Lucien Wald

Hervé Guillard

Hervé

The dynamics of the Mediterranean Sea

come


A low-cost high-quality system for the acquisition and digital processing of images of WEFAX type provided by meteorological geostationary satellites


Florence Goffette-Nagot
email: goffette-nagot@gate.cnrs.fr

Isabelle Reginster

Isabelle Thomas
email: isabelle.thomas@uclouvain.be

A spatial analysis of residential land prices in Belgium: accessibility, linguistic border and environmental amenities

Keywords: land price, accessibility, border effect, environment, Belgium JEL codes: R14, R21, R31 land price, accessibility, border effect, environment, Belgium JEL codes: R14, R21, R31

This paper explores the spatial variation of land prices in Belgium. The originality of the methodology is threefold: (1) to work at the spatial extent of an entire country, (2) to compute several accessibility measures to all jobs and several representations of the environmental amenities and, more importantly, (3) to test the hypothesis that jobs influence land prices only in the same linguistic region. Spatial autocorrelation is accounted for by estimating spatial models. The results show that the linguistic border acts as a strong barrier in the spatial pattern of land prices and that environmental variables have no significant effect at this scale of spatial analysis.

Introduction

Land and housing values vary in space, whatever the spatial scale of analysis. Huge variations can be observed at the scale of a country (e.g., [START_REF] Davis | The price of residential land in large US cities[END_REF][START_REF] Holly | A spatio-temporal model of house prices in the US[END_REF].

Since [START_REF] Alonso | Location and Land Use[END_REF], a large amount of both theoretical and empirical literature has appeared about real estate values and their fall with distance from the central business district (CBD). It has been demonstrated that the land-price gradient is steeper than the housing-price gradient (Fujita, 1989). Of course, this class of models has been subject to many criticisms (notably [START_REF] Hamilton | Wasteful Commuting again[END_REF]; see [START_REF] Ma | Urban spatial change and excess commuting[END_REF] for a recent investigation) but it has also been adapted and tested with success, for instance by incorporating a polycentric form with a number of employment sub-centres, and by considering amenities other than accessibility to the CBD [START_REF] Anas | Urban spatial structure[END_REF][START_REF] Brueckner | Why is central Paris rich and downtown Detroit poor? An amenity-based Theory[END_REF].

Knowledge of the determinants of land prices has recently been improved by careful analyses of the role of "green" amenities in residential choice modelling, and especially in estimating hedonic prices (e.g., [START_REF] Geoghegan | Spatial landscape indices in a hedonic framework: an ecological economics analysis using GIS[END_REF][START_REF] Tyrvainen | Property prices and urban forest amenities[END_REF][START_REF] Irwin | The problem of identifying land use spillovers: measuring the effects of open space on residential property values[END_REF][START_REF] Kestens | The impact of surrounding land use and vegetation on single-family house prices[END_REF][START_REF] Roe | The effects of farmland, farmland preservation, and other neighbourhood amenities on housing values and residential growth[END_REF]. Theoretical models of urban economics incorporating amenities and open space have also been formulated (e.g., [START_REF] Brueckner | Why is central Paris rich and downtown Detroit poor? An amenity-based Theory[END_REF][START_REF] Wu | The influence of public open space on urban spatial structure[END_REF][START_REF] Turner | Landscape preferences and patterns of residential development[END_REF], including calibration of structural equations (e.g., [START_REF] Bates | The public demand for open space: the case of Connecticut communities[END_REF][START_REF] Cheshire | The welfare economics of land use planning[END_REF]. These models show that it is important to account both for distance to economic opportunities and amenities when analysing the urban equilibrium and residential land prices. As both of these types of determinants are correlated, one could possibly capture the effect of the other if the model omitted one aspect of the model. Therefore, empirical studies typically regress housing prices against physical distance from the city centre together with a number of control variables, paying attention to structural and (more recently) environmental attributes, as well as to spatial methodological pitfalls. These analyses are, however, usually restricted to one urban agglomeration (for recent examples see [START_REF] Mcmillen | The number of sub-centres in large urban areas[END_REF][START_REF] Zheng | Land and residential property markets in a booming economy: new evidence from Beijing[END_REF]. The analyses that consider a set of agglomerations are often quite general about the spatial pattern of land prices within each agglomeration (Davis and Palumbo, 2007;[START_REF] Holly | A spatio-temporal model of house prices in the US[END_REF].

The aim of the present paper is to test a general land-price model incorporating all these factors over an entire country. The estimated model covers different employment centres, the decrease of land prices with distance to employment, and environmental amenities. This model is applied to Belgium. Data availability in Belgium limits the statistical approach to data aggregated at the municipality (commune) level. This limits the type of amenities of which impact can be analysed. Amenities with very local effects such as schools or urban parks can not be taken into account at this scale. Furthermore, dealing with residential land prices only gives information as to the impact of amenities that are valued by the households in their residential location, but this does not allow to measure the recreational value of environmental amenities (see [START_REF] Vermeulen | Essays on housing supply, land use regulation and regional labour markets[END_REF].

The spatial variation of housing prices in Belgium has recently led to three publications. [START_REF] Vanneste | Le Logement en Belgique Bruxelles[END_REF] made an impressive exploratory spatial analysis about housing conditions in Belgium, which serves as a useful basis for the understanding of spatial differences in the characteristics of the housing stock; this work however does not provide any multivariate analysis of the spatial variation of land or housing prices. De [START_REF] Bruyne | Explaining the Spatial Variation in Housing Prices: An Economic Geography Approach[END_REF] conducted an econometric analysis of housing prices and not land rents, which is problematic due to the existence of a large amount of unobserved characteristics related to housing quality; furthermore, they account for accessibility to employment located in the two main employment centers for each municipality (Brussels and the province capital) and estimate separate models for Wallonia and Flanders, without considering the interactions between the two regions. Recently, [START_REF] Halleux | The spatial structuring of inter-urban housing markets: application to building sites prepared for self-provided housing[END_REF] studied the average plot price in most of the Belgian communes; he highlights the importance of information availability and demand substitutability for explaining housing markets and focuses on gaining insights into how markets are spatially organised [START_REF] Halleux | The spatial structuring of inter-urban housing markets: application to building sites prepared for self-provided housing[END_REF](Halleux, , p 2144)); this approach however does not account for the effect of accessibility to employment, which is the main determinant of housing and residential land prices in urban economic models.

The present paper improves the existing literature on residential real-estate prices in Belgium by analysing the determinants of residential land prices at a meso-scale level (communes). We focus on the effect of accessibility to employment, taking account of the spatial distribution of all employments. We also consider environmental characteristics and the effect of the linguistic border. The objective is: (1) to explore whether the urban influence is strong enough to explain land prices at the scale of a country; (2) to compare the role of environmental variables and the more traditional effect of transport costs to employment centres; and (3) to measure the effect of the linguistic frontier on land prices. The analysis was conducted at the scale of all 589 communes in Belgium. The methodology relies on three main assumptions:

(i) the cumulative effect of the entire urban network on residential land prices can be explored by using measures of accessibility that take into account all the jobs in Belgium and that consider different types of distance-decay functions; (ii) these employment-accessibility measures account for the existence of the linguistic border; and (iii) detailed measures of land use capture the environmental quality at the scale of the commune.

The paper is organised as follows: the modelling approach is developed in Section 2.

Section 3 describes the available empirical material (area studied, variables, statistical material). Section 4 gives the results of the empirical analysis and the conclusions are reported in Section 5.

Empirical approach

Our method consists of regressing the average residential land prices in each commune on measures of accessibility to employment and environmental characteristics while controlling for a few local variables.

2a. The accessibility of employment

Belgium is a densely-populated country in which commuting flows are large: 67% of the economically active population work in a commune other than that where they live [START_REF] Verhetsel | Pendel in België Deel I: de woonwerkverplaatsingen[END_REF]. The small size of the country and the high population density means that several employment centres are often reachable from a given place. Therefore, it is very likely that residential land prices will be influenced by accessibility to several employment centres.

Hence, it is useful to consider Belgium as a polycentric area.

Recent urban models have considered several employment centres. [START_REF] Anas | Urban spatial structure[END_REF] surveyed the theoretical and empirical literature on polycentric cities comprehensively.

Literature on employment sub-centres often deals with the identification of sub-centres within a metropolitan area: [START_REF] Mcmillen | Nonparametric employment subcenter identification[END_REF] provided both an identification of employment subcentres within the Chicago metropolitan area, and a test of the influence of the distance to these sub-centres on land values. However, rather than considering a finite number of employment centres (that are, in the existing literature, either a priori chosen or endogenously determined, e.g., [START_REF] Mcmillen | The number of sub-centres in large urban areas[END_REF], the approach followed in this paper consists of accounting for the location of all jobs in Belgium and developing indices of accessibility to employment. This seems to offer a comprehensive view of the influence of employment on land prices and avoids any discussion about concentration versus exurbanisation of jobs, or comments about the specialisation of some communes in some economic activities already discussed elsewhere (e.g., [START_REF] Riguelle | Is the Belgian city polycentric? A geographical approach[END_REF][START_REF] Vandermotten | European Planning and the Polycentric Consensus : Wishful Thinking ?[END_REF]. Here all jobs in all communes are considered. This choice imposes that the form of the distance-decay function and its parameters have to be defined a priori. In order to overcome this shortcoming, several accessibility indices were computed; they differ from each other by the shape of the distance-decay function (Section 3d).

