Caroline Zum Kolk

Jacqueline Boucher

Michel Fleury

Eric Bournazel

La Familia Regis Francorum

Jacques Paviot

Monique Sommé

Eliane Viennot

In Frankreich scheint sich die Aussonderung eines Hofstaats für

Holt examines the household of François de Valois, Duke of Alençon

1

Caroline zum Kolk

The Household of the Queen of France in the Sixteenth Century 1

[Page 3 de la première édition dans la revue The Court Historian]

In the long history of the household of the queen of France, the sixteenth century marked a peak: never before or since was its staff so numerous, or was it so deeply involved in life at court. This article will survey the history of the queen's household and its structure and functions in the sixteenth century.

No systematic study has yet been dedicated to the history and function of the household of the queen of France. Of course, the structural and numerical evolution of royal households under the late Valois kings has been described at length and analysed as a rupture with the past, 2 with their increasing number of office holders, the appearance of new titles and the growing effort to define court ceremonial. Nonetheless, few studies highlight the specific role played by the queen and her entourage within this evolution. In general, the queen's household is briefly compared to the king's hôtel, with the conclusion that similar structures defined both institutions. Only the relatively smaller size, the presence of women and the existence of a few specific offices tend to be emphasised. 3 Such an approach makes it impossible to define the contours of the queen's household and understand its importance for the court and its development. In the long history of this institution, the sixteenth century marked a peak. This

[page 4] article will illustrate this evolution through an account of the history of the queen's household, its structure in the sixteenth century and its role in the development of the Renaissance court.

Beginning in the eighth century, more numerous sources make it possible to understand better the role of the queen at court. A crucial text, sometimes considered to be the first ordinance produced by the courts of western Europe, is the De Ordine Palatii by Hincmar of Rheims, written about 882. Hincmar opens his discourse on the organisation of the palace with the notion that 'the king and the queen, with their very noble family, rose above all others'. Thereafter, the author describes the responsibilities of the officers in charge of ecclesiastical and legal aspects of government, as well as the numerous figures who served the royal family. He concludes: '... each one of the officers invoked was independent in his office, had no one above him, and responded directly to the king himself, or, in certain cases, to the queen and the royal family.' 6 . Her day-to-day role included overseeing gifts from visitors and vassals and working with the Chambrier to ensure that the court was well supplied with everything it needed. The queen was thus regarded as an auxiliary of the king, who could act in place of her husband in the control and management of the royal staff.

[page 5]

On the other hand, Hincmar does not provide any information concerning the existence of a household specific to the queen. According to A. Fößel, such a staff must have existed, because a seneschal of Queen Judith, the wife of Louis le Pieux, is named in documents. 7 The queens were also accompanied by a group of military officers, not only for their protection but also as a sign of their dignity. 8 As in the case of the king, they had several nutrii, young noblemen who received their education at the court. When Charlemagne divorced his first wife Desiderata, two of her nutrii prefered to join a nunnery rather than serve the King's second wife. 9

The advent of the Capetian dynasty and the feudal system marked a turning point. The entourage of the king lost its splendour: dukes and counts left the royal court, which was taken over by the nobles of the Ile de France. The responsibilities of major offices and positions were less well defined than in the Carolingian period. 10

Until the twelfth century, documentation emanating from the court is rare, and no document from the chancellery furnishes information about its structure. 11 The king's officers are evoked in the writings of political thinkers such as Suger or Guibert de Nogent, but court literature provides little information regarding the queen's household. Joachim Bumke detected in the works of Chrétien de Troyes the presence of chambriers and seneschals of the queen and concluded that the formation of an autonomous hôtel for the queen occurred in France at a very early date, in the second half of the twelfth century 12 . As we have seen, it had already existed under the Merovingian and Carolingian dynasties, so it would be more appropriate to say it re-occurred.

During these two centuries, several queens left their mark on the politics of their age. Constance d'Arles (circa 984-1032) led a revolt against her son Henri I er ; Alienor d'Aquitaine (1124-1204) played an active role in the government of Aquitaine while she was queen of France (1137-52); and Adèle de Champagne (1150-1206) was co-regent during the absence of her son, Philippe Auguste. She [page 6] seems to have had a personal chancellery at her disposal 13 and left more than 110 charters, one of which concerns one of her ladies-in-waiting, 'domicella Alice'. The Queen offered her a vineyard near Melun. 14 The middle of the thirteenth century is a key moment for the study of the French court. The first ordinance dealing with the responsibilities and privileges of court officers is dated 1261. 15 Thereafter, this kind of document multiplied, especially under the reigns of the grandsons of St Louis. 16 After 1350 a second group of sources came into existence: the états -that is, lists of the members of a household, with information about their duties and wages. 17

The queen's hôtel regularly appears in these sources. Belles-lettres, 1998, p. 913-932. 15 The original is lost but several copies of this document are extant; the oldest, written by a clerk of the chambre des comptes, dates from 1317: Archives Nationales, JJ 57.

16

Most of these documents have been edited by Elisabeth Lalou and can be consulted online (AEdilis, Publications scientifiques, http://www.cn-telma.fr/ordonnances/). 17

The ordinance of 1261 already uses the term état: 'c'est l'ordonnance et estat de l'ostel de la royne'. In the fifteenth and sixteenth centuries, the word tends to designate administrative documents such as lists of the members of a household.

An ordinance of 1286 gives information about the structure of the household of Queen Jeanne de Navarre, wife of Philippe IV 'le Bel', which provided all the [page 7] services and officers that enabled her to move about on her own. 19 Her stable and fourrière included an equerry, a marshal, a fourrier, a palefrenier and several valets; two coches were at the disposal of the queen and her five ladies and maidens-in-waiting. At table, she and her retinue were served by panetiers and échansons. The kitchen was not separated into the cuisine bouche and the cuisine de commun as in the king's household, since the hôtel was not large enough to justify such an arrrangement. The meals of the entire staff were prepared by two cooks, two hasteurs, one saucier, one potier, one poulailler, one fruitier and several valets and aides. A chaplain, an almoner and a chapel cleric attended to the spiritual needs of the Queen and her servants. The chambre aux deniers supervised expenditures under the control of a maître de la chambre, Michel de Morgneval. Like Philippe IV, the Queen had only one maitre d'hôtel, Jean du Chastelier, and two physicians (fisiciens). Her clothes were made by a tailor and a valet couturier and cleaned by two washerwomen along with the household linen. This core staff later grew in size, despite a cut in 1315, when Louis X decided to merge his household with that of his wife in order to limit the number of court servants. 20 This decision was revoked a few month later, and in the following decade, the Queen's hôtel gained in autonomy. An important step was taken in 1316 with the creation of a household specifically for the royal children: prior to this date they had been served by their mother's staff. 21 In that year, the King's hôtel counted 164 members, the Queen's 102 and the household of the children 42.

In the ensuing years, royal households continued to grow, reaching their apogee in 1326: Queen Jeanne d' Evreux then had a staff of 212 people at her disposal. Her household was tightly structured, and, like the King's hôtel, comprised five chambres and six metiers. These increases did not, however, continue. A gap in the documentation between 1326 to 1450 makes it impossible to follow the evolution of the royal hôtels during this period of over a century. However we find that the household of Queen Marie d'Anjou at the end of the Hundred Years War was significantly smaller than that of Jeanne d'Evreux had been in 1326. 22 Between 1326 and 1452 there had been a two-thirds reduction in the number of the queen's servants.

19

The Ordonnance de l'hôtel de Philippe IV le Bel et Jeanne de Navarre (AN JJ 57, f. 1-1, n°142, edited by E. Lalou (http://www.cn-telma.fr/ordonnances/ordonnance1/) specifies that the spouses were regularly apart. 20 Ordonnance de l'hôtel du roi et de la reine, 1315: 'C'est l'ordenance qui est faite pour mettre l'ostel le roy et l'ostel la royne tout à un' (BNF, coll. Clair. 832, p. 646. Edited by E. Lalou (http://www.cn-telma.fr/ordonnances/ordonnance5/).

21

Ordonnance de l'hôtel de Jeanne de Bourgogne, 1316 (BNF ms. fr. nouv. ac. 9175, folio 337).

22

The états from 1452 and 1461 are not complete (the chancellery, the chambre, the kitchen, the fruiterie and the fourrière are missing; they surely existed, though, since Marie d'Anjou lived most of the time apart from the King). Nevertheless, when the staff of the various different departments is compared to 1326, the decline in numbers is obvious.

This unsatisfactory situation ended during the reign of Charles VIII and Anne de Bretagne. Scrutiny of preceding états can help us to understand this change of direction.

[page 8]

The Departments of the Household of the Queen in the Middle Ages (1326-1490) The Expansion of the Queen's Household (1496-1589)

In 1496 a major change occurred with respect to the hôtel of the queen. It began with a spectacular increase in the size of Anne de Bretagne's household. Her staff rose from 88 to 253 servants and stayed at this level until 1498, 24 at which date its size was 88 per cent of that of the King (286 members). Never in its history was the Queen's maison so close in size to that of her husband. France, 1789-1830 (Cambridge, 1988). lead to a reduction in numbers. The largest increase took place between 1494 and 1496: in 1496, the household nearly tripled. How can this sudden expansion be explained? There were several contributory factors. One was the influence of the Italian Renaissance, which saw the birth of courts on the Italian peninsula in which women were present in greater numbers than in the Middle Ages. Their participation in cultural and political life consequently gained in visibility. The wealth and luxury of the Italian courts thereafter became a model for all of Europe, facilitating the integration of women at court. Another factor, politically determined, was that at the end of the Middle Ages the dynastic principle became stronger and was accompanied by a new level of pageantry and pomp. The queen of France, closely associated with royal dignity, was by no means excluded from this rise in dynastic prestige. fundamental role in this new representation of power, and its growth can be observed in most European countries. Increases in the households of the prince, the princess and the other members of the family accompanied this development.

This expansion fulfilled another function, also political: it made it possible to create a large number of positions and offices which could be used to reward clients of the crown. The growing importance of the patronage system contributed to the increasing size and grandeur of the court.

These transformations, effected at the majority of European courts, form the context for the growth of the household of the queen, but cannot, alone, account for it: the households of princesses did not everywhere increase, nor did they necessarily maintain their independence. Several specific cultural and political factors contributed to the development of the queen's household in France.

First of all, France was influenced by the Burgundian court. The memoirs of a lady-in-waiting, Eléonore de Poitiers, 29 written in 1484-7, show that contact between the two courts was close. In the fifteenth century, the Duchess of Burgundy's hôtel was bigger than that of the Queen, its ceremonial more firmly developed 30 . As we shall see, several offices and dignities would be 12

Another influence may have devolved from the Querelle des femmes, which began during the reign of Charles VI and Isabeau de Bavière. This polemical debate about female honour left its mark on ladies of high rank. When Anne de Bretagne settled at the French court, she brought from Brittany six tapestries illustrating Le Livre de la Cité des Dames by Christine de Pizan, 31 the major text in defence of women produced in this debate. Anne developed a form of art patronage which proved her personal commitment to enhacing the role of women and their political status (Figure 1). 32 She he had grown up in a duchy which was not subject to Salic law: women could exercise power there, and Anne intended to preserve her status as a 'ruler in her own right' at the court of France. An impressive and well staffed household was a basic requirement for this. Fortunately, Anne had the example of a predecessor near at hand. Her views coincided with the concept of female honour and dignity held by Anne de Beaujeu, the dominant figure at the French court from 1483 to 1491.

