus of very inten-
	sive research to understand their origin, differentiation
	and functions. Tfh interact with B cells in the secondary
	lymphoid organs leading to B cell differentiation and
	maturation. Tfh are particularly studied in pathological
	contexts such as autoimmune diseases and infection by
	the human immunodeficiency virus (HIV). In the context
	of HIV infection, broadly neutralizing antibodies have
	been identified in a few patients. The generation of
	these broadly neutralizing antibodies requires a long
	and complex maturation of B cells in the secondary
	lymphoid organs. Characterizing Tfh functions and the
	relation with the quality of antibodies in HIV infection
	might help in designing novel immunotherapies and
	vaccination strategies to induce broadly neutralizing
	antibodies. ‡
	LIENS D'INTÉRÊT
	Les auteurs déclarent n'avoir aucun lien d'intérêt concernant les don-
	nées publiées dans cet article.
	RÉFÉRENCES
	1. Mosmann T, Coffman R. TH1 and TH2 cells: different patterns of lymphokine
	secretion lead to different functional properties. Annu Rev Immunol 1989 ;
	7 : 145-73.
	2. Eyerich S, Eyerich K, Pennino D, et al. Th22 cells represent a distinct human
	T cell subset involved in epidermal immunity and remodeling. J Clin Invest
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	3. Zhu J, Yamane H, Paul W. Differentiation of effector CD4 T cell populations.
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4. Linterman MA, Pierson W, Lee SK, et al. Foxp3+ follicular regulatory T cells control the germinal center response. Nat Med 2011 ; 17 : 975-82. 5. Kometani K, Kurosaki T. Differentiation and maintenance of long-lived plasma cells. Curr Opin Immunol 2015 ; 33 : 64-9.

Organes où sont présentés les antigènes et s'initient les réponses immunitaires adaptatives. Les OLII sont généralement constitués d'une zone riche en lymphocytes T et en cellules dendritiques, et d'une zone riche en lymphocytes B, macrophages et cellules dendritiques folliculaires.

Structure se formant dans le ganglion et conduisant à la maturation, la prolifération et la sélection des lymphocytes B. Les CG sont organisés en deux zones : claire, où a lieu le processus de sélection et sombre où les lymphocytes B prolifèrent.

L'enzyme AID (activation-induced cytidine deaminase) induit des mutations au niveau du domaine variable des chaînes lourdes et légères des immunoglubulines permettant, après sélection des lymphocytes B, l'expression d'un BCR et la production d'anticorps de plus haute affinité pour l'antigène.

Patients capables de contrôler naturellement l'infection par le VIH en l'absence de traitement. On distingue les elite controllers dont le virus n'est plus détectable et les patients virémiques qui ne progressent pas ou très lentement vers la maladie malgré une charge virale détectable.

Il s'agit de l'essai RV144, mené conjointement par le gouvernement thaïlandais et l'armée américaine.

L'ADCC est une réaction de lyse cellulaire induite par des anticorps fixés aux cellules exprimant l'antigène spécifiquement reconnu par l'anticorps.


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

LIP ⋆ Sylvain Perifel
email: sylvain.perifel@ens-lyon.fr

The Complexity of two Problems on Arithmetic Circuits

By using arithmetic circuits, encoding multivariate polynomials may be drastically more efficient than writing down the list of monomials. Via the study of two examples, we show however that such an encoding can be hard to handle with a Turing machine even if the degree of the polynomial is low. Namely we show that deciding whether the coefficient of a given monomial is zero is hard for P #P under strong nondeterministic Turing reductions. As a result, this problem does not belong to the polynomial hierarchy unless this hierarchy collapses. For polynomials over fields of characteristic k > 0, this problem is Mod k Pcomplete. This gives a coNP Mod k P algorithm for deciding an upper bound on the degree of a polynomial given by a circuit in fields of characteristic k > 0.

Introduction

Multivariate polynomials are intensively used in computer algebra. The naive way of encoding such polynomials is to write down the list of the coefficients of all monomials, but even low-degree polynomials may have an exponentially large number of such coefficients (consider for example the determinant). That is why this possibility is not always suitable.