There is unfortunately no consensus about the definition and formulation of the concept of accessibility in the literature (for a recent review see [START_REF] Geurs | Accessibility Measures: Review and Applications RIVM Report 408505 006[END_REF][START_REF] Geurs | Accessibility Measures: Review and Applications RIVM Report 408505 006[END_REF]DeVries et al., 2009). [START_REF] Handy | Measuring accessibility: an exploration of issues and alternatives[END_REF] suggest that accessibility should be determined by the spatial distribution of potential destinations, the ease of reaching each destination, and the magnitude, quality, and character of the activities found there. Every definition includes one or more components that affect accessibility (e.g., the location of activities, the system of transport). A simple formulation is adopted here, largely inspired by the critical analysis conducted by [START_REF] Vandenbulcke | Accessibility Indicators to Places and Transport, Politique Scientifique Fédérale et SPF Mobilité et Transports[END_REF]2009). Accessibility is measured as a weighted sum of all employment, weights being given by a distance-decay function that allows for the fact that more distant jobs are less accessible. This accessibility is considered as a measure of potential.

In a highly urbanised country, the straight-line distance between two places gives a reasonable idea of their mutual accessibility. The distances between the centroids of the communes as given by a Geographical Information System (GIS) were computed. Pearson product-moment correlations between time-distances (with and without congestion), real distances along the road network and straight-line distances (in kilometres) were computed for a subset of communes; they were always greater than 0.9 (results available from the authors on request; see [START_REF] Vandenbulcke | Accessibility Indicators to Places and Transport, Politique Scientifique Fédérale et SPF Mobilité et Transports[END_REF] for more details). The simplest system, the straight-line distance, is used here in order to pay more attention to other sensitive aspects in the modelling process.

Three forms of the distance-decay function were used: (i) an inverse of the quadratic distance, (a common form of accessibility measure; e.g., [START_REF] Geurs | Accessibility Measures: Review and Applications RIVM Report 408505 006[END_REF]; (ii) a negative exponential distance function (a classical functional form for the influence of distance to employment on residential real-estate prices in urban models since [START_REF] Kau | Urban land value functions and the price elasticity of demand for housing[END_REF]see Wang, 2009 for a recent example) computed for two different exponents, -0.05 and -0.10, chosen in conformity with existing studies of land prices that show that estimated land price gradients usually lies between 0.2 and 0.10 [START_REF] Colwell | The Structure of Urban Land Prices[END_REF][START_REF] Mcmillen | The return of centralization to Chicago: using repeat sales to identify changes in house price distance gradients[END_REF]; and (iii) an inverse of distance with a threshold, according to which employment is only considered within a given radius around each commune; the chosen radii are 40km and 50km, that correspond to acceptable commuting distances in a denselypopulated country like Belgium. These forms are discussed in more detail in Section 3.

The linguistic border is a reality of life in Belgium (see for example [START_REF] Dujardin | Effet de frontière et interaction spatiale Les migrations alternantes et la frontière linguistique en Belgique[END_REF] for a review as well as Section 3.a hereunder). This has been considered in previous studies conducted in different but related contexts: [START_REF] Klaassen | Measuring psychological distance between the Flemings and the Walloons[END_REF] measured the psychological distance due to the language difference using telephone traffic, and [START_REF] Gérard | Tax Interactions Among Belgian Municipalities: Does Language Matter?[END_REF] compared the intensity of tax interactions among Belgian municipalities within and between linguistic regions. To account for the fact that residential and employment choices are strongly constrained by linguistic considerations, the accessibility measure is considered separately by region (French-speaking Wallonia, Dutch-speaking Flanders, and bilingual Brussels). Access_intra is the accessibility to jobs located within the same region, while Access_inter is the accessibility to jobs located in another region. Accessibility from Wallonia and Flanders to jobs located within the Brussels region (Access_Bxl) is treated separately due to the uniqueness of Brussels (which is bilingual, centrally located and dominates the Belgian urban landscape, see Thisse and Thomas, 2007 and Section 3a). For communes in the Brussels region, the accessibility to jobs within the Brussels region is the Access_intra variable, while Access_Bxl takes the null value. Note that the accessibility values for each commune do not include the commune itself; employment inside the commune is introduced as a separate variable.

2b. Environmental amenities

Several researchers have already studied the impact of environmental characteristics, be they positive like view on mountains or negative such as noise, on residential land prices (e.g., [START_REF] Germino | Estimating visual properties of Rocky Mountain landscapes using GIS[END_REF][START_REF] Reginster | Urban environmental quality in two Belgian cities, evaluated on the basis of residential choices and GIS data[END_REF][START_REF] Dekkers | Monetary valuation of aircraft noise; a hedonic analysis around Amsterdam airport[END_REF]. Some have considered simply the distance to a given amenity, for instance an urban park, a forested area or open water (e.g., [START_REF] Tyrvainen | Property prices and urban forest amenities[END_REF][START_REF] Mahan | Valuing urban wetlands: a property price approach[END_REF]. More sophisticated methodologies take landscape, such as the view from a given property, into account (e.g., [START_REF] Paterson | Out of sight, out of mind? Using GIS to incorporate visibility in hedonic property value models[END_REF][START_REF] Kestens | The impact of surrounding land use and vegetation on single-family house prices[END_REF]. A lot of attention has also been devoted to the impact of land use and open space in the surroundings of properties (e.g., [START_REF] Irwin | The problem of identifying land use spillovers: measuring the effects of open space on residential property values[END_REF].

As in other recent studies, a GIS technology is used to improve many aspects of the hedonic analysis: various indicators of the greenery of an area were computed and introduced into the regression equations. These indicators combine different sources of data (census data and remotely sensed data), land cover (different types of vegetation), and spatial extent (administrative entities or buffers around a location). In particular, land use was measured for each commune both as the average of the commune grid cells and as a commune-level average over circular windows around each commune cell. The first measure is strictly concerned with land use within the commune, whereas the second reflects the environment as perceived at each location in the commune (see Section 3c). The objective here is to see how far land prices are influenced by ecological characteristics compared to the more traditional economic variables.

2c. Econometric model

Land-price equations are estimated on all Belgian communes using three types of explanatory variables: i

i i i i i u Z Y X p Log + + + = γ β α ) ( (1)
where p i is the average unit land price in commune i, X i is a vector of three accessibility variables, Y i is a vector of environmental characteristics, Z i is a vector of control variables and u i is an error term. Because p i is the average of the individual transactions in each commune, the error term is expected to be heteroscedastic and robust variance matrices are estimated using White's correction. There are several reasons why the residuals might be spatially correlated: in particular, unobserved factors that influence land prices are likely to be spatially influenced. Moreover, it is possible that observed land prices in one commune directly influence land prices in the neighbouring communes. For these reasons, the presence of spatial autocorrelation of the residuals was tested (Moran's I) and two spatial models were estimated: a spatial error model and a spatial lag model.

The spatial error model (SEM) explicitly takes a spatially autoregressive component into account in the error term:

ε γ β α + + + = Z Y X ) ( p Log with u W + = ε λ ε (2)
where Log(p) is the n × 1 vector of the logarithms of land prices, X, Y and Z are respectively n × 3, n × j, and n × k matrices of observations of the explanatory variables, W is a n × n spatial weights matrix, u is a n × 1 vector of i.i.d. error terms, ε is an n × 1 vector of spatially autoregressive error terms and λ is the autoregressive parameter to be estimated.

The spatial lag model (LAG) refers to a specification in which land price in a commune i is directly influenced by land prices in neighbouring communes. Formally, in matrix form:

u Z Y X ) ( ) ( + + + + = γ β α ρ p WLog p Log (3)
with the same notation as above, and where ρ is the spatial autoregressive coefficient to be estimated. The spatial-weights matrix is a simple contiguity matrix defined on the basis of communes having common boundaries. This contiguity matrix is then row standardised so that each row sums to 1. An ordinary least squares (OLS) estimation allows us to test for the presence of spatial autocorrelation based on a Lagrange multiplier test statistic for error autocorrelation and lag dependence [START_REF] Anselin | Spatial Econometrics: Methods and Models In Studies in Operational Regional Science Boston[END_REF]. The spatial error model and spatial lag model are estimated and compared below.

The area and the data 3a. The area

Belgium is a small and densely inhabited European country (30,278 sq km and 10 million inhabitants), characterised by a high regional physical diversity and a variety of landscapes.

The natural regions are mainly determined by relief, natural conditions and the nature of the soil: from flatland and low land in the North, through the sandy regions of central Flanders, the hilly regions and loamy plateau of Central Belgium, to a high plateau in the South.

Belgium is on average highly urbanised but contains considerable regional disparities. The population density in the north and the centre is particularly high and exceeds 500 inhabitants/km², while in the southern part, the population density is lower and the people are concentrated in villages and small towns. Even if more than half of the Belgian territory is classified as "urban" and three quarters of the population lives in an urban or suburban commune [START_REF] Luyten | De Belgische Stadsgewesten (the Belgian urban areas)[END_REF], only 20 % of the total surface is covered by buildings or roads. More than half of Belgium is used for agriculture (57%) (27% cropland, 18% grassland and a few permanent crops). Forests and wooded areas cover 20% of the territory; 80% of the forests are located in the southern part of the country.

Belgium is divided into three administrative regions (Flanders, Wallonia and the Brussels Capital Region) (Appendix 1). It is important to note that after federalisation in 1980, town and country planning and environmental planning came under the purview of the regions.

Since then the regions have had their own planning policy. The northern part of the country (Flanders) is Dutch speaking while the southern part (Wallonia) is French speaking. In the centre of the country, the Brussels region is officially bi-lingual with a large proportion of French-speaking inhabitants. Belgium is administratively divided into 589 communes (districts, Nuts5 in the Nomenclature of Territorial Units for Statistics used by Eurostat) that are the units of analysis here. One commune (Herstappe) is however so small that most computed indices have abnormal and missing values (esp. for land prices) and hence "pollute" the results. It is withdrawn from all analyses. Hence, the dataset counts 588 communes.