During her regency, the sister of Charles VIII had to come to terms with violent opposition from those who believed that women were incapable of ruling. However, through her political activitiesand also her patronage of the arts -she managed to impose her authority. In this way, the Duchess assumed 'a role as teacher and mentor to It is this coming together of cultural and historical circumstances that explains the evolution of the household of the queen of France, the growth of which was not only a question of size but also of structural changes and the creation of impressive new offices.

The Divisions and Their Evolution

The general increase in manpower did not affect all the services of the hôtel in a uniform way. Some areas changed considerably; others, hardly at all. Ten major divisions characterise the états de maison of the last Valois queens, arranged according to the prestige of their members and duties.

First named are the dame d'honneur, the gouvernantes and the noblewomen of the queen's retinue, as well as the female personnel that served them (chambermaids, washerwomen, linen maids, nurses). Like their male counterparts, the female nobility can be divided into three groups. The first consists of members of the most important aristocratic families, state servants and clients of the crown ; in addition to women of the high nobility, this group includes the wives of the chancellor, the marshals and other dignitaries of the realm. The second group consists of clients of the queen's family who had followed her to France or who chose to join her after her accession to the throne; this group was sometimes stable (as in the case of the Medici-clients protected by Catherine de' Medici) and sometimes not. The third group consists of personal clients and friends whom she gradually introduced into her household. 34 The women are followed by the most important offices held by men: the chevalier d'honneur and gentilshommes d'honneur, the maîtres d'hôtel and the noblemen who served the queen, her women and guests at table: panetiers, cupbearers, esquires and valets tranchants. Sometimes listed here are the First or Grand écuyer, and esquires of the stable, a fact which underlined the prestige of this office. 35 The third division is the chapel of the queen, with a Grand or Premier Aumonier who directed the almoners, confessor, cantors, chaplains, clerics, musicians and singers. Fourth comes the chancellery; its duties can be interwoven with those of the chambre aux deniers, the queen's treasury, because the secrétaires des finances were sometimes listed here. The chancellery staff was composed of secretaries, clerks and accountants. The counsellors and maîtres des requêtes formed a group of their own.

[page 16] 34 About the role of the household for patronage and the role of the king and the queen in its constitution, see Caroline zum Kolk, 'Les femmes à la cour de France au 16 e siècle: la fonction politique de la maison de Catherine de Médicis (1533-1574)', in E. Santinelli, A. Nayt-Dubois (ed.), Femmes de pouvoir et pouvoir de femmes dans l'Europe occidentale médiévale et moderne, actes de colloque, université de Valenciennes, 6-8 avril 2006 (Valenciennes, 2009).

35

The stables (écurie) and their staff do not appear in all the états de maison. The officers of the Chamber are numerous. In addition to the masters of the chamber, valets, doorkeepers, pages and fourriers, there are the officers responsible for the wardrobe and tapestries (masters and valets of the wardrobe, tailors, tapestry-makers). The doctors in charge of the health of the queen and her staff also appear in this section (physicians, surgeons, apothecaries, barbers).

After the Chamber staff are mentioned three divisions : the sommeliers, the cuisine bouche and the cuisine du commun. The cuisines were each overseen by an equerry. The cuisine bouche prepared meals for the queen and her noble entourage, while the cuisine du commun served the rest for her staff. More than sixteen different specialised roles could be found in the two kitchens.

The ninth division is the fourrière, which organised the journeys of the queen and her household under the direction of the Maréchaux des logis. They are followed by various functions related to hunting and by the Gens de métier, or craftsmen. In this division musicians, painters, stone-carvers, silversmiths and goldsmiths are sometimes listed.

The last division is that of the queen's treasury, the Chambre aux Deniers. Under the direction of the treasurer and the Receveur Général, the secretaries and finance clerks managed the income and expenditure of the queen and looked after her domains; the receivers and secretaries of the chancellery supported them in this task.

If we take all the divisions together, female staff are the most numerous, an average of seventy-one. Next comes the sommeliers (43 in number), the Chamber (41) and the ninth division (Fourrière / crafts, 38 or 39). The other divisions totalled between 20 and 30 employees (in descending order: On the whole, the services responsible for the day-to-day management of the household remained stable or even decreased, except for the fourrière which experienced a slight growth. The increases were mainly in the divisions involved with pageantry (ladies-in-waiting, the Chamber), religion (chapel) and administration (chancellery). It is in these sectors, too, that most of the new offices arose.

The Creation of New Dignities and Offices

The creation of new dignities took place at three crucial points: 1496, 1523-1534 and 1547.

In 1496 Anne de Bretagne integrated into her household three Grands Officiers. The most important new post was that of Chevalier d'honneur. This office had existed at the Burgundian court since 1430. 36 At the court of France, it progressively replaced the office of the Grand maître de l'hôtel de la reine, who in the fifteenth century directed her household. 37 The Grand Maître worked alongside the Chevalier d'honneur until 1523, when he disappeared from the états. This was the first year that one of the Maîtres d'hôtel was named Premier maître d'hôtel in order to support the Chevalier d'honneur in the performance of his multiple duties. The most prestigious task was to accompany the Queen through the palace and to assist her on every occasion. Physical contact was strictly monitored: he could lead the Queen only by touching her arm. 38 The Chevalier d'honneur was senior in rank to all the male officers of the household, and supervised the expenditure of all its divisions. charge of the young noblewomen of the queen's retinue, the demoiselles. She had to oversee their behaviour and education. In so doing, she took upon herself a portion of the [page 19] responsibilities of the Dame d'honneur. The rise in the number of demoiselles led to the expansion of this office: from 1547 Catherine de' Medici employed two Gouvernantes and several Sousgouvernantes.

Among the queen's ladies was a Dame d'atours, whose responsibility was to dress her mistress. The evolution of this office is instructive, since it demonstrates the growing prestige that surrounded members of the royal family. Isabeau de Bavière and Charlotte de Savoie had several femmes and demoiselles d'atours, 47 but most of these women were not noble: their rank and wages were those of a simple chambermaid. The status of this category of women did not evolve in the fifteenth century. It was under the reign of Francis I that physical contact with the queen was given greater importance. Only one lady of high rank, or a relation of the queen, would hold this privilege from this point on. She appeared with the designation 'Dame d'atours' for the first time in 1534.

The third remarkable change concerns the queen's chancellery. Anne de Bretagne had just two secretaries in 1494; in 1496 their number grew to seven, and three counsellors also came to join them. In 1523, Claude de France had five secretaries, four counsellors and two controllers. Three Maîtres des requêtes are listed in the household of Eléonore d'Autriche in 1530, and secretaries are numerous (averaging five for the period between 1530 and 1546), but counsellors are absent from her household. Catherine de' Medici had 8 counsellors in her service in 1547, but their number increased yearly, reaching 20 in 1555, the last year documented by complete états de maison. In these years a Président du Conseil made his appearance for the first time. The number of secretaries increased as well, growing in the same period from 6 to 20 and henceforth overseen by a Premier Secrétaire. The clercs d'offices appeared during these years too (3 in 1547, 6 in 1555); and in 1573 a superintendent. This elaborate structure persisted until the end of the period under scrutiny here.

More than any other household service, the council established a bond between the queen and the great institutions of the kingdom. It consisted of presidents of the various Parliaments of the realm, members of the major royal councils, lawyers and judges. Provincial officers were numerous, too, but of lower rank: provosts, bailiffs, lieutenants, prosecutors and advisors from various cities and regions.

When tracing the evolution of the different divisions of the queen's household, it becomes obvious that the first and second wave of the creation of offices relates to matters of pageantry and the inner organisation of the hôtel. The third relates to the services involved in the management of the queen's patrimony, her diplomacy and political patronage. This third step began before the personal reign of Catherine de' Medici and continued after it; it cannot be explained by the phenomenon of regency but instead seems to be related to the evolution of the court and the status of the queen.

[page 20]

The 'Great Court of Ladies' Of all these changes, one in particular attracted the attention of contemporary observers: the emergence of noblewomen at court. In 1496, the expansion of Anne de Bretagne's household went hand in hand with the increase in the number of ladies and maidens-in-waiting (23 in 1490; 39 in 1496-1498). The Queen insisted on their presence at court -according to Brantôme: 'She [Anne de Bretagne] was the first to begin to gather together the great court of ladies which we still know today, because she had a very great retinue, of both ladies and maidens, and never refused any; and indeed, she asked the gentilshommes, their fathers, who were at court, if they had daughters, and if so who they were, and requested their presence.' 48 This evolution continued under the reign of Francis I. As K. Wilson-Chevalier noted, their presence at court was used by the King to make the behaviour of noblemen more polished and introduce into court society a new civility. 49 Ladies and demoiselles were numerous in the entourage of his wife, his mother and his sister. A new peak was attained in the years 1553-1554, when Catherine de' Medici was regent. The end of the period studied here is marked by a final spectacular increase, beginning in 1579. At the end of the reign of Henri III, 98 women served Louise de Lorraine; the queen mother, Catherine de' Medici, had 112 suivantes. 50 The high visibility of women in the royal entourage provoked contradictory reactions. Brantôme relates a discussion with a prince (whose name he fails to give) who made the mistake of assigning to Francis I the installation of women at court: He much blamed him for two things which have brought harm to the court and the realm, under his reign and those of his successors: first, to have introduced at court large crowds, access to and the constant presence of women; and second, to have summoned, integrated and settled there such a mass of Churchmen. 51

  Caroline zum Kolk, The Household of the Queen of France in the Sixteenth Century Cour de France.fr Documents, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19 e siècle. http://cour-de-france.fr

36Sommé,

  Isabelle du Portugal, p. 289. 37 The presence of an officer of this sort is documented in the hôtels of Isabeau de Bavière, Marguerite d'Autriche and Anne de Bretagne: Maurice Rey, Les finances royales sous Charles VI: les causes du déficit (1388-1413) (Paris, 1965), p. 193; Florence Trombert, 'Une reine de quatre ans à la cour de France: Marguerite d'Autriche, 1484-1485', Geneviève Contamine, Philippe Contamine (ed.), Autour de Marguerite d'Écosse: reines, princesses et dames du . Potter, P.R. Roberts, 'An Englishman's View of the Court of Henri III, 1584-1585: Richard Cook's Description of the Court of France', French History, tome 2, 1988, p. 341; quoted and translated by Chatenet, La cour de France, p. 25.

  The ordinance from 1261 includes measures which targeted less her household than her behaviour: St Louis forbade Marguerite de Provence to invite ladies or noble persons to join her and to prolong conversations with visitors: She must not summon to her presence ladies or other great persons, and when they come [to court] she should not keep them for a long time; she should not let herself talk lightly to so many visitors, but should excuse herself from their presence in honest terms, as is suitable. Studies inMedieval and Renaissance History, 5, 1968, pp. 1-47, and Jean Dufour, 'Le rôle des reines de France aux IXe et Xe siècles', in Comptes rendus de l'Académie des Inscriptions et

	Equally, the King made it clear that the Queen should neither appoint royal officers, nor give them
	orders:
		She should not ask anything of, or give orders to , the Bailly, the Prevost du Roy or others
		who hold an office from the King, and she should not by her authority appoint anybody to
		Baillieships, sergeantships or any other office.
	We have come a long way from the prestigious role assumed by Carolingian queens. The same
	instructions resurfaced sixty years later: Marie de Luxembourg remained subject to them in 1322. 18
	13	Jean Dufour, 'Adèle de Champagne, troisième femme de Louis VII, une reine méconnue', in Marcel Faure, Reines et
	princesses au Moyen Âge: actes du cinquième colloque international de Montpellier Université Paul-Valéry (24-27
	novembre 1999) (Montpellier, 2001), p. 41.
	14	Jean Dufour, 'Adèle de Champagne', p. 37. See also Marion Facinger 'A Study of Medieval Queenship: Capetian
	France (987-1237)', in

Offices of the Household of the Queen (1326 and 1484-1490) 1326 1484 and 1490

  In 1484, Anne de France, duchess of Beaujeu and Bourbon, was regent of the realm. The household of the Queen, Marguerite d'Autriche, was serving a child: Marguerite was only four. Nevertheless, her hôtel, for the first time since 1326, was solidly structured, with ladies and numerous officers present in all of its divisions. With its ninety-four servants, its size had increased, and the number of 23 BNF ms. fr. nouv. ac. 9175. 1326: Jeanne d'Evreux, wife of Charles IV le Bel, f. 341; 1452 and 1461: Marie d'Anjou, wife of Charles VII, f. 347; 1479: Charlotte de Savoie, f. 351; 1484-1490: Marguerite d'Autriche, promised in marriage to Charles VIII, repudiated in 1491, f. 353.Caroline zum Kolk, The Household of the Queen of France in the Sixteenth Century ladies and maids of honour was once again approaching the level of 1326. Only those who worked in the stables were fewer in number, a fact which can be explained by the Queen's youth. Six years later, Marguerite had at her disposal a well furnished écurie and five more maîtres d'hôtel.A comparison between the offices in the maisons of 1326 and of 1484-1490 show an increasing specialisation and a heightened hierarchy:Caroline zum Kolk, The Household of the Queen of France in the Sixteenth Century

	Valets servant le vin en salle		Médecin
					Apothicaire
	Valets of the Queen:			Huissiers de chambre
	Écuyers servants			Huissiers de salle
	Creue puis l'ordonnance							23
					Valets of the Queen:
		Offices	1326	1452	1461 Valets de Chambre 1479	1484	1490
	[page 9]	Dames	12	4	3 Valets de Garde-robe -	8
	[page 10]	Damoiselles	6	13	11	-		10	14
		Métiers
	Paneterie Dames	13	4	4 Femmes mariées 6		7
	Damoiselles			Filles d'honneur
	Échansonnerie Damoiselles des Dames	16	4	5 Femmes de chambre 6	6
	Cuisine Chevaliers	32	-	-Filles de chambre 10		13	13
	Fruiterie Metiers:	7	-	-Femmes des filles 3		4
	Écurie Paneterie	35	2	2 Metiers:	17		5	11
	Fourrière Échansons	11	-	-Paneterie	5		5
	Valets tranchants Cuisine Fruiterie	0	3	3 Échansons Cuisine	1		3
	Total métiers Écurie	114	13	14 Fruiterie	48		43	47
	Fourrière Chambres Valets tranchants			Écurie Fourrière
		Maîtres d'hôtel	3	6	6 Valets tranchants 6		7	13
	Chambres: Chambre aux deniers Maîtres d'hôtel	4	4	Écuyers d'Écurie 5 -Maréchal des logis		2
	Chapelle Chambre aux deniers	15	14	17 Fourriers Tapissier	4		3
	Chancellerie Chapelle:	2	-	-Lavandières	1		2
	Chambre Chapelains	16	-	-	4		4
	Clercs de chapelle			Chambres:
	Total chambres Confesseur	40	24	28 Maîtres d'hôtel 15		18	24
		Divers	40	17	24 Trésorier	7		15	15
	Total état Notaires Chancellerie:	212	71	80 Contrôleur de l'Argenterie 70 94	109
					Chapelle
	Divers:			(no detail available)
	Physiciens
	Huissier d'armes			Chancellerie:
	Sergents d'armes			Secrétaires
	Huissiers de salle			Clercs d'Office
	Portiers
	Valets de porte			Divers:
				8

Cour de France.fr Documents, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19 e siècle. http://cour-de-france.fr 7 Cour de France.fr Documents, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19 e siècle. http://cour-de-france.fr

The Size and Relationship of the Households of the King and Queen of France (1286-1786) 25 Year Household of the king Household of the queen Percentage

  in italics are estimates based on the totals closest to the year in question. For the size of the households see, also: Monique Chatenet, La Cour de France, pp. 26-7; Alexis Diagne-Noël, L'Hôtel de la reine de France à la fin du MoyenÂge, 1261-1422 (Université Paris IV, 1984), p. 32-33; Nicolas Le Roux, 'La Maison du roi sous les premiers Bourbons: institution sociale et outil politique', in Chantal Grell, Benoît Pellistrandi (ed.), Les Cours d'Espagne et de France au XVIIe siècle (Madrid, 2007), pp. 13-40; Jacqueline Boucher, 'L'Évolution de la maison du roi des derniers Valois aux premiers Bourbons', in XVIIe siècle, 1982, n°137, pp. 359-380; Sophie de Laverny, Les domestiques commensaux du roi au XVIIe siècle (Paris, 2002); Mathieu da Vinha, 'La maison d'Anne d'Autriche', in Chantal Grell (ed.), Anne d'Autriche, infante d'Espagne et reine de France (Madrid, 2009); Caroline zum Kolk (ed.), Curia -The Royal Households in the 17th and 18th Centuries: Databases, documents, studies (Research project, Centre de recherche du château de Versailles, http://chateauversailles-recherche.fr/curia/curia.html); Philip Mansel, The Court of

	1286	98	47	48 %
	1315	96	61	64 %
	1490	318	109	34 %
	1494-5	366	88	24 %
	1498	286	253	88 %
	1523	540	285	53 %
	1555	1000	400	40 %
	1572	1064	384	36 %
	1584	1096	374	34 %
	1605	1024

24

Maison of Anne de

Bretagne, 1494: Ms. fr. 8269 (388-389). 1496-1498: BNF, Ms.fr. 8269 (403-408). 25

Numbers between parentheses

The Size of the Household of the Queen of France (1494-1589)

	Eléonore d'Autriche	1533	320 Elisabeth d'Autriche	1573
	Eléonore d'Autriche	1534	315 Elisabeth d'Autriche	1574
	Eléonore d'Autriche	1535	298 Louise de Lorraine	1575
	Eléonore d'Autriche	1536	298 Louise de Lorraine	1576
	Eléonore d'Autriche	1537	297 Louise de Lorraine	1577
	Eléonore d'Autriche	1538	303 Louise de Lorraine	1578
	Eléonore d'Autriche	1539	302 Louise de Lorraine	1579
	Eléonore d'Autriche	1540	306 Louise de Lorraine	1580
	Eléonore d'Autriche	1541	302 Louise de Lorraine	1581
	Eléonore d'Autriche	1542	280 Louise de Lorraine	1582
	The court played a Eléonore d'Autriche	1543	263 Louise de Lorraine	1583
	Eléonore d'Autriche	1544	259 Louise de Lorraine	1584
	[page 12] Eléonore d'Autriche	1545	279 Louise de Lorraine	1585
	Eléonore d'Autriche	1546	268 Louise de Lorraine	1586
	Catherine de'Médici	1547	291 Louise de Lorraine	1587
	Catherine de Médici	1548	297 Louise de Lorraine	1588
	Catherine de' Médici	1549	303 Louise de Lorraine	1589
	Catherine de' Médici	1550	317
	[page 13]
	Queen	Year	Total Queen	Year	Total
	Anne de Bretagne	1494	88	Catherine de' Médici	1551	309
	Anne de Bretagne	1496	253 Catherine de' Médici	1552	340
	Anne de Bretagne	1497	253 Catherine de' Médici	1553	334
	Anne de Bretagne	1498	253 Catherine de' Médici	1554	405
	Claude de France	1523	285 Catherine de' Médici	1555	400
	Eléonore d'Autriche	1530	253 Marie Stuart	1560	289
	Eléonore d'Autriche	1531	272 Elisabeth d'Autriche	1571	339
	Eléonore d'Autriche	1532	311 Elisabeth d'Autriche	1572	384
				11

Documents, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19 e siècle. http://cour-de-france.fr

  Les Vies des Femmes célèbres, Inv. Ms 18, F°1, Musée départemental Dobrée, Nantes, France.Documents, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19Caroline zum Kolk, The Household of the Queen of France in the Sixteenth Century

	[page 14]
		Nantes, France
	[page 15]
	several young ladies of rank, including Marguerite d'Autriche, Louise de Savoie, Anne de Bretagne
	and Diane de Poitiers'. 33
	31	Elizabeth Lestrange, 'Le mécénat d'Anne de Bretagne', in Kathleen Wilson-Chevalier (ed.), Patronnes et mécènes en
	France à la Renaissance (Saint Etienne, 2007), p. 172.

32 Manuscrit Antoine Du Four, e siècle. http://cour-de-france.fr 13 Anne de Bretagne and her ladies-in-waiting receiving from Antoine Dufour the manuscript 'Les vies des femmes célèbres' (Reproduced by kind permission of Musée Départemental Dobrée, Conseil Général de Loire-Atlantique, 33 Elizabeth Lestrange, 'Sainte Anne et le mécénat d'Anne de France', in Wilson-Chevalier, Patronnes et mécènes, p. 137. Cour de France.fr Documents, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19 e siècle. http://cour-de-france.fr 14

Numbers of the Divisions of the Queen's Household (1496-1589)

  Table, cuisine du commun, cuisine bouche, chapel and chancellery). The Treasury was the smallest, with only two employees; the secrétaires des finances were under the chancellery.The period from 1530 to 1588 is well documented, allowing us to make sense of the variations within the different divisions. The expansion of female personnel from 35 to 124 is particularly noteworthy, as are those of the Chapel (from 16 to 40) and the Chancellery (11 to 40). The Chamber (34 to 55) and the Fourrière (crafts department) (27 to 44) increased to a lesser extent. The Chancellery of the queen was volatile: in 1554-5 it had 48 / 47 members; in 1573-4, 34 / 35 members; and at the end of the 1580's its membership varied between 28 and 40.In 1575, when the household of Louise de Lorraine was constituted, two divisions decreased quite significantly: the Sommeliers (44 in1530, 53 in 1574, 26 in 1575, 28 in 1589) and the Table /maîtres d'honneur (31 in 1530, 37 in 1574, 22 in 1575, 29 in 1589). The decrease in the Cuisine du commun (28 in 1530, 22 in 1589) was less dramatic, and no particular variations affected the Cuisine bouche (24 in 1530, 20 in 1589) or the Treasury.

	Average Department	Average
	Women	71
	Table / Maîtres d'hôtel	29
	Chapel	22
	Chancellery	21
	Chamber	41
	Sommeliers	43
	Cuisine bouche	22,5
	Cuisine commun	29
	Fourrière and crafts	38,5
	Treasury	2
	Total	319
	[page 17]

Cour de France.frDocuments, études et ressources scientifiques pour la recherche sur la cour de France, de ses origines au 19 e siècle. http://cour-de-france.fr

Münster, 'Funktionen der dames et demoiselles', p. 342.

'Ce fut la premiere qui commança à dresser la grand Court des Dames, que nous avons veu despuis elle jusques à cest heure; car elle en avoit une très-grande suitte, et de Dames et de filles, et n'en refusa jamais aulcune; tant s'en faut, qu'elle s'enquerroit des gentils-hommes leurs peres qui estoyent à la Court, s'ilz avoyent des filles, et quelles estoyent, et les leur demandoit.' in Brantôme, Recueil des Dames, p. 13.

Kathleen Wilson-Chevalier, 'Feminising the Warrior at Francis I's Fontainebleau', in Philip Ford, Paul White (eds), Masculinities in Sixteenth-Century France: Proceedings of the Eighth Cambridge French RenaissanceColloquium, 5-7 July 2003, (Cambridge, 2006) and online at Cour-de-France.fr: http://cour-de-france.fr/article253.html.

Maison de Catherine de Médici, état de 1547-1585, BNF ms. fr. 7854 (13-36) et étatde 1583, BNF ms. fr. 21451 (363-368).

Brantôme, OEuvres complètes, tome 1, p. 257

The extant états de maison allow us to follow closely the development of sixteenth-century households: no fewer than forty-nine years are well documented. 28 Despite gaps in the record and evidence of occasional downward trends (in 1530, 1544, 1560, 1575), there is a clear pattern of increasing household size. From 1496 to 1589, the average hovers around 319 individuals, while the average number of ladies and maids of honour is 54. The life cycle of the Queen's hôtel becomes apparent: upon its creation (in 1530, 1547, 1571 and 1575), the staff was at first reduced but then grew rapidly, reaching its peak three or four years later. In 1552-4, when Catherine de' Medici controlled the regency, the size of her hôtel underwent an exceptional growth. Moments of economic difficulty or court reorganisation (1544, 1560) could Anne de Bretagne introduced a second dignitary who functioned as a counterpart to the Chevalier d'honneur: her stables were directed by a Grand écuyer who assisted the Queen during her excursions outside the palace. 39 He was second in the hierarchy of the household. At the burial of his mistress, the Grand écuyer walked behind the coffin, the Chevalier d'honneur along the right side of it.

The third office introduced by Anne was that of the Grand Aumônier, which appeared in 1496. He supervised the work of the chapel staff as well as the distribution of alms. 40 The Grand Ecuyer and the Grand Aumônier had existed in the King's household since the Middle Ages and were two of the most prestigious offices of the court. Anne's intention in creating these roles was to place her household on the same level as her husband's.

During the reign of Francis I, changes affected mainly the noble ladies of the household, the dames and demoiselles d'honneur. In 1523 the office of the Première dame d'honneur, the highest of the women in the Queen's retinue, gave way to a new dignity, the Dame d'honneur. 41 From 1523 on, every queen had a dame d'honneur; Louise de Lorraine had two, each of whom served for a six month period. The dame d'honneur supervised the ladies-in-waiting and accompanied the queen wherever she went. 42 Like the Chevalier d'honneur, she was involved in the management of household expenditure. A letter from Catherine de' Médici indicates that her dame d'honneur oversaw the ordering of clothing, medicines and other objects useful to the queen and her ladies. 43 When the queen received visitors, it was she who introduced them and indicated to them their place in the room or at table. 44 At the end of the century, she tasted the queen's meat and began the service of the meal when, followed by a procession of the queen's ladies, she brought the first dish to her table. This duty fulfilled, she had the privilege of eating at her own table, placed near that of the queen. 45 In 1531 another new dignitary, the Gouvernante des filles, appeared in the hôtel of Eléonore d'Autriche. This office had previously existed at the Burgundian Court under the name of mere des filles 46 and should not be confused with that of the governess of the royal children, who was responsible for the household of the royal couple's offspring. The Gouvernante des filles was in 39 Brantôme, Pierre de Bourdeilles, Abbé de, Recueil des Dames, poésies et tombeaux (Paris, 1991) For others, like Blaise de Monluc, the presence of women at court allowed them to act in areas which should normally have been prohibited to them:

[page 21]

Misfortune has it that in France women interfere in too many matters; the king should shut the mouths of women who speak at court; from them come all the stories and malicious gossip. 52 Yet the prominence of women at court did not provoke criticism alone; certain noblemen reacted with enthusiasm, especially amongst the younger generation. Ladies quickly became the main attraction of the court because 'discussions between men about matters of state, hunting and games rapidly become boring, but one is never bored when conversing with honest ladies'. 53 Brantôme reports that if the King left the Queen and ladies to go hunting, or for any other reason, he and his friends were so 'upset, lost and annoyed' that they thereafter referred to 'the court' not as where the King was, but where the Queen and her ladies were. His conclusion is that 'a court without ladies is a court without a court '. 54 This enthusiasm was not without danger for those concerned: Brantôme compares ladies to fortified towns, which, quite naturally, leads him to the conclusion that it 'is allowed to every gallant gentleman to enter there if he can.' 55 Where this desire for conquest was put into practice, there was no major risk for a nobleman but real danger for the noble lady: a bad reputation, or even worse a pregnancy, could destroy the prospect of a suitable marriage and bring discredit on her family. The risks were thus unevenly distributed, and led to warnings such as those that Anne de France had addressed to her daughter Suzanne, in a book of advice on how to preserve oneself from such mishaps and maintain a 'good reputation' at the head of one's household. The Instructions of Anne de France to Her Daughter, 56 written about 1503-5, provides numerous details about how to manage a household. Catherine de' Médici perhaps knew this remarkable account, either personally or via Diane of Poitiers, since the manuscript was in the library of the sénéchalle in her chateau of Anet.

Anne de France lists the qualities which characterise not only a good mistress but also a good ladyin-waiting, since her daughter Suzanne could have been chosen to serve a lady of higher rank. A woman at the head of a hôtel is responsible for the good behaviour of her officers, especially the ladies and demoiselles. She must closely supervise her staff and above all ensure respect for hierarchy and rank. Women must be decent in clothing and speech. They should be occupied in honest pastimes because leisure 'leads to vice and perdition'. Women of loose morals must be Such warnings show how difficult it was to bring together at court men and women little accustomed to such propinquity. It is impossible to outlaw intimate relations between men and women; as E. Viennot stressed, they are tolerated in the end because they make it possible to 'channel and socialise the sexual energy of a group which contains five to ten times more men than women'. 58 Relations may move on to a more intimate level, 'but the rule is clear: nothing must be known, so that everyone's honour is respected and order reigns'.

***

The queens of the sixteenth century managed this sensitive situation well. Their households came to be the epicentre of social and cultural life at court. Scandals were rare; when they occurred, exemplary punishment was meted out. By the spectacular expansion in numbers, the elevation of the hierarchy and the creation of prestigious offices, the household enhanced both the dignity of the queen and the pageantry of the court. The development of separate departments to manage the queen's patrimony, diplomacy and political patronage are an indication of the rising political status of the hôtel and its royal mistress. In the sixteenth century, it could be used as a power base and serve as a 'miniature government' in times of regency. The integration of women at court is another factor which marked the exercise of policy in the French Renaissance: the ladies-in-waiting often functioned as political actors and intermediaries. They also used their position at court for their own family patronage. The queen and her women thus formed a society inspired by the ideals of Christine de Pizan's 'Cité des Dames' and her high ideas about female honour and female responsibilities.


O Amsellem

F Borit

V Guipont

M Jeandin
email: michel.jeandin@ensmp.fr

F Pauchet

A Composite Approach to Al 2 O 3 -based Plasma-Sprayed Coatings

Thermally-sprayed ceramic coatings such as plasma-sprayed alumina show a composite microstructure actually due to the presence of defects such as pores, inter-lamellar and intralamellar cracks. These 2 nd phase-typed features influence the mechanical behavior and electrical insulation of the coating dramatically. In this study, a composite approach to the microstructure of plasma-sprayed alumina was developed for the optimising of component properties such as electrical gaps used in the oil industry. This approach consisted of a Finite Element Analysis (FEA) of thermo-mechanical and electrical properties from simulated microstructures. Series of composite microstructures were tested, i.e that of air plasma-sprayed (APS) alumina basically plus those obtained by addition of glass or resin using co-spraying and impregnation respectively. Various degrees of porosity and cracks could be obtained from different spraying conditions and by subsequent laser surface remelting. Every composite microstructure was studied using quantitative image analysis of series of SEM cross-sections. Electrochemical Impedance Spectroscopy (EIS) in NaCl solution was also performed to characterize the level of connected pores and the resulting electrical insulating properties. From experiments, a Finite Element Model (FEM) based on the actual microstructure was developed. The latter was simulated with involving of all significant features, such as phase distribution, porosity and defects. This simulation was developed to optimize the composite microstructure to meet industrial applications.

Introduction

Thermal spraying is a prominent process for depositing lowcost high-performance dielectric coatings such as those made of pure alumina (Al 2 O 3 ). Al 2 O 3 is used extensively for its electrical insulating properties because of its high dielectric strength [START_REF] Gadow | Combined metallurgical and ceramic coating in the development of tubular ozone generators[END_REF]. Thermal spraying is a deposition process in which molten particles impact at a high velocity, spread and solidify onto a substrate to form thin lamellae. Consequently, thermally-sprayed alumina coatings show a composite microstructure due to the presence of defects such as pores, inter-lamellar and intra-lamellar cracks. These defect origins are multiple. First, build-up defects and gas entrapped generate specific inter-lamellar cracks and pores. Second, intra-lamellar microscopic cracks may be achieved due to rapid solidification after spreading, especially for ceramic materials. The combination of these features generates an interconnected network of pores and cracks. These 2 nd phase-typed features influence the mechanical behavior and electrical insulation of the coating dramatically [START_REF] Pawlowski | The relationship between structure and dielectric properties in plasma sprayed alumina coating[END_REF][START_REF] Beauvais | Influence of defect orientation on electrical insulating properties of plasma-sprayed alumina coatings[END_REF]. In order to know their influence, authors described physical properties as a function of coatings microstructure for the plasma processing [START_REF] Li | Relationships between the Microstructure and properties of thermally sprayed deposits[END_REF][START_REF] Beauvais | Process-Microstructure-Property relationships in controlled atmosphere plasma spraying (CAPS) of ceramics[END_REF][START_REF] Sarikaya | Effect of some parameters on microstructure and hardness of alumina coatings prepared by air plasma spraying[END_REF]. This work was carried out to good deepen into this microstructure-properties approach through the development of coating. This approach has been used in the conventional composite material area and in powder metallurgy. However, for plasma-sprayed coating only few attempts can be noticed because of the intricacy of the involved microstructure [START_REF] Bolelli | Glass-alumina composite coatings by plasma spraying. Part I: Microstructural and mechanical characterization[END_REF][START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF]. This work was based on the study of composite microstructures obtained by air plasma spraying (APS) including those obtained by addition of glass or resin using cospraying and impregnation respectively. Quantitative Image Analysis (QIA) was applied to coatings obtained using different spraying procedures and subsequent laser surface remelting. Electrochemical Impedance Spectroscopy (EIS) was performed to determine the physical and electrical characteristics of the ElectroChemical Interface (ECI). This technique was extensively employed to study corrosion phenomena [START_REF] Creus | Porosity evaluation of protective coatings onto steel, through electrochemical techniques[END_REF][START_REF] Antou | Characterizations of the pore-crack network architecture of thermal-sprayed coatings[END_REF]. EIS was used to correlate the microstructure with coating properties only in the past recent years [START_REF] Saenger | Electrochemical characterization of plasma sprayed WC-Co coatings by impedance techniques[END_REF][START_REF] Zhang | Evaluation of thickness, porosity and pore shape of plasma sprayed TBC by electrochemical impedance spectroscopy[END_REF][START_REF] Byeon | Non-destructive evaluation of degradation in multi-layered thermal barrier coatings by electrochemical impedance spectroscopy[END_REF][START_REF] Liu | An electrochemical impedance spectroscopy study of the corrosion behaviour of PVD coated steels in 0,5 N NaCl aqueous solution: Part I. Establishment of equivalent circuits for EIS data modelling[END_REF]. The objective was to achieve the EIS spectra of thick (400 µm) and highly-dense (porosity <6%) alumina composite coatings. Then, methods can be envisaged to investigate into the thermo-mechanical behavior of the coating i.e., first a model based on a statistical study of composite phases [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Sevostianov | Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the microstructure; cross-property correlations[END_REF] and, second, a model based on the actual coating microstructure [START_REF] Creus | Porosity evaluation of protective coatings onto steel, through electrochemical techniques[END_REF][START_REF] Bolelli | Glass-alumina composite coatings by plasma spraying. Part II: Microstructure-based modelling of mechanical properties[END_REF]. The first model disregards some features of the microstructure. This contribution shows a Finite Element Model (FEM) based on the actual microstructure which allows characterization of all the composite microstructure features and that of thermomechanical properties of coating.

Materials and Processes

Thermal spraying APS alumina composites were sprayed onto grit-blasted stainless steel plates (AISI 304L, 25×30×2 mm 3 , Ra≈ 4µm) with a F4-VB Sulzer Metco torch (Table 1). The scanning step and velocity of the torch were equal to 5mm.pass -1 and 200mm.s -1 respectively. Two air-cooling jets were added close to the torch in order to lower the coating temperature, limit thermal stresses and consequently avoid adhesion problems. Two commercial powders were used, i.e. 105 SFP alumina from Sulzer Metco/Wohlen for alumina and P3000 glass from Potters-Ballotini/Barnsley.

Table 1: Plasma spraying parameters

Post-treatment Impregnation [START_REF] Wielage | Improving the Resistivity of Thermal Sprayed Coatings[END_REF][START_REF] Knuuttila | Sealing of thermal spray coatings by impregnation[END_REF] and laser surface melting of thermal spray coatings were used to seal and/or modify the composite microstructure. The selected sealants were commercial polymers: epoxy and polyurethane resin (Metcoseal URS and ERS). Prior to post-treatment, the as-sprayed samples were cleansed in an ultrasonic bath of ethanol. Laser glazing of plasma-sprayed alumina composite was carried out using a KrF excimer laser radiation (λ=248 nm), operating at 60 Hz. The laser fluence and the number of pulses ranged from 2 to 8 J.cm -1 and from 80 to 500 pulses respectively. The laser beam was shaped using a rectangular mask and focused on the sample surface to achieve laser-treated areas of about 3.5×1.2mm 2 . The velocity of the laser beam was equal to 3cm.s -1 respectively. All experiments were carried out in air.

Quantitative Image Analysis Every Al 2 O 3 composite microstructure was studied using quantitative image analysis of series of SEM images (LEO1450VP). To meet criteria for statistics, 15 SEM digitalized images (764×1024 pixels 2 ) at a magnification of ×1500 were recorded per coating. Back-scattering electrons were used to enhance the image contrast of voids and defects. Cross-sectioned specimens were metallized (Au-Pd film = 3nm) prior to SEM acquisition. Quantitative image analysis (QIA) consisted in multi-stage processing using "MATLAB" software [START_REF] Beauvais | Process-Microstructure-Property relationships in controlled atmosphere plasma spraying (CAPS) of ceramics[END_REF]. X-ray diffractometer and Electron Probe MicroAnalyzer (EPMA) were then used to identify the phases in the composite coating.

Electrochemical Impedance Spectroscopy (EIS)

EIS is an electrochemical method in which an AC signal E(t)=E 0 sin(ωt) is applied to an electrode in an aqueous

Results and discussion

Composite microstructures

It was shown that the microstructure of thermally-sprayed coating can be described as an Al 2 O 3 -based composite with different features related to the matrix and defects (nature, content, orientation). The series of SEM digital images allowed to measure the porosity and crack contents to achieve the quantitative study of the microstructure defects (Figure 2b). Further "crack" images (Figure 2a) were processed by skeletizing, which limited crack defect thickness to only 1 pixel and deleted triple points and their first neighbouring point). From the skeletized "crack" image, both inter lamellar cracks (main orientation <45°) and intra-lamellar cracks (main line orientation >45°) could be discriminated easily using an automatic orientation criterion. X-ray and Electron-Probe Micro-Analysis (EPMA) showed that the matrix coating mainly consisted in cubic γ-Al2O3 with a small amount of body-center α-Al2O3 in unmelted particles. Na impurity which comes from the starting powder was detected in a few particles. Co-spraying process led to a glass-alumina matrix with 20% of glass (Figure 4). The EPMA showed a good building-up of the lamellae of alumina and glass. It can be noticed that the glass splats are much thicker than those of alumina and there were no cracks in the glass phase. On the other hand, the use of sealing or laser post-treatment led to another type of complex composite structure. In the sealing process, vacuum was applied to the sample with resin before sealing to remove the residual air from the pores and improve the impregnation depth. EPMA carbon X-ray map showed that porosity had been impregnated till a depth of 600 µm. As for laser post-treatment, the effects of laser parameter on the surface of alumina coating were studied (Figure 3). Crater formation (figure 3b) and microstructure modification on 10µm depth (figure 3c,3d) were observed from the free surface of the coating. Because of a small wavelength combined to a short pulse duration and a high power density of the excimer laser, only a very thin layer near the material surface could be treated without cracks.

Electrochemical Testing

There are two conventional diagrams for data and results, i.e. the Bode (|Z| and φ vs ω) and Nyquist (Z Im along the Y-axis and Z Re along the X-axis vs ω) plots respectively. Figure 5 shows the typical response of alumina composite coating using both diagrams (Zview®). Three time constants were observed in the frequency range from 0.05 Hz to 13 MHz. Based on the typical spectra shown in Figure 5 and on the morphological structure of the cell system, an EIS equivalent circuit (EC) was proposed. This circuit which is an assembly of circuit elements (resistors, capacitors, inductors, and various forms of impedances) gives, at all frequencies, the same response as the system of the study. Information on the electrochemical cell could therefore be extracted through an appropriate interpretation of the EC variables. Theses variables represent the physical and electrical characteristics of the different defects. Then, C s and R s result in the interface between the ceramic coating and the substrate. The capacitances were replaced by a Constant-Phase-Element (CPE) because of the heterogeneity in the cell system [START_REF] Jiang | Electrochemical Techniques in Studies of Solid ionic Conductors[END_REF]. These measures were duplicated 100 times for good statistics and to take account the corrosion which occurred at the substrate interface during the experiment (Figure 6b-c). Electrochemical results can be correlated with the alumina composite microstructure (2.1). The resistance of the electrolyte in the porosity (R p ) decreased when coating porosity increased (Ref.: A, D, F). EIS in aqueous solution was shown to be powerful to correlate microstructure defects with the electrical properties of composites.

FEM

The microstructure of the composite considered with all the significant features like phase distributions, porosity and defects and related relevant material properties (E ALUMINA = 224GPa). The finite element mesh was generated and refined until it met sufficient microstructure details (Figure 7). In this work, cross-section composites from QIA (see Part 2.2.1) were used to generate microstructure-based finite element meshes. FEM was developed to find a compromise between meshing accuracy and computational time. All simulations were performed in the plane strain hypothesis. The optimal alumina composite representative area (a square image of 71µm in) was determined with successive tests at different image scales. Numerical models succeed in representing anisotropy through the moduli along spray (Es=53GPa) and transverse (Et=96GPa) direction.

Conclusions

Different types of composites were obtained using several processes. APS single spraying and co-spraying led to several alumina-defect composites and alumina-glass-defect composites respectively. In post-processing, the pores were filled with resin and excimer laser was used to modify the surface of as-sprayed alumina composite. Quantitative Image Analysis (QIA) was developed to determine the degree of porosity. EIS allowed the correlating of the microstructure with physical coating properties. It described the coating network architecture especially. Then, a FEM model based on actual microstructure was implemented in order to evaluate the effective elastic moduli and thermal conductivity along the spray and transverse directions of the various alumina composites. All results ascertained the so-called "composite approach" to plasma sprayed materials to establish relationship between microstructure and properties.

Fig. 2 :

 2 Fig. 2: Microstructure analysis a) Q.I.A. processing stages b) Composite porosity and cracks.solution. Current as a function of time, i.e. I(t)=I 0 sin(ωt+φ) is measured to give the impedance at different frequencies, which results in the impedance spectrum Z(ω)=E(t)/I(t). Electrochemical characteristics of the cell are determined from modelling and interpretation of EIS spectra. EIS measurement was performed at room temperature with two different facilities to obtain Z(ω) in a large range of frequencies. A Solartron 1250 Frequency Response Analyzer (FRA) was connected to Solartron 1287 Electrochemical Interface for impedance between f=0.005Hz and 65kHz. A Hewlett-Packard 4192 Electrochemical Interface was used between f=5Hz and 13MHz. Experiments were carried out in a 30g.l -1 NaCl aqueous solution in the 2-electrode mode. The working electrode was the coating and the counter electrode a platinum grid. The electrochemical test cell was a specifically-designed cell which allowed the control of the area coating-electrolyte interface and the electrode distance. The samples were immersed 10min in the solution before measuring to fill the open porosity. Then, a sinusoidal variation of 0.1V was applied and the impedance spectra were achieved after given time intervals.Finite Element Model Elastic properties were studied for all types of sprayed Al 2 O 3 composites. A Finite Element Model (FEM) based on the actual microstructure was developed in Zebulon®. With this FEM code, mapping of digitalized image was obtained onto a finite element mesh.

Fig. 3 :

 3 Fig. 3: Composite microstructures a)As-sprayed alumina surface and cross section b) 2J.cm 2 -500 pulses :Crater formation and crater cross-section c) 6.2J.cm 2 -250 pulses: surface and cross section d) 8J.cm 2 -250 pulses: surface and cross section.

Fig. 4 :

 4 Fig.4: Alumina-glass composite electrochemical interfaces in the system. Coating steel exposed to an aqueous solution has been investigated by several authors[START_REF] Saenger | Electrochemical characterization of plasma sprayed WC-Co coatings by impedance techniques[END_REF][START_REF] Zhang | Evaluation of thickness, porosity and pore shape of plasma sprayed TBC by electrochemical impedance spectroscopy[END_REF][START_REF] Liu | An electrochemical impedance spectroscopy study of the corrosion behaviour of PVD coated steels in 0,5 N NaCl aqueous solution: Part I. Establishment of equivalent circuits for EIS data modelling[END_REF][START_REF] Cottis | Electrochemical Impedance and Noise[END_REF]]. An EC as that shown in Figure6ais generally proposed to describe the different electrochemical interfaces where the substrate is exposed to the electrolyte through permeable defects. The electrolyte is a resistor, represented by R e , in series with the first electrolyte-ceramic interface which consists of a capacitor C c and a resistor R c . C p and R p simulates the electrochemical reaction in the permeable

Fig. 5 :

 5 Fig. 5: Electrochemical response of alumina composite a)Nyquist and b)Bode diagrams

Fig. 6 :

 6 Fig. 6: EIS experiments a)Equivalent circuit b)Table result c)R p measurement, see Table1 for Ref. details.

Fig. 7 :

 7 Fig. 7: FEM experiments, a)Finite element mesh b)Moduli along spray direction function of element density.

Acknowledgments

The authors are indebted to Dr P. Delaporte from LP3 UMR 6182 CNRS-France for the excimer treatment.


Ligorio as a draftsman : Imitation of Antiquity and the Representation of the Body in the Mythological Drawings in the Late Renaissance

The activity of Pirro Ligorio (1512-1582) as a draftsman has never been studied, except by Caterina Volpi in her edition of the "Libro dei disegni", a group of 50 drawings, particularly of mythological subjects, collected in the last book of his forty volumes of the Roman Antiquities ("libri delle Antichità romane"). In fact, until now, the study of the ligorian drawings has been subordinated to the study of the "Antichità romane", that is as a simple iconographical apparatus to the first encyclopedia of classical antiquities representing the "summa" of the literary and archaeological knowledge on the ancient world of the most illustrious erudite and antiquarian of the late sixteenth century. Even if Ligorio himself claims his role as illustrator of antiquities, when he says to have studied mathematic and drawing to be able to express antiquities and ancient buildings in perspective and in profile ("per poter esprimere le cose antiche, come edifici in prospettiva e in profile") 1 , in reality, and especially out of the "Antichità romane", the ligorian drawings have a more complex role and are used to express his own formal and ideological choices and positions. This is a thesis that I have formulated some years ago, when I wrote the recension of Caterina Volpi's book 2 , and that has now begun to convince the scientific community included Caterina Volpi herself. In fact, I have shown that the mythological theme in the "Libro dei disegni" was, in reality, a pretext for Ligorio to explore the human body trough various positions, and that the theme of ancient gymnastic was the key to understand the whole corpus, especially the drawings considered by Catherina Volpi as "anomalous contortions of the body". In fact, the "libro dei disegni" also contains drawings belonging to another corpus, that I have discovered in the Archivio Borromeo and accurately studied 3 , i.e., the corpus of drawings of sport in the ancient world made by Ligorio both to illustrate the famous work "De arte gymnastica (1573)" by Girolamo Mercuriale, the doctor of the cardinal Alessandro Farnese, and to serve as iconographic program for the Ceiling of the "Sala dei Giochi" in the Castle of Alfonso II, Duke of Ferrara, in the context of Counter Reformation 4 . Caterina Volpi has finally admitted the connection between the "libro dei disegni" and the theme of ancient sport but also the influence of Michelangelo' works5 .

Despite this new way in the study of tle Ligorian drawings, it is indispensable to go further both in the study of other significant "corpora" of drawings and in the study of the very complicated relationships between Ligorio and Michelangelo, whom Ligorio both admired and hated. In fact, Ligorio presents him, in his Treaty on the Nobility of arts ("Trattato sulla nobiltà delle arti"), as his "maestro" in drawings : "In painting, Raphael must be followed, in the style and drawing Michelangelo and Polidoro ("Nel dipingere, bisogna seguire Raffaello, nello stile e nel disegno Michelangelo e Polydoro), but in the same work, Ligorio also violently critizes Michelangelo's representation of bodies, especially in the Last Judgment : the forcing of the body, the hands, the arms, the thigs, all made inadequately with every kind of exagerated torsions ("gli sforzamenti del corpo, delle mani, delle braccia e delle coscie, tutti fatti senza proposito e con ogni sorte di storcitura").). Such an ambigous behaviour has to be studied in the context of the new political and cultural context of the Counter Reformation that had forced Ligorio to abandon his projects in the Vatican (Casino of Pius IV) at the arrival of Pius V, and also to leave Rome, even thoug with the hope of coming back, a hope shared by his illustrious protector, the Cardinal Farnese when he recommend Ligorio to Alfonso II d'Este, in 1567 (It will be a delight if you give him back to Rome where he leaves many friends "si compiacerà anche restitutirlo a Roma dove lascia tanti amici").

I have already shown that Ligorio's numerous but vain attempts to return to Rome, strengthened by public declarations of orthodoxy, never prevented him to take position, through his graphic production, against the cultural politic of the Counter Reformation, for example, by rehabilitating, in the Ceiling of the Sala dei Giochi, the gladiators expelled from Mercuriale's book by the Tridentine Censure, and by non hesitating furthermore to put in their hands the swords of duellist, the "detestabilis duellorum usus", strongly repressed by the Church, especially in the Duchy of Ferrara (Decretum de reformatione, XXV,19)6 .

My project is to study, in this new perspective, a corpus of mythological drawings made by Ligorio to illustrate a manuscript containing the Life of Virbius alias Hippolitus, the son of Theseus, that has been examined, but only from an archaeological point of view 7 . The 16 drawings, preserved in the Pierpont Morgan Library (Ms M.A. 542), were commissioned by the cardinal Hippolytus II, uncle of Duke Alfonso II, to be used as cartoon for a tapestry for the Villa d'Este8 . As I have shown, Ligorio sent the manuscript to his patron , in 1569, soon after the conclusion of his last works at the Villa d'Este in Tivoli and before his arrival at the ducal court in Ferrara [START_REF] Vagenheim | Lettre inédite de Pirro Ligorio au cardinal Alexandre Farnèse : « gli abiti delli iddij chiamati Consenti da Marco Varrone[END_REF] .

The study of one drawing belonging to the corpus that is preserved in the Pierpont Morgan Library has already revealed the same ambivalence as in the mythological drawings of the "libro dei disegni" in which the imitation of antiquity serve to explore the human body; the drawing of two fighters, copied from an ancient coin, is simultaneously present in Mercuriale's book on gymnastic and in the fresco representing the sports in Antiquity, in the ceiling of the Castello Estense 10 (see fig. 1 below) : it has convinced me to extend the study to the whole corpus.

In parallel, I intend to conduct a research, that is strongly connected with the former one, i.e., the census of the mythological drawings of Ligorio preserved in the U.S. collections : this research will constitute the first chapter of the first complete catalog ever made of Ligorian mythological drawings dispersed today through various collections, and sometimes wrongly attributed to Ligorio or other artists; in fact, I have shown that a drawing of a sauce-boat preserved in the British Museum, and attributed to Genga [START_REF] Gere | Italian drawings in the BM, Artists working in Rome, London, 1983 : « A boat sauce shaped cup with a figure of Atlas supporting the Firmament by Girolamo Genga ». 12 New York, The[END_REF] , is in reality a drawing of a saltcellar by Ligorio (fig. 2a); here again, as certainly in many others unpublished drawings, Ligorio has chosen a mythological subject -Atlas holding the hearth-to represent the extreme muscular tension of the entire body with a clear reference to the "Atlante Farnese" (fig. 2b).

The catalog will be based on the checklist of Ligorian drawings carefully established by David Coffin, the greatest specialist of Ligorio, and based on existing catalogs of collections 12 and sales 13 ; it has been published as an appendix to his posthumous book with the purpose to serve to such a project 14 .

Fig. 1 :

 1 Fig.1 : the fighters on ancient coins reproduced by Ligorio in the De arte gymnastica of Mercuriale (1573), in the Castello Estense (1570 c.) and in the Ms. M.A. 542 at Pierpont Morgan Library (1569).

Fig. 2 :

 2 Fig.2 : Pirro Ligorio (British Museum) and Atlante Farnèse (Museo archeologico di Napoli).

Turin, Archivio di Stato, Taur.1, f.6r.

Il libro dei disegni di Pirro Ligorio all'archivio di Stato di Torino.A cura di Caterina Volpi, Roma, edizioni dell'Elefante, 1994, 68 ill. "Bulletin de l' Association des historiens de l'art italien",

(1996)(1997), pp. 23-31. 3 Ginette Vagenheim, Some Newly-Discovered Works by Pirro Ligorio, "JWCI", 51, (1988), pp.242-245; for my other works see C.V.

Ginette Vagenheim, Una collaborazione tra antiquario ed erudito : i disegni e le epigrafi di Pirro Ligorio nel De arte gymnastica di Girolamo Mercuriale, in Girolamo Mercuriale e lo spazio scientifico e culturale europeo del '500, (Bibliothèque d'histoire des sciences, 10), Firenze, Olscki, 2008, pp.127-157.

C. Volpi, Sciuratti, mattaccini e giocolieri: Pirro Ligorio, Michelangelo e la critica d'arte della controriforma, in Gli Dei a corte: letteratura e immagini nella Ferrara estense, a cura di Gianni Venturi e Francesca Cappelletti, Firenze, 2009, p.179-205.

G. Vagenheim, Le dessin de l'essercitio gladiatorio de Pirro Ligorio et le De arte gymnastica de Girolamo Mercuriale. De la recherche antiquaire à la propagande de la Contre-Réforme: l'exemple du corps au combat, "Ludica, annali di storia e civiltà del gioco », 3, (1997), pp. 91-100.

R. Lefevre, Pirro Ligorio e la sua Vita di Virbio, dio minore del Nemus aricinum, Roma, 1998.

The complete title is "la Vita di Virbio, detto altrimenti Hippolito figliolo di Theseo, descritta e dissegnata con immitatione dell'antico in sedici historie, da Pirro Ligorio antiquario famoso di sua propria mano, per servitio del Cardinale d'Este II il vecchio che voleva farne fare una tappezeria


Elena Angelini
email: elena.angelini@unisi.it

WARING DECOMPOSITIONS AND IDENTIFIABILITY VIA BERTINI AND MACAULAY2 SOFTWARE

Keywords: Waring decomposition, complex identifiability, real identifiability, Numerical Algebraic Geometry, Hessian criterion. Mathematics Subject Classification (2010): 15A69, 14P05, 14Q99, 13P05, 65H10

Starting from our previous papers [AGMO] and [ABC], we prove the existence of a non-empty Euclidean open subset whose elements are polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2, 3, 3, 3 and rank 6, which are not identifiable over C but are identifiable over R. This result has been obtained via computer-aided procedures suitably adapted to investigate the number of Waring decompositions for general polynomial vectors over the fields of complex and real numbers. Furthermore, by means of the Hessian criterion ([COV]), we prove identifiability over C for polynomial vectors in many cases of sub-generic rank.

Introduction

of powers of the same linear forms. The notions of Waring identifiability and rank extends naturally to the case r > 1.

For instance, if r = d 1 = d 2 = 2, then we deal with the simultaneous diagonalization (by congruence) of a pair of quadrics in P n C : f = (f 1 , f 2 ) admits a unique Waring decomposition with rank k = n + 1, the so called Weierstrass canonical form ( [We]), if and only if the discriminant of the pencil f 1 , f 2 does not vanish. In particular, if f 1 is the Euclidean quadratic form, then we get the diagonalization of f 2 with orthogonal summands, with respect to f 1 ; over R this is possible for any f 2 thanks to the Spectral Theorem. Other known cases of Waring identifiability are the following:

(r, n, d 1 , . . . , d r , k) =          r, 1, d 1 , . . . , d r , 1 1+r r j=1 1+dj dj , d 1 + 1 ≥ k, [ CR
] (4, 2, 2, 2, 2, 2, 4), Veronese [Ve] (2, 2, 2, 3, 4), Roberts [Ro] (3, 2, 3, 3, 4, 7), [AGMO] Regarding the second aspect, we can ask if a general form or polynomial vector, with real coefficients, which is not Waring identifiable, admits a unique decomposition with real linear forms and real scalars. When this happens, we speak about real identifiability. In this direction, in [ABC] we proved that, when r = 1, real identifiability holds in non-trivial Euclidean open subsets in the space of homogeneous polynomials with (n, d, k) ∈ {(2, 7, 12); (2, 8, 15)}. We notice that, in these examples, the number of Waring decompositions for a general form is, respectively, 5 ( [DS]) and 16 ( [RS]).

By combining the simultaneous decomposition problem with the real point of view, we are able to prove the following: Theorem 1.1. There exists a non-trivial Euclidean open subset in the space of real polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2, 3, 3, 3 whose elements have rank 6 and are identifiable over R but not over C.

The real identifiability stated in Theorem 1.1 arises from the computational analysis described in section 3.1.1, realized via the software for Numerical Algebraic Geometry Bertini ( [Be], [BHSW]), suitably coordinated with Matlab, in the spirit of [AGMO] and [ABC]. Our technique is based on homotopy continuation and monodromy loops ( [HOOS]).

By means of a different computational approach developed with the software Macaulay2 ( [M2]), which we outline in section 3.2.1, we are also able to produce many cases of simultaneous Waring identifiability from the complex side, under the assumption of sub-generic rank. Our method is based on the Hessian criterion, originally introduced in [COV] to study the analogous problem in the case of tensors. Our results in this direction are collected in section 3.2.2.

The paper is organized as follows. In section 2 we recall main definitions and known results about Waring decomposition, rank and identifiability, focusing on the complex and real side; moreover we introduce the simultaneous Waring setting, for a detailed description of which we refer to [AGMO]. Section 3 is devoted to our main results and is divided into 2 parts: in section 3.1 we describe the computational approach and the proof for our Theorem 1.1; in section 3.2 we focus on sub-generic rank cases, presenting our procedure and the identifiable cases detected with it.

Preliminaries

Waring decomposition and rank.

Let F be either the complex field C or the real field R. Let (n, d) ∈ N 2 and let F[x 0 , . . . , x n ] d be the space of homogeneous polynomials, called for simplicity forms, of degree d in n + 1 variables over F.

Definition 2.1. A Waring decomposition over F of p ∈ F[x 0 , . . . , x n ] d is given by linear forms i ∈ F[x 0 , . . . , x n ] 1 and scalars λ i ∈ F -{0}, i ∈ {1, . . . , k}, such that (2.1) p = k i=1 λ i d i .
Definition 2.2. Definition 2.5. There exists a unique typical rank over C for forms in C[x 0 , . . . , x n ] d , which we call the generic rank for that space of forms.

One of the most important results for the complex case is the following:

Theorem 2.6 (Alexander-Hirschowitz, 1995, [AH]). The expression for the generic According to [COV], we give the following:

rank of C[x 0 , . . . , x n ] d is k g = n+d
Definition 2.8. Forms in C[x 0 , . . . , x n ] d of rank k < n+d d 1 n+1 are of sub-generic rank.
Remark 2.9. It is well known that it is possible to have more than one typical rank over R for forms in R[x 0 , . . . , x n ] d . The smallest typical rank over R coincides with the generic rank over C, [BT]. Any rank over R between the minimal typical rank and the maximal typical rank is also typical and it is an open problem to determine an expression for the expected maximal typical rank over R in the general case, since only partial results have been obtained, [BBO].

Classical Waring identifiability.

Definition 2.10. A rank-k form p ∈ F[x 0 , . . . , x n ] d is Waring identifiable over F if the presentation (2.1) is unique up to a permutation and scaling of the summands.

Complex identifiability.

Being Waring identifiable is rare for forms of generic rank over C. Namely, as shown in [GM], this happens only in the classically known cases listed in Table 1.

Space of forms Generic rank

Ref. C[x 0 , x 1 ] 2t+1 t + 1 [Sy] C[x 0 , x 1 , x 2 ] 5 7 [Hi] C[x 0 , x 1 , x 2 , x 3 ] 3 5 [Sy]
Table 1. Waring identifiable cases of generic rank over C

On the contrary, Waring identifiability over C is expected for general forms of sub-generic rank. In this direction, in [START_REF] Chiantini | On generic identifiability of symmetric tensors of subgeneric rank[END_REF] it is proved that, if d ≥ 3, then there are no exceptions besides the ones appearing in Table 2.

Space of forms

Sub-generic rank Ref.

C[x 0 , x 1 , x 2 ] 6 9 [AC],[CC] C[x 0 , x 1 , x 2 , x 3 ] 4 8 [CC],[M] C[x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] 3 9 [COV1],[RV]
Table 2. Waring unidentifiable cases of sub-generic rank over C 2.2.2. Real identifiability.

Recent interest has been devoted to the real case ([COV2], [CLQ], [DDL]), which is very important in applications.

Consider a general p ∈ R[x 0 , . . . , x n ] d such that the rank over C equals the rank over R. If p, seen as an element of C[x 0 , . . . , x n ] d , is Waring identifiable over C, then, necessarily, p is Waring identifiable over R. Namely, with the above assumption, the unique complex decomposition of p is actually completely real.

At this point a natural question arises: if p is not Waring identifiable over C, what can one say about the Waring identifiability of p over R? Recently, in [ABC] we proved that real identifiability holds in non-trivial Euclidean open subsets of the spaces of forms collected in Table 3 (in the second column, g stands for generic and sg for sub-generic).

Space of forms

Rank Dec. over C Ref. C[x 0 , x 1 , x 2 ] 7 12 (g) 5 [DS] C[x 0 , x 1 , x 2 ] 8 15 (g) 16 [RS] C[x 0 , x 1 , x 2 ] 6 9 (sg) 2 [AC],[CC] C[x 0 , x 1 , x 2 , x 3 ] 4 8 (sg) 2 [CC],[M] C[x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] 3 9 (sg) 2 [COV1],[RV]
Table 3. Unidentifiable cases over C, identifiable over R in open sets

Moreover in [ABC] we showed that, if there are infinitely many minimal Waring decompositions over C, then it is not possible to find Euclidean open sets of forms in which real identifiability holds.

Therefore, requiring real identifiability of forms is an efficient method to recover Waring identifiability when we deal with finitely many decompositions over C.

Simultaneous Waring setting.

Let n, r ∈ N and let d 1 , . . . , d r ∈ N such that d 1 ≤ . . . ≤ d r .

Definition 2.11. A polynomial vector f = (f 1 , . . . , f r ) is a vector of r forms in n+1 variables over F of degrees d 1 , . . . , d r , i.e. f j ∈ F[x 0 , . . . , x n ] dj for j ∈ {1, . . . , r}.

Definition 2.12. A simultaneous Waring decomposition over

F of f = (f 1 , . . . , f r ) is given by 1 , . . . , k ∈ F[x 0 , . . . , x n ] 1 and (λ j 1 , . . . , λ j k ) ∈ F k -{0} s.t. (2.2) f j = λ j 1 dj 1 + . . . + λ j k dj k
for all j ∈ {1, . . . , r}, or in vector notation

(2.3) f = k i=1 λ 1 i d1 i , . . . , λ r i dr i
.

Remark 2.13. The adjective simultaneous in Definition 2.12 is justified by the fact that f 1 , . . . , f r are decomposed in the sense of Definition 2.1 by means of the same linear forms 1 , . . . , k . For simplicity, throughout the paper, we speak about Waring decomposition also for a polynomial vector.

Remark 2.14. The notions introduced in section 2.1 extend in a natural way to the case r > 1. In particular, referring to Definition 2.2, Definition 2.8 and Definition 2.10, we say that:

• the Waring rank over F of a polynomial vector f = (f 1 , . . . , f r ) is the minimal k appearing in (2.3); • polynomial vectors over C have sub-generic rank k if (2.4) k <     1 n + r r j=1 n + d j d j     ; • a rank-k polynomial vector f = (f 1 , . . . , f r ) is Waring identifiable over F if the expression (2.
3) is unique up to reordering and rescaling.

Unlike what occurs for the case r = 1 (section 2.2.1), in the simultaneous Waring setting, up to now, there are results only on the complex side and it is still an open problem to determine the complete list of identifiable cases of generic rank over C. To the best of our knowledge, the discovered ones are essentially 5 and they are listed in Table 4.

n r d 1 , . . . , d r k Ref. 1 ∀ d 1 + 1 ≥ k 1 1+r r j=1 1+dj dj [CR] ∀ 2 2, 2 n + 1 [We] 2 2 2, 3 4 [Ro] 2 3 3, 3, 4 7 [AGMO] 2 4 2, 2, 2, 2 4 [Ve]
Table 4. Simult. Waring identifiable cases of generic rank over C

Concerning the second line of Table 4, which we mentioned in the Introduction, we stress that, while any quadric of generic rank n + 1 has infinitely many minimal Waring decompositions over C, it becomes identifiable by requiring the simultaneous decomposition with another quadric.

Similarly, the third line of Table 4 reveals that any cubic of generic rank 4, which is not Waring identifiable over C, admits a unique minimal presentation by adding a quadric. This case has a geometric interpretation, close to the previous one, by changing an orthogonal basis with a Parseval frame, in the sense of [CMS]. Indeed, assume that f = (f 1 , f 2 ) is a polynomial vector with

f 1 = x 2 0 + x 2 1 + x 2 2 and f 2 general in C[x 0 , x 1 , x 2 ] 3 . The unique Waring presentation of f consists of 1 , . . . , 4 ∈ C[x 0 , x 1 , x 2 ] 1 and (λ 1 , . . . , λ 4 ) ∈ C 4 -{0} s.t. (f 1 , f 2 ) = 4 i=1 2 i , λ i 3 i .
The 3 × 4 matrix L, whose columns contain the coefficient of 1 , . . . , 4 in the standard basis {x 0 , x 1 , x 2 }, satisfies LL t = I 3 , i.e. the rows of L form an orthonormal set of C 4 . The columns of L yield a Parseval frame of C 3 . Therefore, introducing the simultaneous Waring decomposition over C of more forms with the same (number of) summands, provides another useful technique to recover classical Waring identifiability, when it fails over C.

Recent results

The generic rank case through Numerical Algebraic Geometry.

By combining the points of view described in section 2.2.2 and section 2.3 we get the following: R such that any f ∈ U has rank 6 over C (and over R too) and is identifiable over R but not over C. Theorem 3.1 arises from a computational approach via the software Bertini for Numerical Algebraic Geometry. In this sense, in section 3.1.1 we describe the computer-aided procedure that we implemented to get our result and in section 3.1.2 we outline its proof.

Computational technique.

We consider the polynomial system, equivalent to (2.3), with the assumptions

n = 2, r = 4, d 1 = 2, d 2 = d 3 = d 4 = 3, k = 6: (3.1)        f 1 -λ 1 1 2 1 -. . . -λ 1 6 2 6 = 0 f 2 -λ 2 1 3 1 -. . . -λ 2 6 3 6 = 0 f 3 -λ 3 1 3 1 -. . . -λ 3 6 3 6 = 0 f 4 -λ 4 1 3 1 -. . . -λ 4 6 3 6 = 0 In (3.1), f j ∈ R[x 0 , x 1 , x 2 ] dj is a fixed general form, while i = x 0 + l i 1 x 1 + l i 2 x 2 ∈ C[x 0 , x 1 , x 2 ]
1 and λ j i ∈ C are unknown. By applying the identity principle for polynomials, the j-th equation of (3.1) splits in dj +2 2 conditions, so that we get a square non linear system of order 36, which we denote by F (f1,f2,f3,f4) l 1 1 , l 1 2 , λ 1 1 , λ 2 1 , λ 3 1 , λ 4 1 , . . . , l 6 1 , l 6 2 , λ 1 6 , λ 2 6 , λ 3 6 , λ 4 6 , which is equivalent to (3.1).

We aim to compute the number of solutions of F (f1,f2,f3,f4) in R, that is the number of Waring decompositions over R with 6 summands of the polynomial vector f = (f 1 , f 2 , f 3 , f 4 ).

In practice, to work with general f j 's, we assign random values

l i h , λ j i ∈ R to l i h , λ j i , getting a real vector l 1 1 , l 1 2 , λ 1 1 , λ 2 1 , λ 3 1 , λ 4 1 , . . . , l 6 1 , l 6 2 , λ 1 6 , λ 2 6 , λ 3 6 , λ 4 6
∈ R 36 , called a start-point. By means of F (f1,f2,f3,f4) , we compute the corresponding f 1 , f 2 , f 3 , f 4 , whose coefficients are called start-parameters. The input of our procedure consists of start-point and start-parameters. In particular, the start-point is, by construction, a real solution of F (f 1 ,f 2 ,f 3 ,f 4 ) , i.e. it provides a Waring de-

composition over R with 6 summands of f = f 1 , f 2 , f 3 , f 4 .
Afterwards, we start a triangle-loop, i.e. a loop divided into 3 steps. First, in F (f 1 ,f 2 ,f 3 ,f 4 ) we replace the start-parameters with random complex entries, called final-parameters and we get a new square polynomial system, F 1 , of order 36. We construct a segment homotopy

H 0 : C 36 × [0, 1] → C 36 between F (f 1 ,f 2 ,f 3 ,f 4 ) and F 1 : it provides a path connecting the start-point to a solution of F 1 , called end-point.
The final-parameters and the end-point become, respectively, the start-parameters and the start-point for the second step. Starting from F 1 and proceeding as in the first step, we obtain a polynomial system F 2 and through a segment homotopy

H 1 : C 36 × [0, 1] → C 36
between F 1 and F 2 , the start-point is sent to a solution of F 2 .

Finally, in F 2 the start-parameters are replaced with the entries of f 1 , . . . , f 4 , getting again F (f 1 ,f 2 ,f 3 ,f 4 ) . A segment homotopy

H 2 : C 36 × [0, 1] → C 36 from F 2 to F (f 1 ,f 2 ,f 3 ,f 4 ) sends the start-point of the third step to a solution of F (f 1 ,f 2 ,f 3 ,f 4 ) , which is the output of our algorithm.
At the end of the triangle-loop, we check if the output differs from the initial start-point l

1 1 , l 1 2 , λ 1 1 , λ 2 1 , λ 3 1 , λ 4 1 , . . . , l 6 1 , l 6 2 , λ 1 6 , λ 2 6 , λ 3 6 , λ 4 6
. If this is not the case, then we restart the procedure. Otherwise: if the output is complex but not real, then this procedure suggests that the polynomial vector f = f 1 , f 2 , f 3 , f 4 under investigation is real identifiable, while if the output is real, then identifiability over R fails.

Remark 3.2. This computational technique provides always an answer, since, as we will see in the forthcoming paper [START_REF] Angelini | Catalecticant intersections and unidentifiable cases[END_REF], the general polynomial vector

f = (f 1 , f 2 , f 3 , f 4 ) with f 1 ∈ C[x 0 , x 1 , x 2 ] 2 and f j ∈ C[x 0 , x 1 , x 2 ] 3 , j ∈ {2, 3, 4},
has 2 Waring decompositions over C with rank 6. This fact had previously been checked by means of the computational analysis developed for the perfect cases in [AGMO], from which section 3.1.1 is inspired.

Computational technique.

After establishing the initial data n, r, d 1 , . . . , d r and assigning to k the first value that satisfies (2.4), we define a parametrization of X n d1,...,dr ⊂ P N -1 , where N = r j=1 n+dj dj . Then, we choose k random points g 1 , . . . , g k ∈ X n d1,...,dr , which guarantees that they are general, and we construct the corresponding k(n + r) × N matrix of constants j 1 associated to T . If the rank of j 1 equals k(n + r), then T has the expected dimension (and X n d1,...,dr is not k-defective because of Terracini's Lemma), otherwise we decrease k by 1 and we go back to the choice of points.

By computing the kernel of j 1 we obtain the N -k(n + r) Cartesian equations of T and by partial derivation of them we get the (n + r)

× (N -k(n + r)) Cartesian equations of C k .
In order to determine the dimension of C k at g i , i ∈ {1, . . . , k}, we compute the codimension of T gi C k . It equals the rank of the Jacobian, evaluated at g i , of the Cartesian equations of C k , i.e. of a matrix (n + r) × ((n + r) × (N -k(n + r))), with constant entries. This procedure is an equivalent way to construct the Hessian matrix of the equations defining T , which justifies the expression Hessian criterion. If rank H = dim X n d1,..,dr = n + r -1, then C k has dimension 0 at g i and, by Lemma 3.4, the general complex f = (f 1 , . . . , f r ) ∈ g 1 , . . . , g k of rank k is Waring identifiable over C. Otherwise, other techniques are needed to confirm or deny the identifiability for this particular value of the rank k. Thus, we decrease k by 1 and we restart the procedure from the choice of the k points.

As output of this algorithm, we get the highest value of the sub-generic rank for which the Hessian criterion implies identifiability over C (note that, if identifiability holds for certain k, then the same is true for any k ≤ k, [CO]).

Table of results.

In the following table, we list the results obtained via the computational technique described in section 3.2.1. In the heading of the third column, we denote by g the expected generic rank, given by the formula g = 1 n+r r j=1 n+dj dj

. In particular, we adopt the notation g for the defective cases, with the corresponding bibliographic reference, and g for the cases for whom the defectivity problem is still open. 20 [AB] ≤ 18 3 3 (2, 3, 3) 9 ≤ 7 4 3 (2, 3, 3, 3) 10 ≤ 9

r n (d 1 , . . . , d r ) g k 2 2 (3, 3) 5 [CC] ≤ 4 2 2 (4,
A c k n o w l e d g m e n t s. This paper arises partially from the conference "Waring decompositions and identifiability via Bertini and Macaulay2 softwares", given by the author during the fourteenth International conference on Effective Methods in Algebraic Geometry -MEGA 2017 held at the Castle of Nice University, campus Valrose, in Nice (France), June 12-16 2017. The author would like to thank all the members of the committees for this interesting and stimulating opportunity. This research was partially supported by the Italian GNSAGA-INDAM, by the Italian PRIN2015 -Geometry of Algebraic Varieties (B16J15002000005) and by the Italian "Progetto strategico di ricerca di base Anno 2014 -prof. Giorgio Ottaviani" of the University of Florence.

  for • (n, 2), where k g = n + 1; • (n, d) ∈ {(2, 4), (3, 4), (4, 3), (4, 4)}, where k g = n+d d 1 n+1 + 1. Remark 2.7. As a consequence of Theorem 2.6, if n+d d 1 n+1 ∈ N then the general form in C[x 0 , . . . , x n ] d of generic rank has finitely many minimal Waring decompositions, with some of the exceptions above.

  Theorem 3.1. Let P 35 R be the projective space over R defined by real polynomial vectors with n = 2, r = 4 and d 1 = 2, d 2 = d 3 = d 4 = 3. There exists a non-trivial Euclidean open set U ⊂ P 35

  The minimal k appearing in (2.1) is called the Waring rank of p over F. For simplicity, throughout the paper, we speak about rank.Definition 2.4. A typical rank over F for forms in F[x 0 , . . . , x n ] d is any k such that the set of forms having rank k has positive Euclidean measure.

	Remark 2.3. Any summand in (2.1) has rank 1 over F.

3.1.2. Proof of Theorem 3.1.

We apply the method described in section 3.1.1 to the real polynomial vector f = (f 1 , f 2 , f 3 , f 4 ) arising from (3.1) with start-point For simplicity, the start-point has been divided into 6 blocks, corresponding to the 6 summands in the Waring decomposition of f . In particular, in the each block, the first 2 entries are the coefficients of the linear form, while the other 4 are the scalars appearing in front of the power of this linear form in the decomposition.

The solutions of F (f 1 ,f 2 ,f 3 ,f 4 ) are the start-point and the complex but not real point whose 6 blocks are the following: l 5 1 , l 5 2 , λ 1 5 , λ 2 5 , λ 3 5 , λ 4 5 = -6.783965763898687 • 10 -1 + i 3.078910418302253 • 10 -1 , -3.934665356579083 • 10 -1 + i 6.002384989511068 We notice that l 1 1 , l 1 2 , λ 1 1 , λ 2 1 , λ 3 1 , λ 4 1 , . . . , l 6 1 , l 6 2 , λ 1 6 , λ 2 6 , λ 3 6 , λ 4 6 is self-conjugate, i.e. it is a fixed point of the conjugate operator. Namely, by considering rounding errors, the summands corresponding to the first and the fifth block are one the conjugate of the other and the remaining 4 are real.

If we move f = (f 1 , f 2 , f 3 , f 4 ) in a small Euclidean open subset over R, only one decomposition remains real, because the property of being real is open in the set of Waring decompositions. Therefore there exists a non-trivial Euclidean open subset U in the space of real polynomial vectors under investigation whose elements have one real decomposition plus a complex not real one, as claimed.

Remark 3.3. A result similar to the one stated in Theorem 3.1 may be true for the cases of general real polynomial vectors with finitely many Waring decompositions over C. The interesting fact is that this phenomenon occurs also when the number of decompositions is even.

3.2. The sub-generic rank case through the Hessian criterion.

A sufficient condition for Waring identifiability over C of general polynomial vectors, whose rank satisfies (2.4), is given by the Hessian criterion, which is based on the following generalization of Lemma 5.1 of [START_REF] Chiantini | On generic identifiability of symmetric tensors of subgeneric rank[END_REF] concerning the case r = 1. Lemma 3.4 (Sufficient condition for identifiability of general elements). Assume that the variety X n d1,...,dr = P( r j=1 O P n C (d j )) of rank-1 complex polynomial vectors whose r components depend on n+1 variables and have degrees d 1 ≤ . . . ≤ d r is not k-defective, with k as in (2.4). Let g 1 , . . . , g k ∈ X n d1,...,dr be general polynomial vectors and f ∈ g 1 , . . . , g k general. Let T = T g1 X n d1,..,dr , . . . , T g k X n d1,..,dr and let C k = {g ∈ X n d1,...,dr | T g X n d1,..,dr ⊂ T } be the k-tangential contact locus. If • T has the expected dimension k(dim X n d1,..,dr + 1) = k(n + r) • C k is 0-dimensional at each g i then f is Waring identifiable over C with rank k.

In the following section 3.2.1 we describe the algorithm, based on Lemma 3.4, that we implemented via Macaulay2 to detect identifiable cases of sub-generic rank. With this procedure we are able to produce many identifiable cases, which are listed in section 3.2.2.