Arithmetic circuits can sometimes remedy this situation since their size is in general much smaller than the list of all coefficients. An arithmetic circuit over a field K is a circuit that contains addition and multiplication gates over K (instead of OR and AND gates for usual boolean circuits). With such gates, and with inputs consisting of variables and constants of K, the circuit computes a polynomial (see the preliminaries below for details). For example, the polynomial (x+y) 2 n can be computed by a circuit of size O(n) by "repeated squaring", whereas the number of monomials is exponentially large.

Though compact, this manner of storing data is not always easy to handle. There are efficient randomized algorithms for testing whether a polynomial is identically zero (see e.g. [START_REF] Jacob | Fast probabilistic algorithms for verification of polynomial identities[END_REF]), for example, but many other problems become hard when dealing with arithmetic circuits. One way to simplify them is to consider only circuits of polynomial "formal degree" ( [START_REF] Malod | Polynômes et coefficients[END_REF] or [START_REF] Miller | Efficient parallel evaluation of straight-line code and arithmetic circuits[END_REF], the degree of the polynomial if no cancellation would occur). In this direction, Erich Kaltofen [START_REF] Kaltofen | Greatest common divisors of polynomials given by straight-line programs[END_REF] designed efficient algorithms for the greatest common divisor of two polynomials, for instance.

This paper, in the same spirit as [START_REF] Allender | On the complexity of numerical analysis[END_REF], investigates two natural problems concerning multivariate polynomials given by arithmetic circuits. We are concerned with the complexity of these problems in the usual Turing machine model. One of them is the computation of the degree of a polynomial (Section 5). This problem is efficiently solved by a probabilistic algorithm in the case where the "formal degree" is polynomially bounded; unfortunately, in characteristic 0, the upper bound of [START_REF] Allender | On the complexity of numerical analysis[END_REF] in the counting hierarchy is not improved. However, for polynomials over fields of characteristic k > 0, we obtain the better bound coNP Mod k P . We shall also study the problem of deciding whether the coefficient of a given monomial is zero (Sections 3 and 4). We show that even on polynomially-bounded formal degree instances, this problem is hard, that is, hard for P #P under strong nondeterministic Turing reductions. We also provide a complete characterization of the complexity of this problem when the field is of positive characteristic.

Throughout this paper, we heavily rely on the results of [START_REF] Malod | Polynômes et coefficients[END_REF]. Though stated in an algebraic context in [START_REF] Malod | Polynômes et coefficients[END_REF] (namely Valiant's framework), we show indeed that they can also be useful for proving results in boolean complexity related to the manipulation of polynomials given by arithmetic circuits.

Preliminaries

An arithmetic circuit over a field K is a directed acyclic graph, each of the vertices is of indegree 0 or 2, one and only one vertex being of outdegree 0. This last vertex is called output, and vertices of indegree zero are called inputs. An input is labeled by a variable x i or a constant among {-1, 0, 1} in K. Every other vertex is called a gate and is labeled by + or ×. The polynomial computed by a vertex is defined inductively as follows:

-the polynomial computed by an input is its label (i.e. a constant or a variable); -the polynomial computed by a + gate is the sum of its entries; -the polynomial computed by a × gate is the product of its entries.

The polynomial computed by a circuit is the polynomial computed by the output of the circuit. Note that the only allowed constants of the field are -1, 0 and 1, so the coefficients of our polynomials are in Z (characteristic zero) or in Z k (characteristic k > 0). The size of a circuit C, written |C|, is the number of its vertices. See Figure 1 below for an example where cancellations occur.

x1 × × x2 x3 -1 0 1 @ @ + b b + b b b b + + × × Z Z × × × Figure1. A circuit of size 16 computing the polynomial x 4 1 + x 2 .
Another useful quantity concerning arithmetic circuits is their formal degree. The formal degree of a vertex in an arithmetic circuit is inductively defined as follows:

-the formal degree of an input is 1; -the formal degree of a + gate is the maximum of the formal degrees of its entries; -the formal degree of a × gate is the sum of the formal degrees of its entries.

The formal degree of an arithmetic circuit is the formal degree of its output. We denote the formal degree of a circuit C by deg f (C). Note that one can easily compute the formal degree of a circuit by induction.

Here we are interested in the complexity of handling arithmetic circuits by (usual, boolean) Turing machines. Note that an arithmetic circuit is easily encodable in binary since it merely consists of a labeled directed graph. Hence, such circuits can be manipulated by Turing machines. We shall study two problems: computing the degree of a polynomial, and deciding whether the coefficient of a monomial is zero. We shall define two versions of both problems, a "bounded" version where a part of the input is given in unary, and the unbounded one where the whole input is in binary.

The first language we shall consider concerns the degree of the polynomial. If C is an arithmetic circuit, the degree of the (multivariate) polynomial computed by C is denoted by deg(C). The language DEG is defined as follows:

DEG = {(C, d) : deg(C) ≤ d},
where C is an arithmetic circuit and d an integer. A "bounded" version of DEG is also defined:

DEG b = {(C, 1 d ) : deg(C) ≤ d},
that is, d is given in unary, hence the size of the input is ≥ d.

The second language we shall study concerns the coefficient of a monomial in a polynomial. A monomial m is encoded by the list of the exponents of all the variables in binary. If we denote by coef C (m) the coefficient of the monomial m in the polynomial computed by C, we define

ZMC = {(C, m) : coef C (m) = 0}
(ZMC stands for Zero Monomial Coefficient), and its bounded version:

ZMC b = {(C, m, 1 d ) : coef C (m) = 0 and deg f (C) = d}.
As above, C is an arithmetic circuit, and m a monomial.

Note that these languages only depend on the characteristic of the underlying field, because the only allowed constants in the arithmetic circuits are -1, 0 and 1. When dealing with fields of characteristic k > 0, we will use the superscript k as in ZMC k , ZMC k b , DEG k and DEG k b . We finally briefly recall a few complexity classes; we refer the reader to [START_REF] Chistos | Computational Complexity[END_REF] and [START_REF] Hemaspaandra | The Complexity Theory Companion[END_REF] for instance for further details. If L is a language, we denote by P L (resp. NP L ) the class of languages recognized by a deterministic (resp. nondeterministic) polynomial time Turing machine with oracle L. If C is a set of languages, P C (resp. NP C ) denotes ∪ L∈C P L (resp. ∪ L∈C NP L ).

We define inductively the classes Σ p i as follows:

Σ p 0 = P and Σ p i+1 = NP Σ p i . The polynomial hierarchy is PH = ∪ i≥0 Σ p i .
We shall also encounter probabilistic classes. The class coRP is the set of languages L recognized by a probabilistic Turing machine that doesn't make any mistake on inputs in L, and has a small probability of error for inputs outside of L. More precisely, a language L is in coRP if there exists A ∈ P and a polynomial q(n) such that for all x ∈ {0, 1} * :

-if x ∈ L then Card{y ∈ {0, 1} q(|x|) : (x, y) ∈ A} = 0; -if x ∈ L then Card{y ∈ {0, 1} q(|x|) : (x, y) ∈ A} ≥ 2 q(|x|)-1 .
BPP is analogous but both-sided errors are allowed:

-if x ∈ L then Card{y ∈ {0, 1} q(|x|) : (x, y) ∈ A} ≥ 3/4.2 q(|x|) ; -if x ∈ L then Card{y ∈ {0, 1} q(|x|) : (x, y) ∈ A} < 1/4.2 q(|x|) .
Finally, PP is similar but the threshold is 1/2 and is the same for both accepting and rejecting conditions. The class PP gives rise to the counting hierarchy: CH = P ∪ P PP ∪ P PP PP ∪ . . .

Let us now define "counting classes". The class #P is the set of functions that count the number of accepting paths of a non-deterministic Turing machine working in polynomial time. More precisely, f : {0, 1} * → N is in #P if there exists A ∈ P and a polynomial q(n) such that

f (x) = Card{y ∈ {0, 1} q(|x|) : (x, y) ∈ A}.
The closure of #P under subtraction is called GapP: thus, a function GapP takes its values in Z and is the difference of two #P functions.

An analogous definition provides counting classes modulo k: if k is an integer greater than 1, Mod k P is the set of languages L such that

x ∈ L ⇔ Card{y ∈ {0, 1} q(|x|) : (x, y) ∈ A} ≡ 0 mod k. If k is prime, Mod k P is closed under complement [2]
, so we can take ≡ 0 in the definition instead of ≡ 0.

Let us recall a few inclusions:

P ⊆ NP ⊆ PH ⊆ P #P ⊆ CH, and P ⊆ coRP ⊆ BPP ⊆ Σ p 2 .
We shall also make use of strong nondeterministic Turing reductions, introduced by Long [8]. This is defined as follows:

B ≤ sn T A iff B ∈ (NP A ∩ coNP A ).
The next section studies ZMC (and its bounded version): we first show that over fields of characteristic zero, ZMC b ∈ P #P , ZMC ∈ CH and these languages are outside PH unless it collapses. Then we study fields of characteristic k > 0, in which ZMC k and ZMC k b are Mod k P-complete. Finally, Section 5 deals with DEG and DEG b .

The coefficient of a monomial in characteristic zero

Here we work on polynomials with coefficients in Z, i.e. we do not use modular arithmetic. The following two results are shown. Remark that the hardness result of Proposition 2 is implied by that of Proposition 1. The latter is shown in Section 3.1. The upper bounds of both propositions are proved in Section 3.2.

Hardness for P #P

The language Perm We define the language Perm as a decision problem for the permanent. We recall that the permanent of a matrix M = (m i,j ) 1≤i,j≤n is defined as

per(M ) = σ∈Sn n i=1 m i,σ(i)
where S n is the set of all permutations of {1, . . . , n}. Despite its similarity with the determinant, the permanent seems to be hard to compute (even for 0-1 matrices) since it is #P-complete under counting reductions [START_REF] Valiant | The complexity of computing the permanent[END_REF]. The language Perm is the following:

Perm = {(M, v) : per(M ) = v},
where M is a 0-1 square matrix, and v an integer (this language also appears in [START_REF] Kabanets | Derandomizing polynomial identity tests means proving circuit lower bounds[END_REF]). We show that Perm is P #P -hard under strong nondeterministic Turing reductions, i.e.

P #P ⊆ NP Perm ∩ coNP Perm .
Let L ∈ P #P . Since computing the permanent of a 0-1 matrix is #P-complete, there exists a polynomial-time deterministic oracle Turing machine M that decides L with the help of the oracle per. It is easy to simulate this machine with a nondeterministic one, call it N , that consults the oracle Perm. We proceed as follows. While M does not use its oracle, N simply simulates M in a deterministic manner. As soon as M asks for the value of a permanent, say per(M ) for a matrix M , the machine N guesses the value of this permanent and check for a correct guess thanks to its oracle Perm: only the correct path (corresponding to the true value of the permanent) is continued, whereas the other paths are rejected.

Our machine N thus exactly has one path that behaves like M, the other being rejecting paths. So N recognizes L, and we have shown the inclusion P #P ⊆ NP Perm . The other inclusion P #P ⊆ coNP Perm holds by closure of P #P under complement.

Remark 1. Toda's theorem ( [START_REF] Toda | On the computational power of PP and ⊕P[END_REF]) asserts that PH ⊆ P #P . It yields PH ⊆ NP Perm . This implies that Perm does not lie in the polynomial hierarchy unless it collapses. Indeed, if Perm ∈ Σ p i for some i, then PH ⊆ NP Σ p i = Σ p i+1 , and PH collapses to the (i + 1)-th level.

ZMC b is harder than Perm Remark 2. The following ideas already appear in [START_REF] Valiant | Completeness classes in algebra[END_REF]. We nevertheless detail the reduction here for the sake of completeness.

We now show that ZMC b is harder than Perm. For, if M = (m i,j ) is a matrix (1 ≤ i, j ≤ n) and v an integer, we use the following classical multivariate polynomial, already appearing in e.g. [START_REF] Valiant | Completeness classes in algebra[END_REF] (without the term -vX 1 • • • X n , though):

P (M,v) (X 1 , . . . , X n ) = n j=1 ( n i=1 m i,j X i ) -vX 1 • • • X n .
We claim that the coefficient of the monomial X 1 • • • X n in this polynomial is per(M )v. Indeed, consider the product

( n i=1 m i,1 X i )( n i=1 m i,2 X i ) • • • ( n i=1 m i,n X i ).
In order to develop it, in each factor one has to choose one term of the sum. Since we are concerned with the coefficient of X 1 • • • X n , one has to choose different variables X i for each factor. Therefore, for each permutation σ of {1, . . . , n}, the product yields a term of the form

( i m i,σ(i) )X 1 • • • X n , thus the coefficient of X 1 • • • X n in the product is σ i m i,σ(i) = per(M ). Now, taking the term -vX 1 • • • X n into account, we obtain the announced coefficient per(M ) -v.
From the input (M, v), it is easy to build in polynomial time an arithmetic circuit with polynomially-bounded formal degree1 that computes P (M,v) . Then the language ZMC b enables us to decide Perm by asking whether the coefficient of X 1 • • • X n in P (M,v) is zero. We conclude that ZMC b is harder than Perm for polynomial-time many-one reductions. This shows the hardness results of Propositions 1 and 2. This also implies that ZMC b (and a fortiori ZMC) is not in the polynomial hierarchy unless it collapses.

The help of a #P-oracle

We now show that ZMC b is in P #P . We need the following definition of [10, section 5.2] (also in [START_REF] Malod | Characterizing Valiant's algebraic complexity classes[END_REF]). Definition 1. Let α be a gate whose inputs are β and γ. The gate α is disjoint if the subcircuits associated to β and γ are disjoint.

A circuit is said to be multiplicatively disjoint if all its multiplication gates are disjoint.

Multiplicatively disjoint circuits are useful in the light of the following result of Malod and Portier [11, Lemma 2] (also in [START_REF] Malod | Polynômes et coefficients[END_REF]). In order to show that ZMC b is in P #P , we rely on the proof of Theorem 3 of [10, section 2.4]: the idea is that the coefficient of the monomial m in a multiplicatively disjoint circuit is merely the number of possible "developments" that lead to m. By a "development" we mean a tree which represents the choices in the distribution of × over +. More precisely, if C is a multiplicatively disjoint arithmetic circuit, a development D is a subgraph of C satisfying the following properties:

-every × gate in D has both inputs in D; -every + gate in D has exactly one input in D; -D contains the output gate of the circuit, and every gate in D different from the output gate has at least one child in D.

The monomial computed by a development is merely the product of its variables. If the development contains the constant 0, then the development will be called neutral. Otherwise, if the development contains the constant -1, then it will be called negative, else positive. The following result of [START_REF] Malod | Polynômes et coefficients[END_REF] justifies the introduction of the notion of developments. ⊓ ⊔

For the bounded version ZMC b , the formal degree of the circuit is bounded by the size of the input since it is given in unary. Therefore one can compute in polynomial time an equivalent multiplicatively disjoint circuit by Lemma 1. By Lemma 3, computing the coefficient of m and testing it to zero (i.e. deciding ZMC b ) is done in P #P . This concludes the proof of Proposition 1.

In fact, we obtain a polynomial-time algorithm with only one call to a GapP oracle f . This can be converted into one call to a #P oracle as follows. Let f + and f -be the #P-functions such that f = f +f -and take a polynomial p(n) such that f + (x), f -(x) < 2 p(|x|) for all x. Then the function g defined by g(x) = f + (x)+ 2 p(|x|) f -(x) is in #P and we can recover f from g. We have therefore proved a slightly stronger statement, since only one call to the #P oracle is necessary: ZMC b ∈ P #P [1] .

Unbounded version. However, this construction does not work for the unbounded version, since we cannot transform the original circuit into a multiplicatively disjoint one in polynomial time. We only show here that ZMC is in the counting hierarchy CH.

As in [START_REF] Malod | Polynômes et coefficients[END_REF], we first build a "generic" arithmetic circuit, called G n , whose inputs can be specialized to simulate any other circuit of size ≤ n.

The construction proceeds as follows. We define

G n -n = 1, G n -n-1 = x 1 , . . . , G n 0 = x n .
We then compute every possible sum and product of the previous levels, by adding new variables a l+1,i and b l+1,i :

G n l+1 =   l i=-n a l+1,i G n i     l i=-n b l+1,i G n i   .
We then write G n (y 1 , . . . , y p ) = G n n . The circuit G n is generic in the sense that the computation of any crcuit of size ≤ n can be obtained from G n (y 1 , . . . , y p ) by specializing some variables: here is the precise statement coming from [START_REF] Malod | Polynômes et coefficients[END_REF]Prop. 3].

Lemma 4. Let f (x 1 , . . . , x k ) be a polynomial computed by an arithmetic circuit C of size ≤ n. Then 1. there exist a 1 , . . . , a p among {-1, 0, 1, 2, x 1 , . . . , x k } such that f (x 1 , . . . , x k ) =
G n (a 1 , . . . , a p ); 2. the values of a 1 , . . . , a p can be computed from C in polynomial time.

If C is a fixed circuit and f the polynomial computed by C, let τ be the substitution of the variables y 1 , . . . , y p of G n given by the preceding lemma, that is, the map defined by τ (y i ) = a i . We will write

C = τ (G n ).
If m ′ is a monomial in G n , the image τ (m ′ ) of m ′ by the substitution τ is the product of a (possibly zero) coefficient and a monomial of f ; we will denote by α(m ′ ) the coefficient and by τ ′ (m ′ ) the monomial, that is, τ (m ′ ) = α(m ′ )τ ′ (m ′ ) where α(m ′ ) is an integer. Note that both τ ′ and α are computable in polynomial time.

We thus have:

coef C (m) = m ′ | τ ′ (m ′ )=m α(m ′ )coef Gn (m ′ ).
Malod has explicitely evaluated the coefficients of G n in [10, section 5.2.3] by means of binomial coefficients. More precisely, he shows the following result.

Proposition 3. The coefficient in G n of the monomial m = -n≤i,j≤n a α i,j i,j -n≤i,j≤n b β i,j i,j 1≤i≤n x γ i i is 0 if γ i = n j=-n (α i-n,j + β i-n,j
) for some i ≥ 1, and

n i=1 i-1 j=-n j k=-n α i,k α i,j j k=-n β i,k β i,j otherwise.
Thus, if we are given a circuit C, we compute the substitution τ such that C = τ (G n ) thanks to Lemma 4 and we have:

coef C (m) = m ′ ,τ ′ (m ′ )=m α(m ′ )coef Gn (m ′ ).
Bürgisser shows in [4, Corollary 3.8] that the so-called falling factorials N (N -1) . . . (Nk) are computable bit by bit in CH on input size log N , as well as integer divisions and sums and products exponential in the size of the input (Theorem 3.7). Therefore this sum of products of binomial coefficients is computable bit by bit in the counting hierarchy CH. Now, a coNP computation is enough for testing whether all bits of the coefficients are zero. Therefore ZMC ∈ coNP CH = CH. This concludes the proof of Proposition 2.

Remark 3. That ZMC ∈ CH could also be proven thanks to the result of [START_REF] Allender | On the complexity of numerical analysis[END_REF] that the problem BitSLP is in the counting hierarchy. The problem BitSLP consists in computing the bits of an integer computed by an arithmetic circuit. Indeed, similarly as in the proof of Proposition 2.1 of [START_REF] Allender | On the complexity of numerical analysis[END_REF], one can substitute to the variables x i of our polynomial some integers growing sufficiently fast, for instance 2 2 is 2 , where s is the size of the circuit. Then, when computing the bits of the integer thus computed, there will be no overlap of the coefficients of the polynomial (because they are bounded by 2 2 s in absolute value). We will therefore be able to recover them in CH thanks to BitSLP.

The coefficient of a monomial in positive characteristic

Now we are working over a field of positive characteristic k > 0, that is, the polynomials computed by our circuits now have coefficients in Z k , so we can perform modular arithmetic. This will help a little, since we shall show the following completeness result. Remark 4. Toda's theorem shows that PH ⊆ BPP Mod 2 P . This result is generalized in [START_REF] Toda | Counting classes are at least as hard as the polynomial-time hierarchy[END_REF] to any k ≥ 2, that is PH ⊆ BPP Mod k P , which implies that ZMC k is not in the polynomial hierarchy unless it collapses.

Algorithm

We use the generic polynomial G n of the preceding section and the same notations. Given a circuit C, one can compute in polynomial time the substitution τ such that C = τ (G n ). Now, showing that the coefficients of G n modulo k can be computed in polynomial time is enough in order to show that the coefficients of C can be computed in Mod k P, since the coefficient of m is expressed as a sum of those of

G n : coef C (m) = m ′ ,τ ′ (m ′ )=m α(m ′ )coef Gn (m ′ )
and Mod k P is closed under exponential sum if the index of the summation is computable in polynomial time (as is the case here).

It remains to see why the coefficients of G n are computable in polynomial time modulo k. The main tool here is the following result of Édouard Lucas [START_REF] Lucas | Théorie des fonctions numériques simplement périodiques[END_REF], thanks to which we can efficiently compute binomial coefficients modulo k.

Proposition 5 [START_REF] Lucas | Théorie des fonctions numériques simplement périodiques[END_REF]. Let k be a prime integer. Let N = n i=0 n i k i and M = n i=0 m i k i be two integers written in base k. We have

N M ≡ n i=0 n i m i mod k.
Together with Proposition 3, this proves that the computation of the coefficients of G n modulo k is done in polynomial time. The remark above on the closure of Mod k P under exponential sum shows that ZMC k ∈ Mod k P: this is the first part of Proposition 4.

Lower bound

Let us now show that ZMC k b is hard for Mod k P. We use a reduction from "#Hamilton Cycle", the function that counts the number of Hamilton cycles in a graph. This function is #P-complete under parsimonious reductions, that is, under counting reductions that preserve the number of solutions, see [START_REF] Valiant | The complexity of enumeration and reliability problems[END_REF] (one can also use the proof of [START_REF] Chistos | Computational Complexity[END_REF]Th. 18.2] of #P-completeness of "#Hamilton Path" and then use an easy parsimonious reduction from "#Hamilton Path" to "#Hamilton Cycle"). So it remains complete modulo k, i.e. "#Hamilton Cycle modulo k" is Mod k Pcomplete. The number of Hamilton cycles in a graph of adjacency matrix (x i,j ) 1≤i,j≤n is given by the polynomial

HC n (x i,j ) = σ n i=1 x i,σ(i) ,
where the sum is taken over all n-cycles of S n . Now, as in the case of the permanent, the number of Hamilton cycles is the coefficient of a monomial in a polynomial P that is obtained in polynomial time from the adjacency matrix of the graph: Lemma 5 ([10], Lemma 14). Let the polynomials T p,i,j be defined as T 1,i,j = x i,j y i z j and T p+1,i,j = n k=1 T p,i,k T 1,k,j . Then the polynomial P = T n,1,1 can be computed by an arithmetic circuit of size O(n 4 ) and of formal degree O(n); the circuit itself is computable in time polynomial in n; the coefficient of the monomial y

1 z 1 • • • y n z n in P is HC n .
Since HC n is the coefficient of a monomial in a polynomial P computed by an arithmetic circuit C of polynomial size and polynomial formal degree, the same argument as in the case of the permanent applies: ZMC k b is as hard as counting the number of Hamilton cycles modulo k, i.e. ZMC k b is Mod k P-hard. This concludes the proof of Proposition 4.

Computation of the degree

We now turn to the study of the complexity of DEG and DEG b .

Unbounded version

For the unbounded version, we do not improve the P PP PP PP upper bound of [START_REF] Allender | On the complexity of numerical analysis[END_REF] for fields of characteristic zero. Over a field of positive characteristic, however, Section 4 helps us finding the better bound coNP Mod k P . We first show that DEG is P-hard for logspace reductions.

Lower bound. We show that the "Circuit value problem" (CVP) is easier than DEG. CVP is the language consisting of those boolean circuits whose output is TRUE. This is a P-complete problem under logspace reductions [START_REF] Chistos | Computational Complexity[END_REF]Th. 8.1]. In fact, we can encode a boolean circuit in an arithmetic one, by simulating the boolean operations by polynomials: (x AND y) is xy, (x OR y) is x + yxy, and (NOT x) is 1x. From the boolean circuit B, we construct the arithmetic circuit C 0 that simulates B; then we multiply the output gate of C 0 by a new variable x, and add 1. Call the new circuit C. The polynomial computed by C is 1 if the value of B is FALSE, and x + 1 otherwise. Thus by computing the degree of C, we know the value of B.

Since this procedure uses only logarithmic work space, we conclude that DEG is P-hard for logspace reductions. Since the number of monomials is simply exponential in the size of the circuit, this condition is checkable in coNP with the help of the oracle ZMC. It yields: DEG ∈ coNP ZMC .

In characteristic zero, ZMC is in CH, so DEG ∈ coNP CH = CH.

This does not improve the result of [START_REF] Allender | On the complexity of numerical analysis[END_REF] that DEG is in the counting hierarchy. In characteristic k > 0, however, ZMC k is in Mod k P, so DEG k ∈ coNP Mod k P . Thus we have proved the following result.

Proposition 6. The problem DEG k for fields of characteristic k > 0 is P-hard for logspace reductions and is in coNP Mod k P .

Bounded version

The case of DEG b is easy and uses classical techniques. Let C be a circuit and f the polynomial it computes: we want to know whether the degree of f is ≤ d. Since d is given in unary, one can first build in polynomial time an arithmetic circuit computing all the homogeneous components f i of degree ≤ d of f , as in [START_REF] Bürgisser | Completeness and Reduction in Algebraic Complexity Theory[END_REF]Prop. 5.28] or [10, Lemma 2]. We now simply have to test whether f = d i=0 f i . Testing the equality of two polynomials given by arithmetic circuits is standard by testing the equality at random points, see [START_REF] Jacob | Fast probabilistic algorithms for verification of polynomial identities[END_REF] for example. By working in extensions of the prime field if necessary, it is easy to see that the technique also works in fields of positive characteristic.

As a whole, it yields the following result.

Proposition 7. The problem DEG b is in coRP.

Summary and further research

Here is a summary of our results. The symbol ≤ sn T means "strong nondeterministic Turing reduction", see Section 2. The upper bound for DEG in the counting hierarchy is from [START_REF] Allender | On the complexity of numerical analysis[END_REF].

Proposition 1 .

 1 ZMC b is P #P -complete under strong nondeterministic Turing reductions ≤ sn T . Proposition 2. ZMC is in CH and is P #P -hard under strong nondeterministic Turing reductions ≤ sn T .

Lemma 1 .

 1 Let C be a circuit of size s and formal degree d. There exists a multiplicatively disjoint circuit C ′ of size ≤ ds which computes the same polynomial as C. Furthermore, constructing C ′ from C requires only time polynomial in |C| and d.

Lemma 2 .Lemma 3 .

 23 Let C be a multiplicatively disjoint circuit and m a monomial X α 1 1 • • • X αn n . Let d + (m) and d -(m) be the numbers of positive and negative developments computing m, respectively. Then the coefficient of m in the polynomial computed by C is equal to d + (m)d -(m). We finally have the following result. The function f which, given a multiplicatively disjoint circuit C and a monomial m, computes the coefficient of m, is in GapP. Proof. Checking whether a tree is a development corresponding to a given monomial is in P. Define A + (m) (respectively A -(m)) to be the set of positive (resp. negative) developments of C computing m. The coefficient of m is then |A + (m)| -|A -(m)| and is computable in GapP because deciding A + (m) and A -(m) takes only polynomial time.

Proposition 4 .

 4 ZMC k and ZMC k b for fields of characteristic k > 0 are Mod k P-complete.

  Upper bound. Obviously, both problems DEG and ZMC are related. In fact, we have: deg(C) ≤ d iff (every monomial of degree > d has zero for coefficient).

⋆ UMR 5668 ENS Lyon, CNRS, UCBL, INRIA.

An immediate construction provides a formal degree of roughly

2n for the product, and n log n for vX1 • • • Xn (since |v| ≤ n!, because it is a permanent of a 0-1 n × n matrix). The overall circuit thus has formal degree ≤ 2n(1 + log n) ≤ |C| 2 .

There is still a lot of work to do, namely to reduce the gap between the above results. Possible next steps would be to completely characterize ZMC in characteristic zero, and to find a better upper bound than P PP PP PP for DEG. The authors thank Erich Kaltofen for the suggestion of encoding d in unary in the language DEG b , as well as the anonymous referees for useful remarks.