Because regional differences may matter, three binary variables (Flanders, Wallonia and Brussels) were introduced to indicate which region each commune belongs to. However, to avoid multicollinearity issues due to correlation between population density and the region dummies, only the Wallonia dummy has been retained in the final specifications. This choice is consistent with the economic proximity of Flanders and Brussels, which are closer to each other than to the Walloon region.

Belgium is dominated by two large urban agglomerations where a large proportion of the economic activities take place: Brussels and Antwerp. Brussels is the main employment centre and attracts people from all over the country [START_REF] Thisse | Bruxelles et Wallonie : une lecture de la vie économique urbaine[END_REF]. The urban agglomeration of Brussels has more than 1.5 million inhabitants; it is centrally located and sprawls out to across its administrative border into Wallonia and Flanders [START_REF] Dujardin | Quelles frontières pour Bruxelles? Une mise à jour Reflets et Perpectives de la Vie Economique[END_REF]. Antwerp has half a million inhabitants and is the foremost industrial area and port in the country. In terms of size, Brussels and Antwerp are followed by Ghent (250,000 inhabitants), Liège and Charleroi (approximately 200,000 inhabitants each). The urban network highly structures the economic poles in terms of job locations (see for example [START_REF] Riguelle | Is the Belgian city polycentric? A geographical approach[END_REF]. Whatever their size, major and regional Belgian cities are employment centres. City centres have hence a high concentration of jobs, but also contain deprived populations, while the better-off tend to live in the suburbs (see for example [START_REF] Dujardin | Residential segregation and unemployment: the case of Brussels[END_REF].

This brief description of the geography of Belgium highlights two important characteristics:

(1) Belgium has large regional disparities in landscape, land use and population density; and

(2) the city network plays an important role in the location of socio-economic activities and jobs. These characteristics are expected to appear in the following empirical analysis.

3b. Land price data

Average land prices values are available by commune on the web site of the Belgian National Institute for Statistics (http://statbel.fgov.be/figures/download_fr.asp#hbs). They refer to all properties sold during a year, and to the price declared to the Public Authority. There may be a difference between declared and real price, but any such bias is here assumed to be independent of location. This paper is limited to developable land (here denoted Land).

The advantage of this database is that it is available for the entire country, using the same methodology, and it appears every year. The statistics cover the total number of plots sold, the average price of one square metre and the average size of a plot sold (Area). A mean value was computed for a three year periods (1999)(2000)(2001) to avoid very small numbers and to reduce the effects of discrete extreme values; the choice of these three years make it possible to compare the results with the latest census data (2001). Land is hence the average price of a square metre of developable land sold in 1999, 2000 or 2001 in a commune, whatever the size of the plot. As expected, land price is high in communes encompassing a city centre (CBD), and decreases with distance from this commune. This is especially true for Brussels. There is also a strong North-South division: land is scarcer in densely populated areas (see Figure 1) (Thomas & Vanneste, 2007). Source: Statbel.

3c. Environmental variables

The Some of the data in the CORINE database were aggregated to produce synthetic indicators at the commune level. The following variables were constructed:

• the percentage of each commune which is covered in forest (Forest). This corresponds to the aggregation of three of the CORINE classes: broad-leaved forest, coniferous forest, and mixed forest. This percentage is computed as the percentage of the 250m square cells in each commune which were entirely covered with forest.

• the percentage of land devoted to agriculture (Agriculture). The North Sea coast (which is quite touristy and urbanised) was expected to generate higher land prices. Two coastal variables were computed: Coast indicates the presence (1) or absence (0) of the coast as a border of the commune. Coastprox takes the value 1 in communes not actually on the seafront but located less than 20 km from it, and 0 otherwise. The presence of water (a lake or river) in a commune is also expected to be an attractive amenity. Water takes the value 1 when there is such an amenity in the commune and 0 when it is absent. Finally, Slope is the average gradient of road segments in each commune (Data source [START_REF] Vandenbulcke | Accessibility Indicators to Places and Transport, Politique Scientifique Fédérale et SPF Mobilité et Transports[END_REF]. It is used as a proxy for landscape, on the assumption that hilly landscapes are more attractive to residents.

3d. The accessibility of employment

In line with the regional split in Belgium, the accessibility of employment for a commune i is considered to have three components:

(1) Access

_ intra i = f (d ij ) job j j ∈R i , j≠ i ∑
is the sum of all the jobs located in the same region as i, weighted by a distance-decay function f(d ij ), where ij , is the straight-line distance between the centroids of communes i and j and job j is the number of jobs in commune j.

(

) Access _ inter i = f (d ij ) job j j ∉R i ∑ 2
is the sum of all the jobs located in the other region (Flanders for Wallonia and Wallonia for Flanders and Wallonia and Flanders for Brussels), weighted by a distance-decay function f(d ij ).

(3)

∑ ∈ = B R j j ij job d f bxl Access ) ( _ i
is the sum of all jobs located in the Brussels region (denoted R B ), weighted by a distance-decay function f(d ij ). Note that accessibility of jobs in Brussels for the Brussels communes is measured in variable Access_intra.

Due to the linguistic border, it is expected that the accessibility of employment located in the other linguistic region will have less influence on land prices than employment located in the same region: it is as if the employment in the other linguistic region was further away, or as if the frontier between the two regions had some thickness. This applies to Flanders and Wallonia seen as two "linguistic regions" but not for Brussels that is bi-lingual and that attracts a lot of commuters whatever their place of residence; this is why the Acces_bxl variable is also considered.

This approach forces us to choose the functional form for the distance-decay function and its parameter. Three types of distance-decay functions were computed: inverse of negative powers of distance with parameters -1 and -2, negative exponential functions with parameters -0.10 and -0.05 and inverse distance functions with two different thresholds, as follows:

2

) ( - = ij ij d d f (4.1) ) 05 . 0 exp( ) ( ij ij d d f - = (4.2) ) 10 . 0 exp( ) ( ij ij d d f - = (4.3) 1 ) ( - = ij ij d d f if and only if d ij ≤ 40km (4.4) 1 ) ( - = ij d d f if and only if d ij ≤ 50km (4.5)
However, preliminary multicollinearity tests revealed high variance inflation factors in the model using the inverse quadratic function of distance 4.1. As a result, only the functions 4.2 to 4.5 were retained and are used in the following. They will be introduced into the price equations in turn, and the fits of the regression equations will be compared.

Figure 4 illustrates the effect of these choices on the accessibility measures: the accessibility of jobs located in the same linguistic region (Access_intra), in the other linguistic region (Access_inter), and in the Brussels region (Access_Bxl), when the distance-decay function is either the negative exponential of -0.10 (Equation 4.3), or the inverse of the distance with a 40 km threshold (Equation 4.4). Access_intra is lower in Wallonia than in the two other regions. Access_inter is high of course near the linguistic border but also in Brussels, as the communes in the Brussels region have access both to jobs in Wallonia and in Flanders.

3e. Socio-economic variables

Jobs is the number of jobs in each commune. This variable is obtained from the housing and population census (2001), where each inhabitant had to declare his or her place of work and place of residence. It is used in computing the accessibility variables, and to measure the effect of jobs in the commune itself. Popdens states the population density and hence accounts for the urban effect. It is simply expressed as the total number of inhabitants in a commune divided by its total area, and measures the pressure on land: high densities reinforce competition for land, and is synonymous of high land prices and scarcity of available land plots .

Income is a measure of regional wealth, based on the median declared income in each commune. The data are officially available each year from http://www.statbel.fgov.be/ home_fr.asp. This variable is aimed at accounting for the impact of local wealth on land prices: available income increases the willingness to pay for desirable characteristics in the commune. However, it is also known that high-income households are likely to sort themselves into communes with high land prices (this bi-variate relationship has already been discussed for Belgium by [START_REF] Thomas | Le prix de l'immobilier en Belgique : Un peu de géographie ! Les Echos du Logement[END_REF]. There is therefore a potential endogeneity problem. This is avoided here by taking the average value of households' income over the labour-market area (labour-market areas are delineated for the 22 largest cities in the country).

Tables 1 and2 summarise the main descriptive statistics and the significant spatial autocorrelations for all the variables used in this study. Diagnostics of multicollinearity were performed and all the results presented here are for estimations in which the highest variance inflation factor is below 10 (results available on request).

Insert Tables 1 and2 about here

Results

4a. Comparing spatial models

Estimations were performed with the OLS method (OLS), the spatial error model (SEM) and the spatial lag model (LAG). Table 3 gives the OLS, SEM and LAG estimated coefficients and their standard errors, in each case for two different specifications of the distance-decay function. The Lagrange multiplier test for the two spatial models revealed the presence of spatial autocorrelation, which has to be accounted for to avoid biased or inefficient estimates.

The choice of a spatial model is based on a comparison of likelihoods [START_REF] Anselin | Spatial Econometrics: Methods and Models In Studies in Operational Regional Science Boston[END_REF].

Table 3 shows that, whatever the specification of the accessibility function, the LAG model had the highest likelihood, meaning that the data exhibits a spatial pattern in which land prices are influenced by land prices of neighbouring communes instead of by unobserved characteristics in those communes.

Table 3 shows that the estimates of the SEM model are similar to the OLS estimates. Moran's I for the residuals of these regressions (bottom lines of Table 3), show that the SEM model has a similar spatial structure of residuals to the OLS. On the contrary, the residuals of the LAG model have a very different structure, with a low negative value of the Moran's I, especially in the exponential specification. This results of course from the underlying spatial structure of the LAG model, in which the influence of land prices in neighbouring communes is specifically taken into account, and therefore taken out of the residuals. lower by a factor of two or more in the LAG estimates. This means that ignoring the spillovers of land prices (as occurs in the OLS and SEM models) in neighbouring communes can falsely increase the effects of exogenous variables. Indeed, we know that OLS coefficients are biased when when some spatial autocorrelation of the explained variable is present in the data. The same applies to the accessibility to Brussels (Access_Bxl) and to the intra-regional accessibility (Access_intra). The level of employment in the commune (Jobs) is no longer significant in the LAG model.

Insert Table 3 about here

Two variables (Forest, Agriculture) have an unexpected negative sign in the OLS estimation with inverse distance with a 40 km threshold; in the spatial models (SEM and LAG) they are not significant. This once again means that ignoring the low level of land pressure in neighbouring communes in the OLS regression gives misleading results. These results are in line with those obtained by Möller (2007), in testing a model for residential prices in Germany.

Given the results of this comparison of spatial models, the following discussion will be limited to the spatial LAG model.

4b. Comparing accessibility indices: the distance-decay effect

Regressions were performed to test the stability of the models with respect to the form of the accessibility measure (i.e., the distance-decay effect) and to enhance our understanding of its influence. Table 4 gives the results for the LAG regressions for the four distance-decay functions: two negative exponentials (exponents -0.05 and -0.10) and two inverse distance functions (thresholds 40 and 50 km).

Insert Table 4 about here

The coefficients for intra-regional accessibility (Access_intra) associated with the inverse distance functions are not significant. However, accessibility to Brussels (Access_Bxl) always has a positive significant effect in the regressions, and this effect is larger in the inverse distance specifications than in the exponential specifications (Columns 3 and 4 compared to Columns 1 and 2). The interpretation is twofold. First, the effect of Brussels is so important that it is not sensitive to the way in which the distance-decay is expressed. Second, Access_bxl with the inverse distance-decay functions exhibits a strong spatial differentiation, due to the shrink of the variable beyond the threshold around Brussels (see Figure 4f). This corresponds well to the pattern of land prices around Brussels. The spatial differentiation of Acces_intra with this distance-decay function is much lower (Figure 4d). Consequently, this outweighs the effect of intraregional accessibility. Moreover, the residuals exhibit substantial remaining autocorrelation after the estimation of the LAG model with the inverse distance specification of accessibility, but this is lower in the exponential specification (bottom lines of Table 4). A possible interpretation is that the inverse distance function with a threshold understates job accessibility beyond the threshold. This causes higher residuals beyond the threshold, and these residuals are obviously spatially correlated.

This gives an indication of the form of the accessibility index that best suits the spatial differentiation of land prices in Belgium. The negative exponential function best captures the influence of the spatial distribution of employment on land prices at the scale of the country.

Unsurprisingly, this is the classical functional form used in urban monocentric models.

4c. Environment versus accessibility

As expected, land prices were positively influenced by the presence of the coast (Coast) and by being within 20 km of the coast (Coastprox) (see Table 4). The coefficient of Slope was negative, indicating that hilly areas tend to have low land prices. This may be explained by the fact that steep slopes (such as those in the Ardennes hills) are associated with very low demand for land. Another interpretation is that the straight-line distance considered in our accessibility measures is likely to yield an overestimation of the real level of accessibility in the eastern part of the country. This effect would be captured by the Slope variable, resulting in the observed negative coefficient.

Unexpectedly, the three other environmental variables (Water, Forest and Agriculture) did not influence land prices significantly and their environmental variants (Forest_environt and Agricult_environt) were not significant in any of the models (not shown). At this scale of analysis, the effect of landscape on land prices is not evident; it is probably more significant at an intra-communal level (see for example [START_REF] Kestens | The impact of surrounding land use and vegetation on single-family house prices[END_REF].

Variables that measure the proximity of a commune to economic opportunities had clear effects. Accessibility to jobs in the same region (Access_intra) influenced land prices positively, although jobs in the other region had no significant influence (Access_inter): thus proximity to jobs in Flanders did not affect land prices in Wallonia, and vice versa. This result indicates that the two regions function independently in terms of labour market areas. Regions are isolated from each other due to their linguistic differences. The effect of accessibility to Brussels is a different story: as the principal city, it attracts commuters from far away in the country [START_REF] Verhetsel | Pendel in België Deel I: de woonwerkverplaatsingen[END_REF].

Accessibility to jobs seems to be more important to households than jobs in the commune itself, which did not impact land values in the LAG model. This is not surprising given the intensity of commuting at this level. Similarly, population density in the commune had only a slightly significant effect. More importantly, the average household income in the labour market area (Income) had a very significant effect whatever the estimated model. Wealth supplements proximity to job opportunities in explaining spatial differentials in land prices.

Finally, the regional variable for Wallonia has a negative coefficient. This dummy variable is likely to capture unmeasured variables, such as specific planning or housing policies; the differences between regions are driven by political choices but also by the specific history and geography of the regions. In Wallonia, for instance, there is a longer tradition in renovation support even if recently the difference with Flanders is shrinking [START_REF] Vanneste | Le Logement en Belgique Bruxelles[END_REF].

To assess the relative importance of the various factors affecting land prices, it is possible to calculate the effects of the continuous variables as the changes in the land price in standard deviation that are produced by a one standard deviation change in each of the variables.

Table 5 shows these effects for the variables that are significant in the LAG model estimated with the exponential form of the accessibility measure (exponent -0.10). Although the coefficient of population density was not very significant, a change of one standard deviation in the population density of the commune changed land prices by almost one standard deviation. The effect of accessibility to jobs in the region and of being on the coast were slightly weaker but of the same order of magnitude. The impacts of accessibility to jobs in Brussels, of average income in the commune, and of average hilliness, were about half a standard deviation.

Insert Table 5 about here

In summary, at the level of the 588 communes in Belgium, the traditional determinants of land prices (that is accessibility to jobs and population density), had more influence on the spatial distribution of residential land prices than the environmental variables. The four landuse variables considered in these estimations had no effect on land values.

Conclusion

The objective of this paper was to identify the importance of environmental variables for explaining urban land prices within one country, and more particularly to compare their role to those of more classical variables. The accessibility of jobs was studied in detail. Spatial models were estimated to account for the spatial autocorrelation that is common in hedonic analyses of land prices. A major result is that, at the level of the whole country, variations in land prices mainly result from the role of distance (a factor that depends on the unit transport cost and the trade off between transport and land costs); natural amenities do not have a stable or well-defined role. Environmental and natural variables are probably more important at a local level. We also observe that the border between Wallonia and Flanders acts as a barrier between the labor-markets of the two regions, as revealed by the effect of accessibility of employment on land prices. This result confirms those of Dujardin ( 2001) on commuting flows.

The effect of the linguistic border is distinct from the observation that land rents differ, ceteris paribus, depending on the region, as measured by the effect of the binary variable that contrasts Wallonia to Flanders. Indeed, our results suggest that the structure of land prices depends on the geography of the country, taken as a whole, but also upon the land-use policy:

land-use planning policy in Belgium is different in the North and the South, Wallonia having more space than Flanders, lower population density, a weaker economy, different historical traditions, and a long tradition of housing renovation (by taste or supported by public policy).

Hence, geography, history, economy, linguistic border and land-use policy interact to explain the spatial structure of land prices.

Figure 1 :

 1 Figure 1: Average price (in €1,000) of one square metre of land in 1999-2001 (Land).

  CORINE (Coordination of Information on the Environment, European Commission, 2003) land-cover database provides a detailed inventory of the biophysical land cover in Europe using 44 classes. It is made available by the European Environment Agency (EEA) (http://www.eea.eu.int/products) at a resolution of 250 m grid cells (minimum mapping unit = 25ha), and is based on interpretations of remotely sensed photographs taken in the year 2000.

Figure 2 :

 2 Figure 2: Percentage of 250m square cells in each commune that are covered with forest (noted Forest).

Figure 3 :

 3 Figure 3: Average percentage of forested cells, inside a circular window of radius 20km around each 250m grid cell in the commune (noted Forest_environt).

aFigure 4 :

 4 Figure 4: The accessibility spatial structures for different distance-decay functions and border

Figure 5 Figure 5 :

 55 Figure 5 compares the OLS and LAG residuals. Whereas the spatial pattern of the OLS

  This variable is based on the aggregation of the arable land, permanent crop and pasture classes in the CORINE database(class 2). It is computed as the percentage of the 250m square cells in each commune devoted to agriculture in 2000. Some additional measures of land uses were calculated, taking into account the surroundings of the commune or using different methods of aggregation. The forest environment (Forest_environt) in each commune was calculated. This covers the same classes of forest as Forest, but the aggregation method is different: it is the mean percentage of forest cells, inside

a circular window of radius 20km around each 250m grid cell in the commune. It represents the situation in 1990. Figures

2 and 3

compare the spatial distribution of the Forest and Forest_environt variable. An Agriculture_environt variable was defined for agricultural land use in a similar way.

Table 2 :

 2 Global autocorrelation analysis for the main dependent and independent variables

	Variable	Moran's I	z stat.
	Land price	0.178***	8.990
	Jobs	0.010 n.s.	0.523
	Popdens	0.474***	19.863
	Agriculture	0.099***	3.989
	Forest	0.107***	4.312
	n.s. not significant; * p<0.1; ** p<0.05; *** p<0.01

Table 3 :

 3 OLS, SEM and LAG results for two specifications of the distance-decay function

	Distance-decay function	exp(-0.05 dij) [Equation 4.1]	exp(-0.10 dij) [Equation 4.2]	1/dis. if dij < 40km [Equation 4.3]	1/dis. if dij < 50km [Equation 4.4]
	Access_intra	0.0011**	0.0025**	0.0020	0.0008
		(0.0005)	(0.0011)	(0.0034)	(0.0033)
	Access_inter	-0.0001	-0.0019	-0.0034	-0.0035
		(0.0008)	(0.0024)	(0.0066)	(0.0054)
	Access_Bxl	0.0009**	0.0020***	0.0061***	0.0060***
		(0.0004)	(0.0006)	(0.0020)	(0.0021)
	Jobs	0.0019	0.0022	0.0018	0.0016
		(0.0015)	(0.0017)	(0.0016)	(0.0014)
	Popdens	0.0997**	0.0817*	0.0979*	0.1121*
		(0.0423)		(0.059)	(0.0588)
	Income	0.0455*	0.0631***	0.0659***	0.0688***
		(0.0244)	(0.0236)	(0.0246)	(0.0252)
	Wallonia	-0.4705***	-0.4724***	-0.4861***	-0.4830***
		(0.0760)	(0.0745)	(0.0737)	(0.0744)
	Coast	0.7673***	0.7525***	0.6717***	0.6556***
		(0.1540)	(0.1506)	(0.1523)	(0.1525)
	Coastprox	0.3193***	0.2788**	0.2385**	0.2321**
		(0.1187)	(0.1136)	(0.1114)	(0.1112)
	Slope	-0.0350**	-0.0422***	-0.0409***	-0.0387***
		(0.0139)	(0.0139)	(0.0138)	(0.0137)
	Water	-0.0030	-0.0030	-0.0011	-0.0036
		(0.0355)	(0.0354)	(0.0354)	(0.0356)
	Forests (ln)	-0.0154	-0.0136	-0.0207	-0.0212
		(0.0179)	(0.0177)	(0.0180)	(0.0180)
	Agriculture (ln)	-0.0539	-0.0282	-0.0594	-0.0579
		(0.0621)	(0.0629)	(0.0685)	(0.0687)
	Constant	-2.6977***	-2.9636***	-2.8611***	-2.8767***
		(0.5059)	(0.5113)	(0.5119)	(0.5144)
	σ	0.3999***	0.4003***	0.4006***	0.4003***
		(0.0163)	(0.0164)	(0.0166)	(0.0165)
	ρ	0.4834***	0.4751***	0.5091***	0.5194***
		(0.0512)	(0.0518)	(0.0491)	(0.0488)
	Lagrange multiplier test (ρ=0)	133.811***	118.636***	135.319***	146.093***
	N	588	588	588	588
	Log Likelihood	-309.77673	-309.9085	-312.63358	-312.93498
	Moran's I of residuals	-0.038*	-0.037*	-0.050**	-0.055**
	(z stat.)	-1.461	-1.391	-1.918	-2.112
	* p<0.1, ** p<0.05, *** p<0.01
	Robust standard errors between parentheses.

Table 4 :

 4 Results of the LAG model for four specifications of the distance-decay function Effects of dummy variables are calculated as variations in standard deviation units for X = 1 with respect to X = 0.

	Explanatory variables	Effect on land prices in
		s.d.
	Access_intra	0.886
	Access_Bxl	0.471
	Popdens	0.957
	Income	0.470
	Slope	-0.546
	Wallonia *	-0.161
	Coast*	0.479
	Coastprox*	0.137

*

Table 5 :

 5 Variations in land price in standard deviations for a one standard deviation change in the explained variable -LAG model with

	f	(	d	ij	)	=	exp(	-	0	.	10	d	ij	)

The authors would like to thank the three referees for comments that improved the paper. The funding of this research by a joint grant CNRS / FNRS-CGRI is gratefully aknowledged.


R Ravaud

G Lemarquand
email: guy.lemarquand@ieee.org

V Lemarquand

S Babic

C Akyel

MUTUAL INDUCTANCE AND FORCE EXERTED BETWEEN THICK COILS

We present exact three-dimensional semi-analytical expressions of the force exerted between two coaxial thick coils with rectangular cross-sections. Then, we present a semi-analytical formulation of their mutual inductance. For this purpose, we have to calculate six and seven integrations for calculating the force and the mutual inductance respectively. After mathematical manipulations, we can obtain semi-analytical formulations based on only two integrations. It is to be noted that such integrals can be evaluated numerically as they are smooth and derivable. Then, we compare our results with the filament and the finite element methods. All the results are in excellent agreement.

INTRODUCTION

Various electromagnetic applications are composed of two thick coils that form a loosely coupled transformer. The first coil generates a magnetic field in all points in space, and this magnetic field is partly picked up by the secondary coil. This is an efficient way of transfering power wirelessly. However, a decrease in power transfer efficiency can be caused by a lower mutual inductance between two coils [START_REF] Costa | Responses in transformers built with planar coils inner ring coils excited by square waves[END_REF] [START_REF] Costa | Planar transformers excited by square waves[END_REF]. In other words, it is very useful to know the accurate value of the mutual inductance or the force exerted between two coils. Indeed, the force is directed linked to the mutual inductance as it is proportional to its gradient.

The calculation of mutual inductance for circular coils has been studied by many authors [START_REF] Yang | Magnetic field of cylindrical coils[END_REF]- [START_REF] Babic | Magnetic force calculation between thin coaxial circular coils in air[END_REF]. These papers are generally based on the application of Maxwell's formula, Neumann's formula and the Biot Savart law. By using these approaches, the mutual inductance of circular coils can be expressed in terms of analytical and semi-analytical functions, as the elliptic integrals of the first, second and third kind, the Heummann's Lambda function or the Bessel functions [START_REF] Conway | Inductance calculations for noncoaxial coils using bessel functions[END_REF] [START_REF] Conway | Noncoaxial inductance calculations without the vector potential for axisymmetric coils and planar coils[END_REF]. Such analytical methods are also suitable for calculating the magnetic near-field or far-field from circular cylindrical magnetic sources [START_REF] Selvaggi | Computation of the external magnetic field, near-field or far-field from a circular cylindrical magnetic source using toroidal functions[END_REF] [START_REF] Lemarquand | Annular magnet position sensor[END_REF] or for the determination of the forces exerted between them [START_REF] Ravaud | Force and stiffness of passive magnetic bearings using permanent magnets. part 1: axial magnetization[END_REF]- [START_REF] Lang | Fast calculation method for the forces and stiffnesses of permanent-magnet bearings[END_REF]. On the other hand, authors generally use the finite element method or the boundary element method for solving such magnetostatic problems. However, as stated in [START_REF] Azzerboni | Computation of the magnetic field in massive conductor systems[END_REF], it is interesting to obtain analytical or semi-analytical exact expressions having a lower computational cost for optimization purposes.

In this paper, we propose to replace each coil by a toroidal conductor carrying uniform current volume density for calculating both the force exerted between two thick coils and their mutual inductance. These toroidal conductors are assumed to be perfectly circular and radially centered. We define an equivalent current volume density j which is linked to the number of loops and the coil dimensions. We use the Lorentz force for evaluating the exact axial force exerted between two thick coils carrying uniform current volume densities. We obtain a semi-analytical expression of the force that requires only two numerical integrations. However, its computational cost remains very low compared to the finite element method one as we need only about 0.05 s for calculating the axial force for a given configuration. Then, we compare our semi-analytical approach with the filament method. This comparison is a way for us to verify the accuracy of our semianalytical model and to study the differences that occur between these three methods when two coils are close to each other. The analytical and numerical simulations are in very good agreement. The second part of this paper deals with the analytical calculation of the mutual inductance of two thick coils in air. We use the potential vector and the Stoke's Theoreom for reducing the number of numerical integrations required for evaluating this mutual inductance. We obtain a semi-analytical expression based on two numerical integrations. However, its computational cost remains very low (less than 0.5 s). Furthermore, we do not use any simplifying assumptions in the integral formulations of the mutual inductance. Then, we compare our approach with both the finite element method and a method proposed by Kajikawa [START_REF] Kajikawa | Usable ranges of some expressions for calculation of the self inductance of a circular coil of rectangular cross section[END_REF][28]. Here again, the results are in excellent agreement and show that our approach is exact.

AXIAL FORCE FORMULATION

We present in this section our three-dimensional analytical formulation for calculating the force exerted between two thick coils with rectangular cross sections. It is emphasized here that the two thick coils considered are replaced by two toroidal conductors having uniform current densities. However, for the rest of this paper, we will talk about thick coils rather than toroidal conductors.

Notation and geometry

Let us first consider Fig. 1 where we have represented two thick coils carrying uniform current volume densities:

j 1 = N 1 I 1 (r 2 -r 1 )(z 2 -z 1 ) and j 2 = N 2 I 2 (r 4 -r 3 )(z 4 -z 3 )
. The parameters we use are defined as follows: for the lower coil, (respectively the upper one): r 1 ,r 2 (r 3 ,r 4 ) inner and outer radius [m], z 1 ,z 2 (z 3 ,z 4 ) lower and upper height [m], d = z 3 -z 2 : axial distance between the two thick coils [m].

Expressions of the axial force

The first step for calculating the axial force exerted between two thick coils is to express the magnetic induction field produced by the lower thick coil shown in Fig. 1. By using the Biot-Savart Law, the magnetic induction field B 1 is expressed as follows:

B 1 ( r, z) = µ 0 4π V 1 j 1 dṽ 1 ∧ -∇G( r, r) (1)
The axial force, exerted by coil 1 on coil 2, is derived with the following equation:

F = V 2 j 2 dṽ 2 ∧ B 1 ( r, z) (2)
By inserting (1) in (2), we have:

F = V 2 j 2 dṽ 2 ∧ µ 0 4π V 1 j 1 dṽ 1 ∧ -∇G( r, r) (3)
where the Green's function G( r, r) is defined as follows: For the lower thick coil, the inner radius is r 1 , the outer one is r 2 , the lower height is z 1 , the upper one is z 2 ; for the upper thick coil, the inner radius is r 3 , the outer one is r 4 , the lower height is z 3 , the upper one is z 4 .

G( r, r) = 1 r2 + r2 -2r r cos( θ) + ( z -z) 2 (4)
Therefore, (3) becomes:

F = µ 0 4π V 1 V 2 j 2 dṽ 2 ∧ j 1 dṽ 1 ∧ -∇G( r ′′ , r ′ ) (5)
We obtain the final expression:

F = µ 0 4π V 1 V 2 j 2 ∧ j 1 ∧ -∇G( r ′′ , r ′ ) dṽ 1 dṽ 2 (6)
where dṽ 1 dṽ 2 = rr drd θdzd rd θd z [START_REF] Fawzi | The accurate computation of self and mutual inductances of circular coils[END_REF] In cylindrical coordinates, -∇G( r, r) is reduced to the following form:

-∇G( r, r) = r -r cos(θ) r2 + r2 -2r r cos( θ) + ( z -z) 2 3 2 u r + r sin( θ) r2 + r2 -2r r cos( θ) + ( z -z) 2 3 2 u θ + z -z r2 + r2 -2r r cos( θ) + ( z -z) 2 3 2 u z
By projecting j 1 , j 2 and -∇G( r, r) in cartesian coordinates, we find the following form of the axial force exerted between two thick coils radially centered:

F z = µ 0 j 1 j 2 4π V 1 V 2 z -z cos( θ)r rdrd θdzd rd θd z r2 + r2 -2r r cos( θ) + ( z -z) 2 3 2 (8)
The previous relation can be directly reduced to the following form:

F z = µ 0 j 1 j 2 2 V 1 r 4 r 3 z 4 z 3 z -z cos( θ)r rdrd θdzd rd z r2 + r2 -2r r cos( θ) + ( z -z) 2 3 2 (9)
After integrating with respect to r, z and z, we obtain the following reduced analytical expression of the axial force exerted between two thick coils radially centered:

F z = µ 0 j 1 j 2 2 r 2 r 1 θ 2 θ 1 2 i=1 (-1) i 4 j,k=3
(-1) j+k fgr 2 i cos(θ) cos(θ)drdθ [START_REF] Koledintseva | Mutual external inductance in stripline structures[END_REF]

with f = - αζr i 2 + β 2 ln [α + ζ] g = -z j + γ arctan ζ γ -γ arctan ǫζ αγ + ζ ln [ǫ + α] + ǫ ln [ζ + α] (11)
The parameters α, β, ǫ, γ, ζ are defined in Table 1. It is emphasized here that Eq.( 10) is an exact semi-analytical expression. This implies that it can be used whatever the thick coil dimensions and may be used as a robust tool for evaluating the accuracy of any numerical methods.

Parameters α r 2 + r 2 i -2rr i cos(θ) + (z j -z k ) 2 β r 2 + r 2 i -2rr i cos(θ) ǫ r -r i cos(θ) γ r i sin(θ) ζ z j -z k
Table 1. Definition of the parameters used in ( 10)

COMPARISON OF THE AMPERIAN CURRENT MODEL AND THE FILAMENT METHOD

We present in this section a comparison between the amperian current model and the filament method for calculating the axial force exerted between two thick thick coils carrying uniform current volume densities. We compare two configurations and we discuss the accuracy of each method.

First configuration: two thick coils having the same dimensions

The first configuration consists of two thick coil having the same dimensions and the same uniform current volume densities. We take the following dimensions for our numerical simulations: It is useful to mention that the volume densities j 1 and j 2 correspond to a current I = I 1 = I 2 that equals 1 A. We represent in Fig. 2 the axial force exerted between the two coils versus the axial distance d between them. When d = 0, the two thick coils are in contact. We have also presented the numerical results of our analytical method and the filament method in Table 2. Table 2 and Eq.2 show a very good agreement between the filament method, our amperian current model and the finite element method for calculating the axial force exerted between two thick coils in air. The filament method was employed with 10*10*10*10 filaments and the computational cost was 25 s. However, this computational cost can be reduced by employing 5*5*5*5 filaments and reach only 1.7 s. The computational cost of our semi-analytical method is 0.18 s. 2. Comparison between our analytical approach, the filament method and the finite element method for calculating the axial force (mN) exerted between two thick coils having the same dimensions.

r 1 = r 3 = 0.0875 m r 2 = r 4 = 0.1125 m z 2 -z 1 = z 4 -z 3 = 0.025 m j 1 = j 2 = 320 000 A/m 2 , I= 1

Second configuration: two thick coils having different dimensions

The second configuration consists of two thick coils having different dimensions. We use the following parameters: N 1 = 400 and N 2 = 800 We take the following dimensions: 3. Comparison between our analytical approach, the filament method and the finite element method for calculating the axial force (mN) exerted between two thick coils having different dimensions. It is useful to mention that the volume densities j 1 and j 2 correspond to a current I = I 1 = I 2 that equals 1 A.

r 1 = 0.1 m, r 2 = 0.2 m, r 3 = 0.3 m, r 4 = 0.4 m, z 2 -z 1 = 0.2 m, d [m] Filament
r 1 = 0.1 m, r 2 = 0.2 m, r 3 = 0.3 m, r 4 = 0.4 m, z 2 -z 1 = 0.2 m, z 4 -z 3 = 0.4 m, j 1 = j 2 =
We represent in Fig. 3 the axial force between two thick coils having different dimensions versus the axial displacement d and in Table 2 the numerical results in some points. Table 3 and Fig. 3 show that the three methods presented in this paper are in excellent agreement. Nevertheless, our semi-analytical approach has a lower computational cost than the finite element method or the filament method.

MUTUAL INDUCTANCE OF TWO THICK COILS

Let us first consider the expression of the axial magnetic field produced by the lower thick coil. By considering the number of loops of each thick coil, the axial component of the magnetic field created by the lower coil is expressed as follows:

H z ( r, z) = N 1 I 1 4πS 1 r 2 r 1 θ 2 θ 1 z 2 z 1        (r -r cos( θ))rdrd θdz r2 + r2 -2r r cos( θ) + ( z -z) 2 3 2        (12)
It is emphasized here that the calculation of the force requires the knowledge of the radial component created by the lower thick coil whereas the calculation of the mutual inductance requires the knowledge of the magnetic field axial component created by this lower thick coil. The flux across one elementary loop of the second thick coil whose radius is r is expressed as follows:

ϕ = 2π 0 r 0 H z ( r, z) rd θ (13)
By using the Stoke's Theorem, we can write that:

ϕ = 2π 0 r 0 µ 0 ∇ × A θ rd θ ( 14
)
where A is the vector potential created by the lower thick coil. The previous relation can be transformed as follows:

ϕ(r, z) = 2πµ 0 A θ r, z (15)
We obtain:

ϕ(r, z) = µ 0 N 1 I 1 2S 1    r 2 r 1 θ 2 θ 1 z 2 z 1 cos( θ)rrdrd θdz r2 + r -2rr cos( θ) + ( z -z) 2    (16)
The total magnetic flux across the second thick coil is given as follows:

φ = µ 0 N 1 N 2 I 1 2S 1 S 2 r 4 r 3 z 4 z 3    r 2 r 1 θ 2 θ 1 z 2 z 1 cos( θ)rrdrd θdz r2 + r -2rr cos( θ) + ( z -z) 2    drd z (17)
The mutual inductance can be deducted from the previous expression as follows:

M = φ I 1 = µ 0 N 1 N 2 2S 1 S 2 r 4 r 3 z 4 z 3    r 2 r 1 θ 2 θ 1 z 2 z 1 cos( θ)rrdrd θdz r2 + r -2rr cos( θ) + ( z -z) 2    drd z
(18) After integrating with respect to r, z and z, we obtain a semi-analytical expression based on only two numerical integrations:

M = µ 0 8 N 1 N 2 S 1 S 2 2π 0 r 4 r 3 2 i=1 (-1) (i) 4 j,k=3 (-1) (j+k) rf (r i , z j , z k , cos(θ))drdθ (19) with f (r i , z j , z k , x) = 2 3 2r 2 i -r i rx + r 2 (2 -3x 2 ) -ǫ(z j -z k ) 2 -2r 2 (z j -z k ) ln [z j -z k + ǫ] +r 2 i (z j -z k ) + 2r i rx(z j -z k ) -4rx r 2 (-ϑ 2 )(z j -z k ) arctan r i -rx r 2 (1 -x 2 ) +r 2 (2x 2 -1)(z j -z k ) ln r 2 i + r 2 -2r i rx -2rx r 2 ϑ 2 + (z j -z k ) 2 ln [r i -rx + ǫ] + r 2 ϑ 2x -2x 3 -ϑ + 2x 2 ϑ (z j -z k ) ln [A 1 ] + r 2 ϑ -2x + 2x 3 -ϑ + 2x 2 ϑ (z j -z k ) ln [A 2 ]
where We represent in Fig. 4 the mutual inductance versus the axial distance d with our analytical method and in Table 4 the numerical results obtained with the finite element method, the approach of Kajikawa and our analytical approach. Table 4 and Fig. 4 clearly show that all the results are in excellent agreement. This confirms the accuracy of our analytical approach.

A 1 = r i rϑ 2 + r 2 (+x 2 -1)(ϑ -x) -ϑ(z j -z k )(z j -z k + ǫ) r 2 (+2x -2x 3 -ϑ + 2x 2 ϑ) (+r i + r(ǫ -x)) (z j -z k ) 3 A 2 = r i rϑ 2 -r 2 (-x 2 -1)(ϑ + x) -ϑ(z j -z k )(z j -z k + ǫ) r 2 (-2x -2x 3 -ϑ + 2x 2 ϑ) (-r i + r(ǫ + x)) (z j -z k ) 3 (20)
= r 3 =0.025 m, r 2 = r 4 =0.03 m, z 2 -z 1 = z 4 -z 3 =0.025 m, N 1 = N 2 =200. d [m]

CONCLUSION

We have presented exact semi-analytical expressions of the force exerted between two thick coils carrying uniform current volume densities and of their mutual inductance. For this purpose, we have replaced each thick coil by a toroidal conductor having uniform current volume density. Our expressions are based on two numerical integrations of continuous and derivable functions. Consequently, their computational cost remain very low compared to the finite element method and are also lower than the filament method. This exact expression can be used for calculating the force exerted between two thick coils in air as our results are in excellent agreement with the filament method and the finite element method.

Figure 1 .

 1 Figure1. Geometry considered: two thick coils carrying uniform current volume densities. For the lower thick coil, the inner radius is r 1 , the outer one is r 2 , the lower height is z 1 , the upper one is z 2 ; for the upper thick coil, the inner radius is r 3 , the outer one is r 4 , the lower height is z 3 , the upper one is z 4 .

  A, N = 200 turns d = axial distance between the two thick coils [m]

Figure 2 .

 2 Figure 2. Representation of the axial force exerted between two thick coils versus the axial distance d = z 3 -z 2 . We take the following dimensions: r 1 = r 3 = 0.0875 m, r 2 = r 4 = 0.1125 m, z 2 -z 1 = z 4 -z 3 = 0.025 m, j 1 = j 2 = 320 000 A/m 2 . Line = our analytical method, Points = Filament method

Figure 3 .

 3 Figure 3. Representation of the axial force exerted between two toroidal conductors versus the axial distance d. We take the following dimensions: r 1 = r 3 =0.0875 m, r 2 = r 4 =0.1125 m, z 2 -z 1 = z 4 -z 3 =0.025 m, j 1 = j 2 =320 000 A/m 2 . Line = our analytical method, Points = filament method.

z 4 -

 4 z 3 = 0.4 m, j 1 = j 2 = 20 000 A/m 2 , d = axial distance between the two thick coils[m]

Figure 4 .

 4 Figure 4. Representation of the mutual inductance between two thick coils versus the axial distance d; r 1 = r 3 =0.025 m, r 2 = r 4 =0.03 m, z 2 -z 1 = z 4 -z 3 =0.025 m, N 1 = N 2 =200.

  r 1 =0.025 m, r 2 =0.03 m, r 3 =0.025 m, r 4 =0.03 m, z 2 -z 1 =0.025 m, z 4 -z 3 =0.025 m, N 1 =200, N 2 =200

Table 4 .

 4 Comparison between our analytical model, the approach ofKajikawa[28] and the finite element method for calculating the mutual inductance (mH).

		Kajikawa amperian current model
	0	0.776	0.7753904
	0.005	0.571	0.5712872
	0.01	0.435	0.4348317
	0.02	0.267	0.2667799
	0.03	0.173	0.1729528
	0.05	0.0823	0.0823239
	0.1	0.021	0.0209659
	0.15	0.00803	0.0080293
	0.2	0.00386	0.0038579
	0.3	0.0013	0.00130199
	0.4	0.000586	0.0005860
	0.5	0.000312	0.0003119


Yohann Genzmer

Ï C Lo

Laboratoire I R M A Teyssier

EXISTENCE OF NON-ALGEBRAIC SINGULARITIES OF DIFFERENTIAL EQUATION

An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some local chart by a differential equation with algebraic coefficients. We show that there exist at least countably many saddle-node singularities of the complex plane that are not algebraizable.

near an isolated singularity, which can be conveniently located at (0, 0) by translation. The coefficients A and B are germs of a holomorphic function with a common zero at (0, 0) and no common factor. We denote by λ 1 and λ 2 the eigenvalues of the linear part of the equation at (0, 0). We will always assume that at least one of those is non-zero, say λ 2 = 0, and set λ := λ1 λ2 . We recall the following classical result :

Theorem. (Poincaré and Dulac [START_REF] Dulac | Sur les points singuliers d'une équation différentielle[END_REF]) If λ / ∈ R ≤0 then there exist two polynomials P, Q such that the previous differential equation is orbitally equivalent through a local analytic change of coordinates to P (x, y) dy = Q (x, y) dx .

If moreover λ /

∈ N ∪ 1/N =0 then we can choose P (x, y) = x and Q (x, y) = λy ( i.e. the equation is linearizable).

We recall that two germs of differential equation are orbitally equivalent when there exists a germ of biholomorphism conjugating their solutions. It thus turns out that a generic equation is orbitally equivalent to a linear, or at least algebraic, equation. Up to now an open question regarded whether every differential equation is algebraic in some local chart. Such an equation will be called algebraizable. Geometrically, it is equivalent to ask if any germ of a singularity of foliation in the complex plane can be realized as some singularity of a foliation of CP 2 . We aim to prove that it is not so in the case of a saddle-node (λ = 0), as was expected in [START_REF] Martinet | Classification analytique des équations différentielles non linéaires résonnantes du premier ordre[END_REF] for non-linearizable resonant singularities. Notice that these equations are nonetheless formally algebraizable.

Theorem 1. There exist at least countably many non-equivalent saddle-node equations

x 2 dy = (y + h.o.t) dx (0.2) which are not algebraizable.

Our proof is based on Martinet-Ramis' theorem about orbital classification of such equations, stating that the equivalence classes of all equations (0.2) under the action of local changes of coordinates is in one-to-one correspondence with the space of germs C {h}. We will give a more precise statement in Section 3. Our argument boils down to the following : since the space of orbitally equivalent saddle-node equations is in one-to-one correspondence with a functional space of germs and since this space is "big" then the trace of all algebraic equations should reasonably be "meagre", for instance in the sense of Baire. Many problems arise immediately, one of those being that C {h} cannot be endowed with a topology which would make it a Baire space while at the same time preserving the "nice" structure of the set of algebraic equations. Another problem lies in the fact that C {h} might not be objectively "big" as it can be the range of a continuous map R → C {h} and thus set-theoretically equivalent to the field of scalars. Hence both set theory and topology are not sufficient to guarantee that the heuristics works, and we must consider "analytic Baire properties". What makes things work is the fact that Martinet-Ramis' invariant of classification is analytic with respect to the equation, as was already known. The main part of our proof regarding this Baire analyticity property deals with showing that Dulac's prenormalization procedure is analytic too.

What is actually expected is that the typical saddle-node equation is non-algebraizable, i.e. the set of non-algebraizable equations is a G δ -dense subset of all saddle-node equations, not only that the image of those non-algebraizable equations is a G δdense subset of the space of invariants (which is what we prove here). To do so one must consider a finer topology on spaces of germs than the ones used presently and study analyticity and openness of maps from and into these spaces. This requires a lot of additional technical work and is currently being carried out. The authors nonetheless believe this stronger result to be true.

A topology on

C {z 1 , z 2 , . . . , z n }
In the sequel we use bold-typed letters to indicate multi-variables z := (z 1 , . . . , z n ) ∈ C n or multi-indices J := (j 1 , . . . , j n ) ∈ N n . We use the standard notations J! := ℓ (j ℓ !), |J| := ℓ j ℓ and z J := ℓ z j ℓ ℓ . 1.1. Norm on C {z}.

Let us endow the topological space C {z} with the norm

||f || := J |a J | J! where f (z) = J a J z J .
F (x + h) = F (x) + L (h) + o (h) .
(3) A map G : C {w} → C {z} is said to be strongly analytic if the image of any analytic family of C {w} with a lower bounded radius of convergence is an analytic family of C {z} with a lower bounded radius of convergence.

Proposition 3. If F is strongly analytic then it is analytic.

Notice that there exist analytic maps which are not strongly analytic: the obstruction comes simply from the non-existence of local uniform lower bound for the radius of convergence of series on any open ball of C {z} for ||.||. The following example, due to J. Duval, illustrates that fact.

Example 4. Consider the family of compact sets for ε > 0

K ε := D\ {0 < Im (z) < ε}
which is the union of two simply connected, compact and connected sets K + ε and K - ε such that, say, K ± ε intersects ±iR >0 . According to Runge's approximation theorem there exists a sequence of polynomials (P ε n ) n∈N which is a uniform approximation of the function defined by x ∈ K + ε → 1 x and x ∈ K - ε → 1. There exists a slowly converging sequence ε n > 0 such that sup x∈D |P εn n (x)| ≤ √ n. We now form the sequence P n := P εn n and consider the map:

F : x ∈ C → j∈N P j (x) j z j .
The reader can easily prove that F (x) ∈ C {z} for all x ∈ C and that its radius of convergence is |x| if Im (x) > 0 and equals 1 otherwise. As a consequence F cannot be strongly analytic, as (x, z) → F (x, z) is analytic on no neighbourhood of (0, 0), whereas x → F (x) is analytic, for

F (x + h) -F (x) -h j∈N jP ′ j (x) P j (x) j-1 z j ≤ C |h| 2 j∈N √ j j j!
if we require that x belong to a smaller disc rD, 0 < r < 1, thanks to Cauchy's formula as will be detailed further down.

Proof. In the proof we assume that n = p = 1 : the general case can be treated in much the same way. Since analyticity is a local property, we can also perform the proof in a neighbourhood of 0 ∈ C. Let us write F (x) (z) = j≥0 f j (x) z j . Since F (x) (z) is analytic as a map of two variables, the series F (x) are convergent on a common open disc centered at x = 0 with radius 2ρ. The Cauchy formula ensures that for any j

f j (x) = (-1) j 2iπ ˆγ F (x) (ξ) ξ j+1 dξ
for any loop γ in the disc of convergence. Substituting γ := {|ξ| = ρ} yields

ρ j+1 |f j (x)| ≤ ||F (x)|| ∞,D(0,ρ) ,
Since F (x) (z) is bounded on D (0, β) × D (0, ρ) for some β, there exists a positive number C such that for any j

|f j (x)| ≤ C ρ j . Hence on a disc D(0, β ′ ) with β ′ < β we have a control of the second derivative of the components of f j (x) f (2) j (x) ≤ C ′ ρ j .
As a consequence, we have on a yet smaller disc :

f j (x + h) -f j (x) -hf (1) j (x) ≤ C ′′ 1 ρ j |h| 2 . Defining D x F (h) as h j≥0 f (1) j (x)
, which is a convergent series, yields

||F (x + h) -F (x) -D x F (h)|| ≤ C ′′ e 1 ρ |h| 2 ,
which ensures the analyticity of F .

Analytical Baire property of C {z}

We haven't been able to find a suitable "nice" and reasonably interesting topology on C {z} in order to obtain a Baire space, and surely it is not possible to do so if we agree on what "interesting topology" might be... We can prove that (C {z} , ||•||) is not Baire. But we can also prove that this space cannot be covered by countably many analytic subspaces, which is the purpose of this paragraph.

Definition 5.

(1) An analytic subspace of C {z} is the range of an analytic map

F : Ω ⊂ C p → C {z}. (2)
We say that C {z} is an analytic Baire space if it cannot be the union of a countable analytic subspaces.

Our main result is the following Theorem 6. C {z} is an analytic Baire space.

Annoying facts about C {z}.

We begin with proving the following

Lemma 7. C {z} is in one-to-one correspondence with C.
This result is a consequence of the existence of a "Peano-curve" in C {z} for some relatively natural topology.

Proof. The space C {z} is naturally a subset of C N , which can be endowed with the product topology. The induced topology on C {z} makes this space a connected and locally connected topological space. Moreover for any (p, r) ∈ N × Q the subset of C {z} defined by

A p,r :=    f (h) = j≥0 a j h j : |a j | ≤ pr j    is compact. The union N×Q A p,
r covers the whole C {z}, which means the latter is σ-compact for the topology under consideration. A theorem of Hahn, Mazurkievicz, Menger, Moore and Sirpieński [START_REF] Law | Oeuvres choisies. Tome III. Théorie des ensembles et ses applications, travaux des années 1930-1966[END_REF] states precisely that the continuous images of [0, 1] are the compact, connected and locally connected spaces. Therefore C {z} is a continuous image of R, and obviously of C, for the above not-too-pathological product topology. A weaker consequence is that from a purely set-theoretical point of view C and C {z} are in one-to-one correspondence.

Now we show that

Lemma 8. (C {z} , ||•||) is not a Baire space.
Proof. We consider the following example due to R. Schäfke. Consider the subspaces

M N := a j z j : |a j | ≤ N j , N ∈ N . Obviously C {z} = ∪ N M N . Moreover M N = ∩ j |a j | ≤ N j is closed as the asso- ciation f → f (j) (0)
is continuous, and its interior is empty as the example 4 shows that no neighbourhood of f ∈ C {z} may admit a uniform lower bound for the radius of convergence.

As an inductive space C {z} can also be endowed with the inductive topology : this space becomes complete but not Baire. In particular this topology cannot be induced by a metric.

Preliminaries.

In order to prove Theorem 6 we will need to eventually locate the proof within a Baire space to get a contradiction. Let A be the subspace of C {z} defined by

A :=    f (z) = j≥0 a j z j : |a j | is bounded    together with the norm ||•|| ∞ : ||f || ∞ := sup j |a j | . (A, ||•|| ∞
) is a complete metric space and is thus a Baire space because it is isometric to a subspace of C N formed by all bounded sequences equipped with the sup-norm. Proof. Suppose that for any p ∈ N there exists a non-trivial relation Λ p := (λ 1,p , . . . , λ n,p ) = 0 for the family ϕ p := (J p (f 1 ) , . . . , J p (f n )), that is

J p   n j=1 λ j,p f j   = 0 .
Up to rescalling Λ p one can suppose that it belongs to the unit sphere of C n and so consider some adherence value (λ 1,∞ , . . . , λ n,∞ ) . Because if J k+1 (f ) = 0 then J k (f ) = 0, by taking the limit p → ∞ while fixing an arbitrary k we obtain that Λ ∞ is a non-trivial relation for ϕ k by continuity of f → J k (f ), and thus is a non-trivial relation for (

f 1 , • • • , f n ).
According to this lemma, if F of maximal rank at x, i.e. its rank is equal to the dimension of the source space, there exists N ∈ N such that the function J N F is of maximal rank. Since the space of polynomials of maximal degree N is of finite dimension, the function J N R is locally one-to-one around x. So is the application F . Hence the Corollary 11. Let F : Ω ⊂ C n → C {z}.

(1) If D x F is of rank n then F is locally one-to-one near x.

(2) If D x F is of maximal rank p < n then there exists a smooth hypersurface S of dimension p at x such that F | S is of rank p and has the same image as F .

Proof. The second part of the corollary is proved using the same result in finite dimension : indeed, if the range of F were some finite dimensional vector space, one could choose for S the hypersurface

x i1 = • • • = x in-p = 0 where D (xj 1 ,...,xj p ) F is of rank p with {1, • • • , n} = {i 1 , .
. . , i n-p } ∪ {j 1 , . . . , j p } . Now if the range of F were C {z}, one applies this argument to J N F for all N big enough.

The key point to Theorem 3 is the following proposition :

Proposition 12. Let F : Ω → C {z} be continuous, analytic and one-to-one on an open set Ω ⊂ C n . Let E < C {z} be any subspace of infinite dimension and suppose that D x F is of rank n for some x ∈ Ω. Then there exist δ in E and ε > 0 such that for any 0 < |t| < ε the germ F (x) + tδ does not belong to F (Ω).

Proof. Suppose the claim is false and fix δ ∈ E\ {0}. There exists a sequence (u n ) n∈N ⊂ C n such that, for n large enough, x + u n ∈ Ω and

F (x + u n ) = F (x) + δ n .
Any accumulation point u of u n satisfies F (x + u) = F (x). Because F is one-toone u must vanish, which in turn implies that u n converges towards zero. Besides, the definition of differentiability we use implies

o (u n ) = D x F (u n ) - δ n = ||u n || D x F u n ||u n || - δ n ||u n || . Dividing by ||u n || yields D x F un ||un|| -δ n||un|| = o (1)
. Now by compactness of the unit sphere of C n we can assume that un ||un|| tends to some u = 0 when n tends to infinity. Hence δ n||un|| has to tend to someλδ as n tends to infinity and, according to the rank assumption, λ = 0. As a matter of consequence

D x F (u) = λδ,
which cannot be possible for every δ in E, for the image of the differential map D x F is finite dimensional.

2.3.

Analytical Baire property of C {z} : proof of Theorem 6.

We show here that C {z} has an analytical Baire property by supposing on the contrary that C {z} is a countable union of analytic sets :

C {z} = n∈N j∈N F j,n (Ω j,n ) ,
where F j,n is a differentiable function defined on an open set Ω j,n of C n . Taking if necessary a finite covering of each Ω j,n , one can assume that F j,n is of rank n on Ω j,n . Indeed the set of points where F j,n is not of maximal rank is an analytical subset Σ j,n of Ω j,n locally closed. The analytical set Σ j,n admits a decomposition Σ j,n = ∪C k where each cell C k is biholomorphic to an open set of some C p with 0 ≤ p < n [START_REF] Coste | Ensembles semi-algébriques[END_REF]. Hence we get the following decomposition

F j,n (Ω j,n ) = F j,n (Ω j,n \Σ j,n ) k F j,n (C k ) .
If the rank p of F j,n is strictly smaller than n on Ω j,n \Σ j,n then one can find a finer covering of Ω j,n \Σ j,n = k B j,n,k and a family of smooth hypersurfaces S j,n,k ⊂ B j,n,k of dimension p such that the rank of F j,n | S j,n,k is n and F j,n | B j,n,k and F j,n | S j,n,k has the same image . Now on each cell C k one can seek the points where F j,n | C k is not of maximal rank and do the same procedure as above. This construction stops after finitely many steps since at each stage the dimension of the open set we consider is strictly less than that of the previous stage. Finally since any open set of C p is a countable union of compact sets, we obtain the following decomposition

C {h} = n∈N j∈N q∈N R j,n (K j,n,q ) ,
where Ω j,n = q∈N K j,n,q and each K j,n,q is a full compact subset of some C p with p ≤ n. The set R j,n (K j,n,q ) is compact and therefore closed for the topology induced by (||•|| k ) k>0 . According to Lemma 9 the set R j,n (K j,n,q ) ∩ A is also closed in A for the norm ||•|| ∞ . It is besides of empty interior : since A is infinite dimensional if R j,n (x) belongs to A we can invoke Proposition 12 to obtain δ ∈ A such that for t small enough

R j,n (x) + tδ / ∈ R j,n (Ω j,n ) ,
which ensures that any small ball for the norm ||•|| ