Dr Madina Rival
email: madina.rival@cnam.fr

Dr Madina

Keywords: lobbying, political action, taxonomy, multivariate data analysis, cross-country comparisons

Lobbying by businesses is a practice that has become increasingly widespread today, but hardly studied in Europe. This paper endeavours to determine whether corporate political operations differ significantly from one country to the other. To address this question in Europe, lobbying by French and UK firms is analysed by creating a database from a number of the biggest financial newspapers in both countries over the last years. Multivariate data analysis techniques are used to model corporate political actions in France and the UK. Our results suggest that there is only a few lobbying practices. Three types of lobbying practices can indeed be found in France as well as in the UK. This could suggest that firms' reaction to their political environment is less determined by country specific factors, although national specifics exist, than by the particular problem try to address..

Lobbying by firms to influence public decision : is it a legal or an illegal networking ?

The cases of France and the United kingdom

The word "lobbying" originally comes from General Grant who had lived for a while in a hotel after the White House had burst into fire. At that time, pressure groups would gather in the lobby of the hotel to pressure him [START_REF] Décaudin | Encyclopédie de gestion[END_REF]. In this paper, lobbying is defined as those political actions of firms that are meant to influence public decisions including, inter alia, laws, regulations or other things [START_REF] Farnel | Le lobbying : stratégies et techniques d'intervention[END_REF].

Nowadays, a new strategic environment has emerged that has strengthened the need for lobbying : business globalisation, the European construction [START_REF] Jean | Lobbying communautaire : stratégies et modèles[END_REF] and the increasing power of local actors. All these factors are conducive to more competition between firms and more regulation from public decision makers.

Against this background, the study of European firms political actions over the last years is of pressing interest. Two countries with opposite traditions regarding lobbying legitimacy, the UK and France, are analysed. In the UK, lobbying is a regular device used by firms and openly resorted to while, in France, it is a fairly new tool whose legitimacy is, so far, not this well established. Indeed, in the UK, the Crown by tradition is supposed to hear the views of all members of the society. That is why, lobbyists are legally heard by the Commons. Forty cabinets of lobbying operate in London and are managed by MPs or journalists. MPs are allowed to be employed by those cabinets. Conversely, the Le Chapelier law of 1791 used to forbid lobbying in France until 1884, when a new law was passed in Parliament. Nowadays, a number of firms specialized in lobbying exist but they are neither well known nor well regarded.

The remainder of the paper is set out as follows:

the first section presents a taxonomy of corporate political actions; the second section describe the database of political actions in France and Britain that has been created; the third section identifies two models of corporate political action in France and the UK, compares them and conclude.

The taxonomy of lobbying : a theoretical construct

A taxonomy of firms lobbying operations is first presented which will be used to describe, in the second section, French and British corporate political actions and to model their lobbying behaviour in the third section.

Existing lobbying taxonomies

Taxonomies on corporate political actions have already been outlined in the United States. [START_REF] Oberman | Strategy and tactic choice in an institutional resource context[END_REF] presents them and concludes that they do not define clearly the underlying corporate political strategies. Some taxonomies classify corporate political actions using as a sole classification criterion the modalities used by the firm to influence its target. [START_REF] Parsons | Politics and social structure[END_REF] and [START_REF] Mcmillan | Strategy formulation : Political concepts[END_REF] have worked on such taxonomies. These studies however do not analyse global political strategies.

Conversely, other studies are too global and lack precision to describe fully lobbying actions. For example, [START_REF] Handler | Business in politics : campaign strategies of corporate political action committees[END_REF] have worked on financial contributions of political action committees in the United States. They partition these actions in two models : "ideological" actions and "pragmatic" actions.

Likewise, Yoffie (1987) comes up with five global corporate political strategies : the "free rider", the "follower", the "private goods", the "leader" and the "entrepreneur" strategies. Both studies have a point in common: they use a general strategic vocabulary without investigating the specificity of the political market.

Against this backdrop, this paper tries to build a new taxonomy of corporate political actions based on existing taxonomies. The main reference will be the study of [START_REF] Epstein | The corporation in American politics[END_REF] that defines five descriptive variables for lobbying : goals, actors, resources, management and tactics. This paper will use these variables but complements them with new elements.

Our taxonomy

General characteristics on the firm

These four characteristics are essentially useful to describe the data sample.

Size. This paper chooses the standard weighing of firms according to their turnover. Other classifications include the number of employees but these figures can differ from one country to the other according to different regulations. This paper break downs the turnover of firms in ten levels following the INSEE standard.

Legal status. This variable has not exactly the same signification in each country and it is difficult to compare the status of French and UK firms. Nevertheless, it is useful to describe the sample in each country. For example, it can be assumed that a limited liability firm will not have the same type of political action than an unlimited liability company according to the different levels of liability of owners.

Industrial branch. [START_REF] Schmalensee | Do markets differs much ?[END_REF] has shown that the industrial branch can explain a fraction of corporate performance. The analysis of the industrial activity of firms is based on national classification systems in France and the UK. They have common levels and both are in line with the FTSE Global Classification system.

This international system provides a clear structure for cross-country sectoral analysis. This paper breaks down this classification in 17 levels plus one specific level for "Holdings".

Listing of the firm on stock markets. This variable is linked to the legal status of the firm. It is used to investigate whether the type of corporate governance influences lobbying.

General stance of the firm regarding lobbying

General behaviour towards public decision makers. According to Miles et Snow (1978), firms can pay attention to their strategic environment in various ways : they can have a "prospective" behaviour, which means that they act before the evolution of the environment; they can have a "defensive" behaviour which means that they act only after a modification of the environment ; they can have a mixed behaviour. [START_REF] Demil | Stratégie de pionnier et de suiveur : une application à un processus réglementaire[END_REF] has applied this classification to corporate political actions to investigate whether this impacts results positively. In this paper, we use this classification as well with different names which are more suitable for lobbying: "active", if lobbying is done to trigger the public decision (according to the "prospective way") ; "anticipating" if the public decision is already under way but the firm can still influence it (according to the "mixed way") ; "passive" if the public decision has already been taken (according to the "defensive way").

Frequency of relations with public decision makers.

There are two strategies: on based on long run committed cooperation with public decision makers ("continuous" relations) or contacts on a case by case basis when necessary them only when it is necessary ("case by case" relations). The first strategy creates a confidence climate between lobbyists and public decision makers. Furthermore, lobbyists have time to learn the best practices to influence political action. Nevertheless continuous relations are more costly (financial, organisational, etc.) than case by case relations.

The political action studied

Causes triggering the lobbying action. [START_REF] Getz | Selecting corporate political tactics[END_REF] classifies the goals of lobbying in four categories : social, economic, political and technical issues. This classification is difficult to apply here, but this paper is aware that each firm has different reasons to lobby. Usually, the relevance of the problem is specific to the lobbyist firm or to a group of firms. But, occasionally, the lobbyist may argue that questions of public relevance have to been solved (for example, environmental or employment issues). Furthermore, firms can expect different results from the political action : some want to minimize a loss if the public decision can be opposed to the firm's interest; other firms want to maximize a gain if the public decision can be positive to the firm's interest. Lastly, corporate political strategies can be focused on a single issue or, conversely, on a number of problems simultaneously.

The lobbyists in practice. First, the lobbying firm can have partners for its political action or act alone.

Collective action has advantages such as a greater weight in negotiations with public decisions makers, economies of scale and efficiency. It has more legitimacy if it is a structured organization such as an industrial branch union. Nevertheless, collective action entails costs, namely: lack of autonomy in decisions, costs of controlling the behaviour of other firms in the union, etc. For instance, [START_REF] Mclaughlin | Corporate lobbying in the European Community[END_REF] have shown that the size of pressure groups could contribute to explain the results of their lobbying and that there is an optimal size for the pressure group. Moreover, this paper distinguishes whether the lobbyist in practice is internal to the firm (such as the "public affairs" directorate studied in France by Attarca, 1998) or external to the firm (such as specialised consultancies studied by Lamarque, 1996). The current issue is to balance profits and losses of these two solutions. For example, an external lobbyist is more experienced but can originate agency theory-related problems (developed by [START_REF] Jensen | Theory of the firm, managerial behavior, agency cost and ownership structure[END_REF].

The lobbying target. Public decision makers who are targeted are obviously at the core of the description of corporate political actions. [START_REF] Getz | Selecting corporate political tactics[END_REF] found 48 types of targets according to four main dimensions : whether public decision makers are elected, the localisation of the target (international, national, regional, local), the political level of the target (executive power, legislative power, judicial power), the links between the target and the firm (insider relationships, outsider relationships). This paper resorts to some of the variables of this study: the political level of the target (instead of the judicial level, this paper will consider consultative level such as "autorités administratives indépendantes") and the localisation of the target (this paper will not consider the regional level but adds a European level). Furthermore, this paper will take into account to the type of decisions made by the target: be they regulation decisions which a high level of enforcement, such as laws or mere decisions of control authorities (such as those of "autorités administratives indépendantes").

The management of lobbying. [START_REF] Corrado | Media for manager[END_REF] has built a taxonomy thereof along different variables : the communication with the target that can be direct (e.g. a meeting) or not, the modalities used to influence the target (relationships, financial), whether the action is legal or not. This paper will only use the first two variables because it is difficult to obtain information about illegal practices. Turning to the modalities used to influence the target, this paper adds a third one : the legal modality (e.g. a firm can sue a public decision).

Moreover, this paper will take account of the length of the corporate political action which can be short or long.

Eventually, this paper, based on Jacomet's classification of corporate political actions (2000) underlines the attitude of the lobbyist firm to the target which can be pressing (if the lobbyist has a real bargaining power) or cooperative (if this is not the case).

Table 1 summarises our taxonomy of corporate political actions.

The management of lobbying

The attitude to the target Pressing, Cooperative

The lobbying networks studied in France and Britain: description of the data sample

The data sample of corporate political actions related to lobbying has been created by reading the biggest financial newspapers in France and the UK, namely : La Tribune and Les Echos in France, the Financial Times in the UK. In each country, this paper only took into account the lobbying of firms with a French or a British parent company.

It turns out that there is less corporate political actions described per year in newspapers in France than in the UK. For example, in 2000, 169 lobbying actions initiated by British firms were found but only 47 by French firms. To have more comparable data samples in terms of degrees of freedom, this paper studies the case of France from 1995 to 2000 (resulting in 510 corporate political actions) and of the UK only in 2000 (leading to 160 corporate political actions).

General characteristics

The French and British samples exhibit roughly similar characteristics.

First, very large companies dominate the samples since about 50 percent of lobbyist firms have a turnover above 100 000 GB Pounds. 40 percent of the remainder companies are still fairly large firms (with a turnover above 1000 GBP).

Second, manufacturing firms dominate with a share of 30 percent of both samples. This branch spans from agricultural-related industries to electronics.

Third, the firms of the sample are not listed on stock markets in most cases. Nevertheless, there is a difference here between the two samples : British companies are quoted more often tan French companies (a third for the UK against a quarter for France).

Finally, it is difficult to compare the legal status of French and British firms. In France, 75 percent of the firms have the a status of "Société Anonyme". "Private" societies dominate the British sample (2 out of 3).

General stance towards lobbying

In the same way, usual corporate strategies are mainly common for French and British samples.

More than 50 percent of firms have an "anticipating" behaviour towards public decision makers. The remainder is more or less equally distributed between the active and passive stances.

Furthermore, in both samples continuous relations and case by case relations with public decision makers are equally distributed.

Causes triggering the lobbying action

About 75 percent of firms lobby to solve a private problem in both countries. One could expect such a result since a political action is a costly process. Both in France and the UK, about half of corporate political actions are triggered to reduce a loss and half to maximise a gain. Finally, in most cases, there is a single issue for one political action, while 30 percent of individuals have multiple objectives.

With regard to the lobbyists in practice, the target of lobbying and the management of lobbying, the main results are the following.

In most cases the lobbyists are collective and structured organizations (more than 75 percent in both samples). Nevertheless, notice that individual actions are more developed in the UK (13 percent of the sample) than in France (6.3 percent of the sample). By contrast, the externalisation choice of the lobbyist differs as 80 percent of French corporate political actions are operated inside the firm whereas lobbying is mainly done outside British firms (two-thirds). This could be the consequence of a more transparent attitude towards lobbying in the British legal system.

Regarding the target of lobbying, there are still a number of differences between the two countries.

First, the political level of the target is the executive power for more than 90percent of British firms and amounts to only 61.3percent for French individuals. A caveat applies to these figures : they do not mean that the legislative power is not targeted by UK firms. It could be even the opposite: for example, formal relationships between British MPs and companies exist and are not reported in the newspapers used to create the database.

Second, targets for French actions are mainly national (about 75 percent) and less European (12.4 percent) whereas only 60 percent of British actions are national and 24.9 percent European. Local action does not exist in the UK. The decision level of the target is partitioned in both samples into 2/3 for the regulation authorities and 1/3 for the control authorities.

Eventually, the management of the lobbying differs between the French firms to the British firms with regard to the length of the action and the attitude to the target but the modalities used to influence the target and the communication with the target are the same. Most British firms operate lengthy actions (2/3) whereas this is the case of less than half of French firms. The attitude to the target is mainly cooperative in the UK and pressing in France. It may be a cultural point. In both samples, the corporate political action is managed mainly indirectly and by relationships. By simultaneously minimising the within-class variance of the data and maximising the between-class variance of the data, one can, by construction, define homogeneous classes in a statistical sense whereby specific types of lobbying can be defined along them. This paper will take significant variables (according to test statistics which are χ2 distributed) to describe the lobbying type of each class revealed by the data analysis.

Table 2. A comparison between the French sample and the UK sample

Points in common

The French lobbying model

Figure 1. The French lobbying model (1995)(1996)(1997)(1998)(1999)(2000) Results are calculated over the 510 individuals of the sample The data analysis found five classes in the French sample. The average population is 102 individuals for each class but corporate political actions are not equally distributed across the five classes : there is a very large class (class 2) with 42.2 percent of the sample and a small class (class 4) with 6.3 percent of the sample.

Some variables seem particularly discriminating in this respect (the classes differing according to the modalities of these variables) : general stance of the firm regarding lobbying, and especially general behaviour towards public decision makers; the target of the lobbying (the three variables are important but the one most significant is the decision level of the target); the lobbyists in practice. By contrast, variables concerning the causes the lobbying action and the management of lobbying are less discriminating. Regarding the general characteristics of the firms, only the branch of industry may to some extent have a little bearing on the French model of lobbying.

Class 3 (Cl 3) with 88 individuals can be called the "international corporate political action type" since 91.5 percent of international targeted actions belong to this class. The target of the lobbying is really significant in this class : it acts mainly at the consultative level as a control authority. The relevance of the lobbying problem is specific too : 100 percent of the political actions with a public relevant issue belong to this class. The communication with the target is direct for more than 90 percent of the lobbying actions in this class.

"Hyparlo" Case (10/11/1999)

The French supermarkets firm "Hyparlo" is a member of a professional business union (the national Federation of Trade and Distribution firms) that lobbies the World Trade Organization. Its aim is to defend consumers' food quality standards during the Millennium Round.

Class 1 (Cl 1) with 96 individuals can be called the "European corporate political action type" because of the importance of the European localised target. A specificity has to be underlined here, the relevance of the problem is both private and public. Furthermore, another significant mode is the absence of information on the issue of the externalisation of the lobbying action.

"Europe 1" Case (30/07/1998)

The French communication firm "Europe1" has started to lobby the European Commission to lower broadcasting costs during the Soccer World Cup of 1998. It operates within the European Broadcasting Union to defend at the same time consumers' and professionals' rights.

Class 2 and class 5 lobbying actions are both managed at the national level (90 percent of the targets are nationally based in class 2 and 80 percent in class 5). As the target is national it mainly acts through regulation authorities in both cases. To discriminate further these two classes, other (less significant) variables have to be compared. Table 3 shows us the main differences between class 2 and class 5.

Table 3. The French model : a comparison between class 2 and class 5

Variable Mode for class 2 Mode for class 5 General behaviour towards public decision makers

Anticipating and passive Active

Frequency of relations with public decision makers

Case by case Continuous

The expected result

Minimise a loss Maximise a gain

The number of problems to solve Single Multiple The attitude to the target Pressing Cooperative

Finally, regarding those differences, class 2 (Cl 2) with its 215 individuals may be called the "national and imposed corporate political action type".

"Canal plus" Case (06/02/1995)

The French group "Canal Plus" lobbies against the policy of the French Ministry of Communication to minimise a loss, in particular regarding regulations related to TV advertising. Indeed, they are different between private and public channels.

By contrast, class 5 (Cl 5) with its 79 individuals may be called the "national voluntarist corporate political action type".

"Société Générale" Case (26/03/1997)

The French Bank "Société Générale" belongs to the Parisian financial markets organization "Paris

Europlace". This organization promotes Paris in many dimensions as a financial centre, through active political actions on the French government. Members include all categories of players in financial circlese.

Class 4 (Cl 4) with 32 individuals can be called the "local corporate political action type" since 96.9 percent of the targets in this class are local based. It is strongly linked with a rather "dynamic" general stance regarding strategy with an active general behaviour towards public decision makers. The political level of the local target is mainly legislative (81.3 percent of the class 4). Within this class, lobbying firms want in general to solve multiple issues by a cooperative attitude to the public decision makers.

"Sabaté Diosos" Case (15/11/1995)

Sabaté Diosos is a small firm located in south of France. It is a member alongside fifteen similar firms of a regional lobby. This regional lobby expects local MPs to take action contributing to their economic development but multiple issues are often discussed. The data analysis found four classes in the British sample. The average population is 42.3 individuals for each class but corporate political actions are not equally distributed across the four classes : there is a very large class (class 1) with 48.5 percent of the sample and three smaller classes (from 11.8 percent of the sample for class 2 to 22.5 percent of the sample for class 3).

Some variables seem particularly discriminating in this respect (the classes differing according to the modalities of these variables): general stance of the firm regarding lobbying, and especially general behaviour towards public decision makers; the lobbyists in practice (the type of partnership for the lobbying action as much as the externalisation of the lobbying action); the management of lobbying (and especially the communication with the target).

Class 1 (Cl 1) with 82 individuals can be called the "European corporate political action type" (95.24 percent of the European targeted actions belong to this class). A specificity has to be underlined here regarding the lobbyists in practice: about 100 percent of the lobbying actions in this class are led by collective and structured groups using an external lobbyist. General stance of the firm regarding lobbying is "dynamic" with an anticipating behaviour and continuous relations with public decision makers. A the target is European it often acts through control authorities. Within this class, lobbying firms want in general to solve a single issue (to minimise a loss) by an indirect and long run action. This type of partnership is strongly linked with an internal lobbyist. Other specificities have to be underlined :

the relevance of the problem can be public, the targets can be control authorities, the modalities used to influence the target can be financial. Furthermore, the short run action is managed directly by pressing the target.

"BOC" Case (18/02/2000)

The UK government considers significant concession on the climate change levy following fierce lobbying by big industrial companies that were excluded from a discount deal struck. One of the fiercest campaigns is waged by the industrial gas manufacturers "BOC", "Air Products" and "Messer Griesheim". Sir David John, "BOC"'s chairman has written to Tony Blair and Gordon Brown.

Class 2 and class 4 both belong to the "national and structured corporate political action type". They have other points in common: case by case relations with public decision makers, a private issue that is to maximise a gain, a target acting as a regulation authority. To discriminate further these two classes, other (less significant) variables have to be compared. Table 4 shows us the main differences between class 2 and class 4. The British firm "Eidos Interactive" belongs to the "Entertainment and Leisure Software Publishers

Association" (ELPSA). ELSPA lobbies the UK government to introduce a levy on blank CDs, DVDs, CD writers and DVD writers to reduce pirates' profits. Under ELSPA's proposals, the funds would be channels directly to enforcement agencies, to help them to fight piracy.

By contrast, class 4 (Cl 4) with its 29 individuals may be called the "structured and imposed corporate political action type" at the national level.

"Corus" Case (10/11/2000)

"Corus" is opposed to the UK chancellor's proposals to revive inner cities, said too limited in scale to provide a significant boost to the construction and house building sectors. This UK firm acts within the Construction Products Association. The organization tries to solve multiple issues, to build a "growing, profitable and sustainable future for the construction products industry".

An attempt to compare the French and British lobbying styles : is it a legal or an illegal networking ?

Global comparison of the two models

The geography of corporate political actions of the French and British models differ. On the one hand, national actions stands for more than 50 percent in each case. But, on the other, British lobbying actions are only managed at the European and national levels while French lobbying actions can reach international or local targets too. In Britain, there is absolutely no local action and very few international actions (albeit not enough to make a class). As a consequence, there are more European political actions in the British model (48.5 percent) than in the French model (18.8 percent). Moreover, the variable "type of partnership for the lobbying action" is discriminating enough to be a key characteristic of the British model at the national level (opposing structured actions vs. non structured actions) but not in the case of the French model. Furthermore, at the national level, a third class can be identified in Britain (and not in France), the "non structured actions". In French model, non structured actions are allocated in two other classes, namely at the international level (40.7percent of individual actions and 36.2 percent of non structured collective action) and at the national level (especially in imposed actions: 43.8percent of individual actions and 53.2percent of collective and non-structured actions).

Those differences may be explained by the political history of each country. Lobbying is wellestablished in the UK as well as in the European Union. That is why British firms prefer political action at the European level which is more similar to their political action at the national level than international or local lobbying. Moreover, it is difficult for French firms to lobby without a collective organization since it is more difficult for single firms to resort to lobbying alone, which is less recognised in France than in the UK, than for British firms. As a consequence, there is no choice in France as far as partnership type is concerned when it comes to lobbying.

Finally, in both countries, at the European and national level, three corporate political action style (referred to as "European action", "voluntarist action" and "imposed action") stand out. It is worthwhile checking further whether they are exactly the same in France and in Britain (see infra.).

Comparison of the actions at the European level in the two models

For most discriminating variables, European corporate political action styles show common features in France and in the UK: an anticipating general stance towards public decision makers, with both private and public natures of problems, collective and structured actions, indirect communication with the target. We studied the cases of "Europe 1" and "Vosper Thornycroft" to illustrate this kind of lobbying.

These strategic choices are perhaps linked to European institutions. Indeed, the European Commission has more input into decision-making since it prepares laws and regulations that are subsequently passed by the European Parliament and the Council and, as such, is a better target for lobbyists. At the same time, the European Union is opened to lobbyists actions. That is why it is easier for firms to have continuous and active relations with European institutions.

Nevertheless, some (less discriminating) variables differ: the expected result, the externalisation of lobbyist action, the attitude of the firm to the target. For example, French firm do not publicly reveal their choice of the lobbyist whereas British firms openly use outside lobbyists. That may be linked to the lack of legitimacy of lobbying in France.

Comparison of the actions at the national level in the two models

At the national level, a corporate political action of voluntarist type ( illustrated by the "Société Générale" and " Eidos Interactive" cases) has been identified for class 5 in France and class 2 in Britain. In both cases, these lobbying styles exhibit an active general behaviour towards public decision makers, to maximise a gain , by a collective and structured action. Firms try to reach regulation authorities by both direct and indirect ways of communication in the long run. Nevertheless, there are three mains differences between French and UK voluntarist lobbying styles. UK corporate political actions are of a case by case nature, led by external lobbyists mainly to solve one single issue. On the contrary, French corporate political actions are continuous, led by internal lobbyists to solve multiples problems. This feature may be accounted for by the fact that British lobbying is more institutionalised than the French one.

Turning to the imposed corporate political action identified for class 2 in France ("Canal Plus" case") and class 4 in Britain ("Corus" case) a number of points in common can be singled out: a passive (or anticipating) general behaviour towards publics decision makers with only case by case relations, structured and collective actions led internally in most cases, regulation level are targeted and communication is indirect. That may imply that this kind of lobbying is caused by a lack of firms' resources but not by differences in national political histories.

Differences exist too. British firms whose lobbying is "imposed" focus on a single issue, to maximise a gain via a short run and cooperative action. French firms whose lobbying is "imposed" address multiple issues, to minimise a loss, by a long-run and, in half of the cases, pressing action.

Conclusion : is it possible to control lobbying networks ?

To conclude, this paper has modelled corporate political actions of French (1995French ( -2000) ) and British firms (2000). Multivariate data analysis over 679 French and British firms shows that a few kinds of lobbying practices exist in each country : five on four geographical levels in France ("international action", "European action", "national and imposed action", "national and voluntarist action", local action") against four on two geographical levels in the UK ("European action", "national and non structured action", "national structured and imposed action", "national structured and voluntarist action").

Overall, general patterns of lobbying actions are different for French and UK firms but some types of lobbying are common: "European action", "national imposed action" and "national voluntarist action". This may imply that lobbying patterns can be explained by reasons that are more problem-specific than countryspecific. Nevertheless, national specifics exist. We expect to investigate them further by studying more detailed cases in the furure.

Looking ahead, it would be interesting to investigate whether these lobbying actions have an impact on the profitability of firms which undertake them. Such a study could show whether lobby is rational or not.

  Differences -Size -Industrial branch -General stance of the firm regarding lobbying -Causes triggering the lobbying action -The type of partnership for the lobbying action -The decision level of the target -The communication with the target -The modalities used to influence the target -Legal status -Listing on stock markets -The externalisation of the lobbying action -The political level of the target -The geographical level of the target -The length of the action -The attitude to the target Lobbying networks in France and the UK: models and comparison Multivariate data analysis techniques (multiple correspondences analysis and automatic classification) are used to model corporate political actions in the French (510 individuals from 1995 to 2000) and British samples (169 individuals for 2000).

"

  Vosper Thornycroft" belongs to the Shipbuilders and Shiprepairers Association. Nick Gainger, director of the SSA lobbies the European Commission to reprieve national subsidies. He argues that, given a temporary reprieve (as little as three years), merchant shipyards could become independent of government support. Class 2, class 3 and class 4 are managed at the national level (100 percent of the targets are nationally based in class 2 and in class 4, about 80 percent in class 3).Nevertheless, structured corporate political actions (individual or collective) define class 3 whereas structured (and collective) corporate political actions define class 2 and 4. Class 3 (Cl 3) with 38 individuals can be called the "national and non structured corporate political action type" since 86.4percent of individual actions and 100 percent of non structured and collective actions belong to this class. Finally, about 90 percent of the lobbying actions in this class are non structured actions.

Table 1 .

 1 The taxonomy of corporate political actions

	Category	Variable	Modalities
	General characteristics	Size	10 levels of turnover
	of the firm	Legal status	Particular to each country
		Industrial branch	17 levels
		Listing on stock markets	Yes,
			No
	General stance of the	General behaviour towards public	Active,
	firm regarding	decision makers	Anticipating,
	lobbying		Passive
		Frequency of relations with	Continuous,
		public decision makers	Case by case
	Causes triggering the	The relevance of the problem	Private,
	lobbying action		Public
		The expected result	Maximise a gain,
			Minimise a loss
		The number of problems to	Single,
		solve	Multiple
	The lobbyists in	The type of partnership for the	Individual action,
	practice	lobbying action	Collective and structured action,
			Collective but non -structured action
		The externalisation choice of the	Internal,
		lobbyist	External
	The lobbying target	The political level of the target	Executive,
			Legislative,
			Non political
		The geographical level of the	Local,
		target	National,
			European,
			International
		The decision level of the target	General,
			Particular
		The length of the action	Short,
			Long
		The communication with the	Direct,
		target	Indirect
		The modalities used to influence	Relationships,
		the target	Financial,
			Legal

Table 4 .

 4 The British model : a comparison between class 2 and class 4

	Variable	Mode for class 2	Mode for class 4
	General behaviour towards public decision makers	Active	Passive
	The number of problems to solve	Single	Multiple
	The externalisation choice of the lobbyist	External	Internal
	The length of the action	Long	Short
	The communication with the target	Both	Indirect
	Finally, regarding those differences, class 2 (Cl 2) with its 20 individuals may be called the
	"structured and voluntarist corporate political action type" at the national level.
	"Eidos Interactive" Case (07/11/2000)


Madina Rival

Do corporate political strategies differ from one country to the next ? The case of France and Britain

Keywords: lobbying, political action, typology, variance analysis

Lobbying by businesses is a practice becoming more and more common today, but little studied in Europe. This study seeks in particular to determine if corporate political actions differ from one country to the next. To answer this question in Europe, it presents the way in which French and British firms have conducted their lobbying operations by constructing and analysing a base of gifts given in reference newspapers during the last years. Then, variance analysis is used to modelize corporate political strategies in both countries.

The word lobbying first comes from General Grant who had settled in an hotel after a fire at the White House. Pressure groups had to wait for in the lobby of this hotel to met him [START_REF] Décaudin | Lobbying , Encyclopédie de gestion[END_REF]. In this paper we chose to define lobbying as corporate political action to influence public decisions such as laws, regulations or other things [START_REF] Farnel | Le lobbying : stratégies et techniques d'intervention[END_REF].

Today, we can notice the new strategic environment of firms that reinforces the need for lobbying : globalisation in business, European building (Jean, 1922) and decentralization in each country mean more competition between firms and more regulation from public decision makers.

That is why we found interested to study how European firms have managed their political actions during the last years. We will study particularly the case of two different countries regarding the legitimacy of lobbying : in Britain, political action is part from corporate strategies and done openly, in France it is a new practice not very well established.

The typology of corporate political actions used: a theorical building

We have first to define a typology to describe the way in which firms conduct their lobbying. It will be the easier then to compare French and British corporate political strategies.

Existing typologies on lobbying

Typologies on corporate political actions have already been built in United States. [START_REF] Oberman | Strategy and tactic choice in an institutional resource context[END_REF] tried to list them and found that they were confused. Actually, some typologies classify corporate political actions only according to the way of influencing used by the firm. These studies don't analyse global political strategies. [START_REF] Parsons | Politics and social structure[END_REF] and MacMillan (1978) have worked on such typologies. They explain that corporate political action can be managed in a positive way or in a negative way. At the opposite, some studies are to global and miss precision to describe lobbying. For example, [START_REF] Handler | Business in politics : campaign strategies of corporate political action committees[END_REF] have worked on financial contributions of political action committees in United States. They part these action in to models : ideological actions and practical actions. In the same way Yoffie (1987) has found five global corporate political strategies : the free rider strategy, the follower, the private goods, the leader and the contractor strategies. Both studies have a common point : they use a general strategic vocabulary without wondering about the specificity of political market.

That is why we will try to build our own typology of corporate political actions based on different existing typology. Our main reference will be the study of [START_REF] Epstein | The corporation in American politics[END_REF] that define five variables for lobbying : goals, actors, resources, management and tactics. We will use that idea with new elements.

Our typology

1.2.1

General points on firm These four points are more useful to describe the samples studied than to build models of lobbying. They will enable us to answer a question : which firms lobby in France and in Britain ?

• Size

As it is usually done, we chose to weigh up the size of a firm according to its turnover. An other way could have been the number of employees but these figures can differ from one country to the next according to different regulations. Furthermore, we classified the turnovers of firms in ten levels to make comparisons easier.

• Juridical status This variable has not exactly the same signification in each country and it will be difficult to compare it between France and Britain for example. Nevertheless, it is useful to describe the sample in each country. For example, we can think that a limited firm will not have the same type of political action than an unlimited company regarding the different levels of responsibility of the owners.

• Branch of industry [START_REF] Schmalensee | Do markets differs much ?[END_REF] has shown that the branch of industry could explain a part of corporate performance. So, we have to take care of it here. We have based our analysis of the industrial activity of firms on National classification systems in France and Britain. They have common levels and are both are suitable with the FTSE Global Classification system. This international system provides a clear structure for sector analysis across country boundaries. It has shown to have the highest correlation of individual stocks to their sectors out of all index. We summarized it in 17 levels of industries plus one specific level for holding activities.

• Public quotation

The public quotation is linked to the juridical status of the firm but we wanted to stress it in a special point. As a matter of fact, it could be interested to weigh up the power of shareholders in lobbying.

1.2.2

Usual corporate political strategy • General behaviour towards public decision makers According Miles et Snow (1978), firms can pay attention to their strategic environment in various ways :

-they can have a "prospective" behaviour, that means they act before the evolution of this environment -they can have a "defensive" behaviour that means they act only after a modification of their environment -they can have a mixed behavior. [START_REF] Demil | Stratégie de pionnier et de suiveur : une application à un processus réglementaire[END_REF] has applied this classification to corporate political actions to see if one had better results. We will based too on this classification to define the general behaviour of studied firms towards public decision makers. It will be called :

-active if lobbying is done to begin a public decision (according to the "prospective way") -anticipating if public decision is already began but the firm can yet lobby (according to the "mixed way") -passive if public decision has already been taken (according to the "defensive way"). • Frequency of relations with public decision makers For this point, two strategies exist for firms : they can have a long time cooperation with public decision makers (continuous relations) or they can contact them only when it is necessary (appropriate relations). The first strategy allows confidence between lobbyists and public decision makers. Furthermore, lobbyists have time to learn best practices in political action. Nevertheless continuous relations have more costs for the firm than appropriate relations.

1.2.3

The political action studied • The reasons for lobbying [START_REF] Getz | Selecting corporate political tactics[END_REF] classified the goals of lobbying in four categories : social issues, economical issues, political issues, technical issues. This classification is difficult to apply here, but we are aware that each firm has different reasons for lobbying. Usually, the relevance of the problem is specific to the lobbyist firm or to a group of firms. But, some times, the lobbyist will argue that public relevant questions have to been solved (for example environmental issues or employment issues). Furthermore, firms can expect different results from the political action :

-some want to minimize a loss if the public decision can be opposed to firm's interests -some want to maximize profits if the public decision can be positive for the firm's interests. At the end, we noted that corporate political strategies can be concentrated around a single issue or, at the opposite, try to solve a lot of problems at the same time.

• The actors of lobbying First, we will wonder if the lobbyist firm has partners for its political action or if it acts alone. Collective action has advantages : more weigh in discussion with public decisions makers, scale economies and efficiency. It has more legitimacy if it is a structured organization such as industrial branch groups for instance. Nevertheless, collective action has costs : lack of autonomy in decisions, costs for controlling other lobbyist firms. For instance, McLaughlin, Jordan & Maloney (1993) have shown that the size of pressure groups explained the results of their lobbying.

Then, we will see if the real lobbyist is inside the firm (such as the "public affairs" departments studies in France by Attarca in 1998) or outside the firm (such as specialised consulting firms studies by Lamarque in 1996). The current issue is to balance profits and losses of the two ways of lobbying. For example, an outside lobbyist is more experienced but can create agency issues (developed by [START_REF] Jensen | Theory of the firm, managrial behavior, agency cost and ownershipsructure[END_REF]).

• The traget of lobbying

The public decision makers who are targeted by the lobbying are obviously at the centre of the description of corporate political actions. [START_REF] Getz | Selecting corporate political tactics[END_REF] found 48 sorts of targets using four dimensions : the election or not of the public decision makers, the localisation of the target (international, national, regional, local), the action of the target (executive, legislative, judicial), the links between the target and the firm (inside relationships, outside relationships). We will based some variables on this study : the action of the target (instead of judicial level we will consider consultative level such as regulation committees) and the localisation of the target (we will not consider regional level but we will add a European level). Furthermore, we will pay attention to the kind of decisions made by the target : are there regulation decisions (with a text to be enforced such as laws) or only control decision (such as administrative control for some industry branches) ?

• The management of lobbying [START_REF] Corrado | Media for manager[END_REF] has built a typology for managements of lobbying with different variables : the way of communication with the target that could be direct (a meeting) or not, the way of influencing the target (by relationship ways or financial ways), allowed practices or not. We will only use the first two variables because it is difficult to obtain information about illegal practices. For the way of influencing target, we will ad a third way : the juridical way (a firm can suit a public decision for example). Then, we will note the length of the corporate political action that can be short or long. At the end, we will base on Jacomet classification of corporate political actions (2000). He underlines the standpoint of the lobbyist firm towards the target : it can be pressing (if the lobbyist dominates public decision makers) or cooperative (if there is more compettiton).

The Table 1 presents our typology of corporate political actions.

General points (passive variables)

Public quotation

Yes, no General behaviour towards public decision makers

Active, anticipating, passive

Usual corporate political strategy (active variables) Frequency of relations with public decision makers

Continuous, appropriate

The relevance of the problem

Private, public, both The expected result To increase profits, to reduce a loss

The reasons for lobbying (active variables)

The number of problems to solve

Single, multiple

The partnership for the action

Individual action, collective structured action, collective non structured action

The actors of lobbying (active variables)

The choice for the lobbyist

Inside the firm, outside the firm The action of the target

Executive, legislative, consultative, several levels The localisation of the target

Local, national, european, international

The target of lobbying (active variables)

The decision level of the target Regulation, control, both The length of the action

Short, long

The way of communication with the target

Direct, indirect, both

The way of influencing the target

By relationship ways, by financial ways, by juridical ways, mixed ways

The management of lobbying (active variables)

The standpoint towards the target Pressing, cooperative, both

The political actions studied in France an Britain: description of the sample

The individuals of the sample that is to say corporate political actions) have been selected by reading references newspapers : La Tribune and Les Echos in France, the Financial Times in Britain. In Each country, we only took into account the lobbying of firms with a French or a British mother house.

There were less corporate political actions described in newspapers in France than in Britain. For example, in 2000, we could analyse 169 actions managed by British firms but only 47 managed by French firms. As a consequence, we studied France from 1995 to 2000 (that means 510 corporate political actions) and Britain only in 2000 (160 individuals).

General points

Both French and British sample show mainly the same general points.

First, we are studying very large companies since about 50 % of lobbyist firms have a turnover higher than 100 000 GB Pounds. 40 % of other companies are still large firms (turnover >1000 GBP).

Second, we are studying firms from manufacturing industries (30 % of both sample). We can notice that this branch of industry is not really precise, it goes from agricultural industries to electronic industries. Then part of firms work in services or finance.

Third, the firms of the sample are not quoted in most cases. Nevertheless, there is a difference here between the two samples : British companies are quoted more often tan French companies (1/3 for Britain ; 1/4 for France). Actually , it is well known that the British financial market is more open than the French one.

Finally, it is difficult to compare juridical status between France and Britain. In France, 75 % of individuals have the juridical status of "Société Anonyme". Private societies dominate the British sample (2/3).

Usual corporate political strategy

In the same way, usual corporate strategies are mainly common for French and British samples.

More of 50 % of individuals have an "anticipating" general behaviour towards public decision makers. Other individuals are more or less equally parted between active and passive behaviours.

Furthermore, in both sample there are about as many continuous relations as appropriate relations with public decision makers.

The political action studied

The modes for the political action studied differ more from the French sample to the British sample, except for the whole category of the reasons for lobbying.

Indeed, about 75 % of companies lobby to solve a private problem in both countries. We could wait for such a result since a political action is costly for the firm that tries to reduce positive externalities of their actions. Then, in France as in Britain about 50 % of corporate political actions are managed to reduce a loss and 50 % to increase profits. Finally, in most cases there is a single issue for one political action whereas 30 % of individuals have multiple issues.

For other categories of variable (the actors of lobbying, the target of lobbying, the management of lobbying) there are some common points and some differences between the French and the British sample.

In most cases the actors of lobbying are collective structured organization (more than 75 % of both sample). Nevertheless notice that individual actions are more developed in Britain (13 % of the sample) than in France (6.27 % of the sample). On the opposite, the choice for the lobbyist differs from one case to the other : 80 % of French corporate political actions are lead inside he firm whereas lobbying is mainly done outside the British firm (2/3). This could be the consequence of more transparency in the British system.

Regarding the target of lobbying, there are still a lot of differences between the two cases. First, the action of the target is executive for more than 90 % of the British individuals and only for 61.33 % of the French individuals. We have to take care with these figures : they don't mean that there is no relationship between legislative targets and firms in Britain. It could even be the opposite : for example, relationships between British MPs and companies are officially managed ; consequently there is no need for lobbying at this level. Second, targets for French actions are mainly national (about 75 %) and less European (12.38 %) whereas only 60 % of British actions are national and 24.85 % European. Local action doesn't exist in Britain. The decision level of the target is parted in both sample from 2/3 for regulation and 1/3 control.

In the end, the management of the lobbying differs from the French firms to the British firms according to the length of the action and the standpoint towards the target but the way of influencing the target and the way of communication with the target are common. Most British firms lead long actions (2/3) whereas less than 50 % for French firms. The standpoint towards the target is mainly cooperative in Britain and pressing in France. It may be a cultural point. In both sample, the corporate political action is managed directly and by relationship ways. We have used data analysis methodologies such as multiple correspondences analysis and classification to modelize corporate political actions for the French sample (510 individuals from 1995 to 2000) and for the British sample (169 individuals for 2000). We will take significant variables (regarding test values and khi 2) to describe the lobbying type of each class given by data analysis.

The French lobbying model

Data analysis found a five classes model for the French sample.

Some variables seem more significant to describe this model : usual corporate political strategy and specially general behaviour towards public decision makers ; the target of the lobbying (the three variables are important but the more significant is the decision level of the target) ; the choice for the lobbyist. Variables about the reasons for lobbying and the management of lobbying are less significant. Regarding General points on firms, only the branch of industry could have little signification in the model.

The average population is 102 individuals in each class but corporate political actions are not equally parted between the five classes : there is a large class (class 2) with 42.16 % of the sample and a thin class (class 4) with 6.27 % of the sample. Each class is homogeneous, that is why we can define five kinds of lobbying according to theses fives classes.

Class 1 (Cl 1) with 96 individuals could be called the European corporate political action type because of the domination of the European localised target. A specificity has to be underlined here, the relevance of the problem is both private and public. Furthermore, another significant mode is the non answer at the question choice for the lobbyist. Class 3 (Cl 3) with 88 individuals could be called the international corporate political action type since 91.49 % of international targeted actions are in this class. The target of the lobbying is really significant in this class : it acts mainly at the consultative level to control. The Relevance of the lobbying problem is specific too : 100 % of the political actions led for a public relevant issue are in this class. The way of communication with the target is direct for more than 90 % of the lobbying actions in this class. Class 4 (Cl 4) with 32 individuals could be called the local corporate political action type since 96.88 % of the targets are local localised in this class. It is strongly linked with a dynamic usual corporate political strategy (active general behaviour towards public decision makers). The local target acts mainly at the legislative level (81.25 % of the class 4). Lobbyists firms want to solve multiple issues by a cooperative standpoint towards the public decision makers. Classe 2 and class 5 are both managed at the national level (90 % of the targets are national localised in class 2 and 80 % in class 5). As the target is national it mainly acts for regulations in both cases. That is why we have to compare other significant variables to look for differences between these two lobbying styles. Table 3 shows us the main differences between class 2 and class 5.

Table 3. The French model : a comparison between class 2 and class 5

Variable

Mode for class 2 Mode for class 5 General behaviour towards public decision makers

Anticipating and passive

Active

Frequency of relations with public decision makers

Appropriate Continuous

The expected result To reduce a loss To increase profits The number of problems to solve

Single Multiple

The standpoint towards the target

Pressing Cooperative

Finally, regarding those differences, class 2 (Cl 2) with its 215 individuals could be called the national subjected corporate political action type. On the opposite, class 5 (Cl 5) with its 79 individuals could be called the national built corporate political action type

The British lobbying model

Data analysis found a four classes model for the British sample.

Some variables seem more significant to describe this model : usual corporate political strategy and specially general behaviour towards public decision makers ; the actors of lobbying regarding the partnership for the action as much as the choice for the lobbyist ; the management of lobbying (and specially the way of communication with the target).

The average population is 42.25 individuals in each class but corporate political actions are not equally parted between the four classes : there is a large class (class 1) with 48.5 % of the sample and three other class less important (from 11.8 % of the sample for class 2 to 22.5 % of the sample for class 3). Each class is homogeneous, that is why we can define four kinds of lobbying according to theses four classes.

Class 1 (Cl 1) with 82 individuals could be called the European corporate political action type (95.24 % of the European targeted actions are in this class). A specificity has to be underlined here about the actors of the lobbying : about 100 % of the lobbying actions in this class are led by ac collective structured organization using an outside lobbyist. Usual corporate political strategy is dynamic with an anticipating behaviour and continuous relations with public decision makers. We can notice that the European target acts more to control firms than to regulate them. Furthermore, the action is long, managed indirectly, to solve a single issue that is to reduce a loss. Classe 2, class 3 and class 4 are all managed at the national level (100 % of the targets are national localised in class 2 and in class 4, about 80 % in class 3). First, we can non structured corporate political action (should it be individual or collective) in class 3 from structured (and so collective) corporate political actions in class 2 and 4. So, class 3 (Cl 3) with 38 individuals stands for the national non structured corporate political action type since 86.36 % of individual actions and 100 % of non structured collective actions are in this class. Finally, about 90 % of the lobbying actions in this class are non structured. That may explain the choice for the lobbyist mainly inside the firm. In this kind of lobbying, some specificities have to be underlined : the relevance of the issue can be public, the target can have control functions, the way of influencing his target can be a financial way. Furthermore, action is short and managed directly by pressing the target. Class 4 (Cl 4) with 32 individuals could be called the local corporate political action type since 96.88 % of the targets are local localised in this class. It is strongly linked with a dynamic usual corporate political strategy (active general behaviour towards public decision makers). The local target acts mainly at the legislative level (81.25 % of the class 4). Lobbyists firms want to solve multiple issues by a cooperative standpoint towards the public decision makers. At the end, we have to define class 2 and class 4 which stand both for national and collective structured corporate political actions. They have common points : appropriate relations with public decision makers, a private issue that is to increase profits, a target acting at the regulation level. That is why we have to compare other significant variables to look for differences between these two lobbying styles. Table 4 shows us the main differences between class 2 and class 4.

Active Passive

The number of problems to solve

Single Multiple

The choice for the lobbyist

Outside the firm Inside the firm

The length of the action

Long Short

The way of communication with the target

Both Indirect

Finally, regarding those differences, class 2 (Cl 2) with its 20 individuals could be called the national collective structured built corporate political action type. On the opposite, class 4 (Cl 5) with its 29 individuals could be called the national collective structured subjected corporate political action type.

We will wonder in a last part if those kind of lobbying are the same than those called nation subjected and national built for the French sample.

An attempt to compare French and British lobbying styles

3.3.1

Regarding the whole model First, localisation of corporate political actions differs from the French model to the British model studied here. On the one hand, national actions stands for more than 50 % in each case. But on the other hand, British lobbying actions are only managed at European and national levels whereas French lobbying actions can reach international or local targets too. In Britain, there is no local action and there are few international actions (not enough to make a class). As a consequence, there are more European political actions in the British model (48.5 %) than in the French model (18.8 %).

Second, the variable "partnership for the action" is significant to build the British model at the national level (structured actions / non structured actions) but not for the French model.

Finally, we have define two national corporate political action style as "built action " and "subjected action" in both models. We will the have to check if they are exactly the same in France and in Britain.

3.3.2

Regarding national action A national built corporate political action has been defined for class 5 in France and class 2 in Britain. In both cases these lobbying styles show an active general behaviour towards public decision makers, to increase profits, by a collective structured action. Firms try to reach regulation level targets by at the same time direct and indirect ways of communication in a long action. Nevertheless, there are three mains differences between French built lobbying style and British built lobbying style. British corporate political actions are appropriate, led by outside lobbyists to solve single issue. On the contrary, French corporate political actions are continuous, led by inside lobbyists to solve multiples problems. Indeed, we know that British lobbying is more institutionalised than French lobbying. It could be an explanation.

Then, a national subjected corporate political action has been defined for class 2 in France and class 4 in Britain. We can underline common points for the two models : passive (or anticipating) general behaviour towards publics decision makers with only appropriate relations, structured collective actions led inside the firm in most cases, regulation level targets and indirect ways of communication. Differences exist too : British subjected lobbying firms have a single issue, to increase profits by a short and cooperative action ; French subjected lobbying firms have multiple issues, to reduce a loss, by a long and some times pressing action.

Furthermore, we have defined a third national action style in Britain, the non structured actions. It doesn't appear as a specific class in France. In French model, non structured actions are managed at international level (40.63 % of individual actions and 36.17 % of non structured collective action), as well as at national level (specially in subjected actions).

3.3.3

Regarding European action For most significant variables, European corporate political action styles show common points in France and in Britain : an anticipating general behaviour towards public decision makers, a problem both private and public, collective structured actions, indirect ways of communication with the target. These strategic choices are probably consequences of European institutions.

Nevertheless, some less significant variables differ from French model to British model : the expected result, the choice for lobbyist, the standpoint toward the target. For example, French firm don't publicly reveal their choice for the lobbyist whereas British firms officially use outside lobbyists. That may be linked to the lack of legitimacy for lobbying in France.

In a nutshell, we have been able here to modelize corporate political actions observed for French firms (1995)(1996)(1997)(1998)(1999)(2000) and for British firms (2000).

The whole model shows differences. Nevertheless some kind of lobbying can be found in France as well as in Britain : European action, national subjected action, national built action.

Another question should now be answered : do those different corporate political strategies have different results regarding companies performance?

  for lobbying -The partnership for the action -The decision level of the target -The way of communication with the target -The way of influencing the target -Juridical status -Public quotation -The choice for the lobbyist -The action of the target -The localisation of the target -The length of the action -The standpoint towards the target 3 Corporate political action in France and in Britain : models and comparison

Table 1 .

 1 The typology of corporate political actions

	Category	Variable	Mode
		Size	10 levels of
			turnover
		Juridical status	Peculiar to each
			country
		Branch of industry 17 levels according
			to the FTSE
			classification
			system,
			one specific level
			for holding
			activities

Table 2 .

 2 A comparison between French sample and Britain sample

Table 4 .

 4 The British model : A comparison between class 2 and class 4

Variable Mode for class 2 Mode for class 5

	General behaviour
	towards public
	decision makers


David Sauzin
email: sauzin@imcce.fr

Initiation to mould calculus through the example of saddle-node singularities

This article proposes an initiation to Écalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field. This is illustrated on the case of saddle-node singularities, generated by two-dimensional vector fields which are formally conjugate to Euler's vector field x 2 ∂ ∂x +(x+y) ∂ ∂y , and for which the formal normalisation proves to be resurgent in 1/x.

Introduction

This article is a survey of a part of a longer article [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] which aims at presenting in a systematic way some aspects of Écalle's theory of moulds, with the example of the classification of saddle-node singularities as a red thread.

Mould calculus was developed by J. Écalle in relation with his Resurgence theory almost thirty years ago [START_REF] Écalle | Les fonctions résurgentes[END_REF] and the application to the saddle-node problem was indicated in [START_REF] Écalle | Cinq applications des fonctions résurgentes[END_REF] in concise manner.

Here, we omit much of the material of [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] and present the arguments in a different order, trying to explain the formal part of Écalle's method in fewer pages and hoping to arouse the reader's interest by this illustration of mould calculus.

1 Saddle-node singularities Germs of holomorphic singular foliations of (C 2 , 0) are defined by analytic differential equations of the form P (x, y)dy -Q(x, y)dx = 0 with P and Q ∈ C{x, y} both vanishing at the origin. Classifying such singular foliations means describing the conjugacy classes under the action of the group of germs of analytic invertible transformations of (C 2 , 0); this is equivalent to classifying the corresponding singular vector fields P (x, y) ∂ ∂x + Q(x, y) ∂ ∂y up to time-change. We shall examine the case in which the foliation is assumed to be formally conjugate to the standard saddle-node foliation, defined by x 2 dyydx = 0. This is the simplest case for differential equations whose 1-jets admit 0 and 1 as eigenvalues (the possible formal normal forms are x p+1 dy -(1 + λx)dx = 0, with p ∈ N * and λ ∈ C, and y dx = 0).

In this case, by a classical theorem of Dulac ( [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], [START_REF] Moussu | Singularités d'équations différentielles holomorphes en dimension deux[END_REF]), the foliation can be analytically reduced to the form x 2 dy -A(x, y)dx = 0, with A(x, y) ∈ C{x, y}, A(0, y) = y, ∂ 2 A ∂x∂y (0, 0) = 0.

(1)

Moreover, the vector fields corresponding to the foliation and to the normal form,

X = x 2 ∂ ∂x + A(x, y) ∂ ∂y and X 0 = x 2 ∂ ∂x + y ∂ ∂y , (2)
are themselves formally conjugate: there is a unique formal transformation of the form θ(x, y) = x, ϕ(x, y) , ϕ(x, y) = y

+ n≥0 ϕ n (x)y n , ϕ n (x) ∈ xC[[x]]. (3)
such that

X = θ * X 0 (4)
(no time-change is needed here). We call it the formal normalisation of the foliation (or of X itself). Thus, in this case, our problem of analytic classification boils down to describing the analytic conjugacy classes of vector fields of the form X under the action of the group of "fibred" transformations (leaving x unchanged), knowing that they are all formally conjugate of one another (since all of them are formally conjugate to X 0 ).

Our starting point will be the data ( 1)-( 2). We shall study the formal conjugacies (3) by means of Écalle's "moulds" and see how this leads to the "resurgent" character of the formal series which appear. (A similar study could be performed for the more general normal forms, with any p and λ.) We shall not give here the complete resurgent solution of the problem of analytic classification; the reader is referred to [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] for this and for the comparison with Martinet-Ramis's solution [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF].

Formal separatrix, formal integral

Observe that our foliations always have an analytic separatrix (a leaf passing through the origin), namely the curve {x = 0} in Dulac coordinates. For the vector field X, this corresponds to the solution z → (0, ue z ) (with an arbitrary constant of integration u ∈ C).

The "formal curve" {y = ϕ 0 (x)} is to be considered as a "formal separatrix" of our foliation (and as a formal centre manifold of X), since it is the image by θ of {y = 0} which is the other separatrix of the normal form. A time-parametrisation of the corresponding integral curve of X 0 is z → (x, y) = (-1/z, 0). The formal series ϕ 0 can thus be obtained by setting ϕ 0 (z) = ϕ 0 (-1/z) and looking for a formal solution z → -1/z, ϕ 0 (z) of X, i.e. the formal series ϕ 0 must solve the non-linear differential equation

d Y dz = A(-1/z, Y ). (5)
More generally, if we set

Y (z, u) = u e z + n≥0 u n e nz ϕ n (z), ϕ n (z) = ϕ n (-1/z) ∈ z -1 C[[z -1 ]], (6)
we get what Écalle calls a "formal integral" of X, a formal object containing a free parameter and solving (5) (this is equivalent to finding a formal transformation of the form (3) which solves (4)). One can find the formal series ϕ n by solving the ordinary differential equations obtained by expanding [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF] in powers of u. The simple and famous example of Euler's equation, for which A(x, y) = x + y, shows that the above formal series can be divergent. Indeed, the equation for ϕ 0 (x) is then x 2 dϕ0 dx = x+ϕ 0 and one finds ϕ 0 (x) = -n≥1 (n-1)!x n . Since the equation is affine, there are no other non-trivial series in this case: ϕ(x, y) boils down to y + ϕ 0 (x).

It is not by studying the differential equation ( 5) that we shall get information on the formal transformation θ, but rather by directly working on the conjugacy equation (3).

The formal normalisation as an operator

Let A = C[[x, y]] and let ν denote the standard valuation (thus ν(f ) ∈ N is the "order" of f , with the convention that ν(x m y n ) = m + n). We denote by M the maximal ideal of A , consisting of all formal series without constant term, i.e.

M = { f ∈ A | ν(f ) ∈ N * }.
We are given analytic vector fields X and X 0 as in [START_REF] Écalle | Cinq applications des fonctions résurgentes[END_REF], which are operators of the C-algebra A , more precisely C-derivations (in fact, they are derivations of C{x, y}, but we begin by forgetting analyticity). We shall look for the formal normalisation θ through its "substitution operator", which is the operator Θ ∈

End C A defined by Θf = f • θ, f ∈ A . (7)
There is in fact a one-to-one correspondence beween "substituable" pairs of formal series, i.e. pairs θ = (θ 1 , θ 2 ) ∈ M × M, and operators of A which are formally continuous algebra homomorphisms, i.e. operators Θ ∈ End C A such that Θ(f g) = (Θf )(Θg) for any f, g ∈ A and ν(Θf n ) → ∞ for any sequence (f n ) of A such that ν(f n ) → ∞ (the proof is easy, see e.g. [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF]; one goes from Θ to θ simply by setting θ 1 = Θx and θ 2 = Θy).

Equation ( 4) can be written

(Xf ) • θ = X 0 (f • θ), f ∈ A ,
and thus rephrased in terms of the substitution operator Θ as

ΘX = X 0 Θ. ( 8
)
Thus, looking for a formal invertible transformation solution of the conjugacy equation (4) is equivalent to looking for a formally continuous algebra automorphism solution of (8).

The formal normalisation as a mould expansion

The general strategy for finding a normalising operator by mould calculus consists in constructing it from the "building blocks" of the object one wishes to normalise. This implies that we shall restrict our attention to those operators of A which are obtained by combining the homogeneous components of X in all possible ways. . . Since we are interested in fibred transformations, it is relevant to consider the homogeneous components of X relatively to the variable y only and to view

A = C[[x]
] as a ring of scalars; we thus write

X = X 0 + n∈N a n B n , B n = y n+1 ∂ ∂y , (9)
with coefficients a n ∈ A stemming from the Taylor expansion

A(x, y) = y + n∈N a n (x)y n+1 , N = { n ∈ Z | n ≥ -1 }. ( 10
)
Homogeneity here means that each a n B n sends Ay k in Ay k+n for all k ∈ N: the component a n B n is homogeneous of degree n (this was the reason for shifting the index n by one unit in the Taylor expansion), while X 0 is homogeneous of degree 0.

We shall look for a solution of (8) among all the operators of the form

Θ = r≥0 n1,...,nr∈N V n1,...,nr B nr • • • B n1 . (11)
Here (V n1,...,nr ) is a collection of coefficients in A, to be chosen in such a way that formula (11) has a meaning as a formally continuous operator of A and defines an automorphism solving [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF]. The above summation is better understood as a summation over N • , the free monoid consisting of all words ω (of any length r) the letters of which are taken in the alphabet N. We thus set

B ω = B nr • • • B n1 , ω = (n 1 , . . . , n r ) ∈ N • , (12)
and B ∅ = Id for the only word of zero length (ω = ∅), and rewrite formula (11) as

Θ = ω∈N • V ω B ω . (13)
An operator Θ defined by such a formula is called a mould expansion, or a mouldcomould contraction. Here the comould is the map

ω ∈ N • → B ω ∈ End C A ,
that we decided to define from the homogeneous components of X, and the mould is the map

ω ∈ N • → V ω ∈ A,
that we must find so as to satisfy the aforementioned requirements.

The general framework for mould-comould contractions

The general theory of moulds and comoulds requires:

-an alphabet N, which is simply a non-empty set, often with a structure of commutative semigroup (since it appears in practice as a set of possible degrees of homogeneity);

-a commutative C-algebra A (the unit of which we denote by 1), in which moulds take their values;

-an A-algebra F (the unit of which we denote by Id), possibly non-commutative, in which comoulds take their values. We also assume that a complete ring pseudovaluation val : F → Z ∪ {∞} is given. 1 In the saddle-node case we can choose for

F a certain A-subalgebra of End A A , with A = C[[x]] and A = A[[y]] = C[[x, y]]
(the fact that we deal with operators which commute with the multiplication by an element of A reflects the fibred character over x of the situation), defined as the set F A,ν of operators admitting a valuation with respect to the valuation ν of A , i.e. the set of all Θ ∈ End A A for which there exists δ ∈ Z such that ν(Θf ) ≥ ν(f ) + δ for all f ∈ A . This way, the valuation 2 ν of A induces a complete ring pseudovaluation val on

F A,ν , namely val (Θ) = inf f ∈A \{0} {ν(Θf ) -ν(f )}.
One can thus safely speak of "formally summable" families in

F : for instance, if for any δ ∈ Z the set { ω ∈ N • | val (V ω B ω ) ≤ δ } is finite, then the family (V ω B ω
) is formally summable and formula (13) defines an element of F . We repeat our definitions in this general context:

-a mould is any map N • → A (we usually denote by M • the mould whose value on the word ω is M ω ), -a comould is any map N • → F (we usually denote by B • the comould whose value on the word ω is B ω ), -a mould expansion is the result of the contraction of a mould M • and a comould B • such that the family (M ω B ω ) ω∈N • is formally summable in F ; we usually use the short-hand notation

Θ = M • B • .
The monoid law in N • is the concatenation, denoted by , which allows us to define mould multiplication by the formula

P • = M • × N • : ω → P ω = ω=ω 1 ω 2 M ω 1 N ω 2 . 1 This means that val (Θ) = ∞ ⇐⇒ Θ = 0, val (Θ 1 -Θ 2 ) ≥ min val (Θ 1 ) , val (Θ 2 ) , val (Θ 1 Θ 2 ) ≥ val (Θ 1 ) + val (Θ 2 ) and the distance (Θ 1 , Θ 2 ) → 2 -val(Θ 2 -Θ 1 ) is complete.
2 For technical reasons, rather than the standard valuation on C[[x, y]], we shall use another monomial valuation, defined by ν(x m y n ) = 4m + n (see Section 6).

The space of moulds is in fact an A-algebra. Correspondingly, if a comould B • is mutiplicative, in the sense that B ω 1 ω 2 = B ω 2 B ω 1 for any ω 1 , ω 2 ∈ N • (as is obviously the case for a comould defined as in (12)), then

(M • × N • ) B • = N • B • M • B •
as soon as both expressions in the right-hand side are formally summable.

One can easily check that a mould M • has a multiplicative inverse if and only if M ∅ is invertible in A. In this case, if (M ω B ω ) is formally summable and B • is multiplicative, then M • B ω is invertible in F . We do not develop farther the theory here and prefer to return to the formal normalisation of our saddle-node singularity X.

Solution of the formal conjugacy problem

Let us use the A-algebra F = F A,ν defined in the previous section, to which each operator B n = y n+1 ∂ y obviously belongs (B n is A-linear and val (B n ) = n), and thus also the B ω 's defined by (12). Formula (9) can be written X -X 0 = J • a B • , with

J ω a = a n1 if ω = (n 1 ) 0 if r(ω) = 1 ( 14
)
and the conjugacy equation ( 8) is equivalent to

Θ(X -X 0 ) = [X 0 , Θ]. ( 15
) Lemma 6.1. Let V • be a mould such that the family (V ω B ω ) is formally summable in F . Then [X 0 , V • B • ] = (x 2 ∂ x V • + ∇V • )B • ,
where the mould ∇V • is defined by ∇V ∅ = 0 and

∇V ω = (n 1 + • • • + n r )V ω
for ω = (n 1 , . . . , n r ) non-empty.

Proof. The operator B n1,...,nr is homogeneous of degree

n 1 + • • • + n r . One can check that, if Θ ∈ End A (A[[y]]) is homogeneous of degree n ∈ Z, then [y∂ y , Θ] = nΘ.
Indeed, by A-linearity and formal continuity, it is sufficient to check that both operators act the same way on a monomial y k ; but Θy k = β k y k+n with a

β k ∈ A, thus y∂ y Θy k = (k + n)β k y k+n = (k + n)Θy k while Θy∂ y y k = kΘy k . Since X 0 = x 2 ∂ x + y∂ y and x 2 ∂ x commutes with the B n 's, it follows that [X 0 , V ω B ω ] = x 2 ∂ x V ω + (n 1 + • • • + n r )V ω )B ω , ω = (n 1 , . . . , n r ) ∈ N • .
The conlusion follows by formal continuity.

Looking for a solution of the form Θ = V • B • for equation (15), we are thus led to the mould equation

x 2 ∂ x V • + ∇V • = J • a × V • . ( 16
)
Lemma 6.2. Equation (16) has a unique solution V • such that V ∅ = 1 and

V ω ∈ xC[[x]] for every non-empty ω ∈ N • . Moreover, V n1,...,nr ∈ x ⌈r/2⌉ C[[x]], (17)
where ⌈s⌉ denotes, for any s ∈ R, the least integer not smaller than s.

Proof. Let us perform the change of variable z = -1/x and set ∂ = d dz and a n (z) = a n (-1/z). Observe that

a n ∈ z -1 C[[z -1 ]], a 0 ∈ z -2 C[[z -1 ]], (18)
as a consequence of (1). The equation for V ω (z) = V ω (-1/z), with ω = (n 1 , . . . , n r ), r ≥ 1, is

(∂ + n 1 + • • • + n r ) V n1,...,nr = a n1 V n2,...,nr . (19)
On the one hand, ∂ + µ is an invertible operator of C[[z -1 ]] for any µ ∈ C * and the inverse operator

(∂ + µ) -1 = r≥0 µ -r-1 (-∂) r (20) leaves z -1 C[[z -1 ]] invariant; on the other hand, when µ = 0, ∂ induces an isomor- phism z -2 C[[z -1 ]] → z -1 C[[z -1 ]]. For r = 1, equation (19) has a unique solution in z -1 C[[z -1 ]], because the right- hand side is a n1 , element of z -1 C[[z -1 ]], and even of z -2 C[[z -1 ]] when n 1 = 0.
By induction, for r ≥ 2, we get a right-hand side in z -2 C[[z -1 ]] and a unique solution V ω in z -1 C[[z -1 ]] for ω = (n 1 , . . . , n r ) ∈ N r . Moreover, with the notation 'ω = (n 2 , . . . , n r ), we have

v( V ω ) ≥ α ω + v( V 'ω ), with α ω = 0 if n 1 + • • • + n r = 0 and n 1 = 0, 1 if n 1 + • • • + n r = 0 or n 1 = 0, where v denotes the standard valuation of C[[z -1 ]]. Thus v( V ω ) ≥ card R ω , with R ω = { i ∈ [1, r] | n i + • • • + n r = 0 or n i = 0 } for r ≥ 1.
Let us check that card R ω ≥ ⌈r/2⌉. This stems from the fact that if

i ∈ R ω , i ≥ 2, then i -1 ∈ R ω (indeed, in that case n i-1 + • • • + n r = n i-1
), and that R ω has at least one element, namely r. The inequality is thus true for r = 1 or 2; by induction, if r ≥ 3, then R ω ∩ [3, r] = R "ω with "ω = (n 3 , . . . , n r ) and either

2 ∈ R ω , or 2 ∈ R ω and 1 ∈ R ω , thus card R ω ≥ 1 + card R "ω .
The formal summability of the family (V ω B ω ) follows from (17) if we use the modified valuation of footnote 2. Indeed, one gets val (V n1,...,nr B n1,...,nr ) ≥ n 1 + • • • + n r + 2r. The n i 's may be negative but they are always ≥ -1, thus

n 1 + • • • + n r + r ≥ 0. Therefore, for any δ > 0, the condition n 1 + • • • + n r + 2r ≤ δ implies r ≤ δ and (n i + 1) = n 1 + • • •+ n r + r ≤ δ.
Since this condition is fulfilled only a finite number of times, the summability follows.

Setting Θ = V • B • , we thus get a solution of the conjugacy equation ( 8) and Θ is a continuous operator of A .

But is Θ an algebra automorphism? If one takes for granted the existence of a unique θ of the form (3) which solves (4) (and this is not hard to check), then one can easily identify the operator Θ that we just defined with the substitution operator corresponding to θ and Θ is thus an algebra automorphism. But it is possible to prove directly this fact by checking a certain symmetry property of the mould V • , called symmetrality.

Cosymmetrality and symmetrality

Let us return for a while to the general context of Section 5, with an alphabet N and a commutative C-algebra A, focusing on the case where B • is the multiplicative comould generated by a family of A-linear derivations (B n ) n∈N of a commutative algebra A .

We thus assume that A is a commutative A-algebra, on which a complete ring pseudovaluation ν is given, that F = F A,ν and that, for each n ∈ N, we are given B n ∈ F satisfying the Leibniz rule

B n (f g) = (B n f )g + f (B n g), f, g ∈ A .
This property can be rewritten

σ(B n ) = B n ⊗ Id + Id ⊗B n ,
with the notation σ : End A A → Bil A (A × A , A ) for the composition with the multiplication of A and Θ 1 ⊗ Θ 2 (f, g) := (Θ 1 f )(Θ 2 g) for any Θ 1 , Θ 2 ∈ End A A . We now consider the comould defined by

B ∅ = Id, B ω = B nr • • • B n1 for non-empty ω = (n 1 , . . . , n r ) ∈ N • .
One can easily check, by iteration of the Leibniz rule, that

σ B (n1,n2) = B (n1,n2) ⊗ B ∅ + B (n1) ⊗ B (n2) + B (n2) ⊗ B (n1) + B ∅ ⊗ B (n1,n2) , σ B (n1,n2,n3) =B (n1,n2,n3) ⊗ B ∅ + B (n1,n2) ⊗ B (n3) + B (n1,n3) ⊗ B (n2) + B (n2,n3) ⊗ B (n1) + B (n3) ⊗ B (n1,n2) + B (n2) ⊗ B (n1,n3) + B (n1) ⊗ B (n2,n3) + B ∅ ⊗ B (n1,n2,n3) ,
the general formula being

σ(B ω ) = ω 1 ,ω 2 ∈N • sh ω 1 , ω 2 ω B ω 1 ⊗ B ω 2 , ( 21
)
where sh ω 1 , ω 2 ω denotes the number of obtaining ω by shuffling of ω 1 and ω 2 : it is the number of permutations σ such that one can write ω 1 = (ω 1 , . . . , ω ℓ ), ω 2 = (ω ℓ+1 , . . . , ω r ) and ω = (ω σ(1) , . . . , ω σ(r) ) with the property σ(1) < • • • < σ(ℓ) and σ(ℓ + 1) < • • • < σ(r) (thus it is non-zero only if ω can be obtained by interdigitating the letters of ω 1 and those of ω 2 while preserving their internal order in ω 1 or ω 2 ).

Any comould satisfying ( 21) is said to be cosymmetral. Dually, one says that a mould M • is symmetral if and only if M ∅ = 1 and, for any two non-empty words ω 1 , ω 2 ,

ω∈N • sh ω 1 , ω 2 ω M ω = M ω 1 M ω 2 . ( 22
)
One says that a mould M • is alternal if and only if M ∅ = 0 and, for any two non-empty words ω 1 , ω 2 , the above sum vanishes. Suppose that B • is cosymmetral and that the family

(M ω B ω ) is formally summable, with Θ = M • B • . It is easy to check that M • symmetral ⇒ σ(Θ) = Θ ⊗ Θ, M • alternal ⇒ σ(Θ) = Θ ⊗ Id + Id ⊗Θ.
In other words, contraction with a symmetral mould yields an automorphism, while contraction with an alternal mould yields a derivation.

Symmetral and alternal moulds satisfy many stability properties (see [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] §5). Here we just mention that the multiplicative inverse M • of a symmetral mould M • is the symmetral mould defined by M n1,...,nr = (-1) r M nr,...,n1 .

(

Since the multiplicative comould generated by a family of derivations is cosymmetral, in the case of the saddle-node it is sufficient to check the following lemma to prove that the Θ defined in Section 6 is indeed an automorphism: Lemma 7.1. The mould V • defined by Lemma 6.2 is symmetral.

Proof. We must show that

V α V β = γ∈N • sh α, β γ V γ , α, β ∈ N • . (24)
Since V ∅ = 1, this is obviously true for α or β = ∅. We now argue by induction on r = r(α) + r(β). We thus suppose that r ≥ 1 and, without loss of generality, both of α and β non-empty. With the notations

d = x 2 d dx , α = α 1 + • • • + α r(α) and β = β 1 + • • • + β r(β) , we compute A := (d + α + β ) γ sh α, β γ V γ = γ =∅ sh α, β γ (d + γ )V γ = γ =∅ sh α, β γ a γ1 V 'γ ,
using the notations γ = γ 1 + • • • + γ s and 'γ = (γ 2 , . . . , γ s ) for any non-empty γ = (γ 1 , . . . , γ s ) (with the help (16) for the last identity). Splitting the last summation according to the value of γ 1 , which must be α 1 or β 1 , we get

A = δ sh 'α, β δ a α1 V δ + δ sh α, 'β δ a β1 V δ = a α1 V 'α • V β + V α • a β1 V 'β
(using the induction hypothesis), hence, using again (16),

A = (d + α )V α • V β + V α • (d + β )V β = (d + α + β )(V α V β ).
We conclude that both sides of (24) must coincide, because d + α + β is invertible if α + β = 0 and both of them belong to xC[[x]], thus even if α + β = 0 the desired conclusion holds.

Resurgence of the formal conjugacy

At this stage, we have found formal series V ω ∈ C[[x]] which determine a formally continuous algebra automorphism Θ = V • B • conjugating X 0 and X. Since Θx = x, we deduce that Θ is the substitution operator associated with the formal transformation θ(x, y) = x, ϕ(x, y) where ϕ = Θy. This is what was announced in [START_REF] Écalle | The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms[END_REF].

The components ϕ n (x) of ϕ(x, y) are easily computed: on checks by induction that

B ω y = β ω y n1+•••+nr+1 , ω = (n 1 , . . . , n r ), r ≥ 1, (25)
with Similarly, if we define V -• as the multiplicative inverse of V • with the help of formula (23), then Θ -1 = V -• B • is the substitution operator of a formal transformation (x, y) → x, ψ(x, y) , which is nothing but θ -1 , and

β ω = 1 if r = 1, β ω = (n 1 + 1)(n 1 + n 2 + 1) • • • (n 1 + • • • + n r-1 + 1) if r ≥ 2; one has β ω = 0 whenever n 1 + • • • + n r ≤ -2 (since (
ψ(x, y) = Θ -1 y = y + n≥0 ψ n (x)y n , (27)
where each coefficient can be represented as a formally convergent series

ψ n = n1+•••+nr+1=n β ω V -ω .
These are remarkably explicit formulas, quite different from what one would have obtained by solving directly the differential equation ( 5) for ϕ 0 (z) = ϕ 0 (-1/z) for instance.

An advantage of these formulas is that they allow to prove the resurgent character with respect to the variable z = -1/x of all the formal series which appear in our problem. We recall that a formal series

ϕ(z) = n≥0 c n z -n-1 ∈ z -1 C[[z -1 ]] is said to be resurgent if its formal Borel transform ϕ(ζ) = n≥0 c n ζ n n! ∈ C[[ζ]]
has positive radius of convergence and defines a holomorphic function of ζ which admits an analytic continuation along all the paths starting in its disc of convergence and lying in C \ Z (see [START_REF] Écalle | Les fonctions résurgentes[END_REF], or [START_REF] Sauzin | Resurgent functions and splitting problems[END_REF], or [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] §8). This property is stable by multiplication, the Borel transform of the Cauchy product ϕ • ψ being the convolution product ϕ * ψ defined by

( ϕ * ψ)(ζ) = ζ 0 ϕ(ζ 1 ) ψ(ζ -ζ 1 ) for |ζ| small enough (28)
(with ϕ and ψ denoting the Borel transforms of ϕ and ψ respectively).

A simple example is provided by the formal series V ω (z) = V ω (-1/z): indeed, the Borel transforms of the convergent series a n (z) = a n (-1/z) are entire functions a n (ζ) and the Borel counterpart of ∂ = d dz is multiplication by -ζ, hence (19) yields

V n1 (ζ) = - 1 ζ -n 1 a n1 (ζ) V n1,n2 (ζ) = 1 ζ -(n 1 + n 2 ) a n1 * V n2 . . . V n1,...,nr = (-1) r 1 ζ - ∧ n 1 a n1 * 1 ζ - ∧ n 2 a n2 * • • • 1 ζ - ∧ n r a nr • • • with ∧ n i = n i + • • • + n r .
Since the a n 's are entire functions of ζ, the function V n1,...,nr is holomorphic with ∧ n 1 , . . . , ∧ n r as only possible singularities. Moreover, the above formula is sufficiently explicit to make it possible to give majorant series arguments so as to prove the uniform convergence of the series of holomorphic functions [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] is devoted to this task); in other words, the formal series ϕ n and ψ n are resurgent with respect to z = -1/x.

ϕ n = n1+•••+nr +1=n β ω V ω , ψ n = n1+•••+nr +1=n β ω V - ω ( §8 of

Conclusion

9.1. The reader is referred to [START_REF] Écalle | Cinq applications des fonctions résurgentes[END_REF] or [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] §9-11 for the resurgent approach to the question of the analytic classification of saddle-node singularities, which one can develop once the analytic structure of the functions ϕ n (ζ) is clear. This approach relies on the use of Écalle's alien calculus: the singularities in the ζ-plane are controlled through operators ∆ m , m ∈ Z * , called alien derivations; mould calculus allows one to summarize the singular structure of the functions V ω in the simple equation

∆ m V • = V • × V • (m),
where V • (m) is an alternal scalar-valued mould (the V ω 's are complex numbers), with n 1 + • • • + n r = m ⇒ V n1,...,nr (m) = 0. This leads to Écalle's "Bridge Equation" for Θ, which gives the alien derivatives of all the resurgent functions ϕ n and in which the resurgent solution of the analytic classification problem is subsumed. 9.2. Another application of mould calculus in Resurgence theory is the use of the formal series V ω to construct "resurgence monomials" U ω which behave as simply as possible under alien derivation. This allows one to prove that alien derivations generate an infinite-dimensional free Lie algebra. 9.3. Other problems of formal normalisation can be handled by an approach similar to the one explained in Section 4. See [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF] §13 for the simple case of the linearisation of a vector field with non-resonant spectrum; much more complicated situations (taking into account resonances) are considered in [START_REF] Écalle | The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms[END_REF] or [START_REF] Écalle | Correction and linearization of resonant vector fields and diffeomorphisms[END_REF]. The strategy always begins by expanding the object to study in a sum of homogeneous components B n and considering the corresponding multiplicative comould B • .

It must be mentioned that, when this is applied to the analysis of a local diffeomorphism, rather than a local vector field, the operators B n are no longer derivations, but they still satisfy a kind of modified Leibniz rule:

σ(B n ) = B n ⊗ Id + n ′ +n ′′ =n B n ′ ⊗ B n ′′ + Id ⊗B n .
The resulting comould is not cosymmetral but cosymmetrel, a property which involves "contracting shuffling coefficients" instead of the shuffling coefficients sh ω 1 , ω 2 ω . Correspondingly, the relevant symmetry properties for the moulds to be contracted into B • are "symmetrelity" and "alternelity".

The theory can thus be extended so as to treat on an equal footing vector fields and diffeomorphisms.

  25) holds a priori in the fraction field C((y)) but B ω y belongs to C[[y]]), hence ϕ(x, y) = Θy = y + n≥0 ϕ n (x)y n , ϕ n = r≥1, ω∈N r n1+•••+nr +1=n β ω V ω (26) (in the series giving ϕ n , there are only finitely many terms for each r, (17) thus yields its formal convergence in xC[[x]]).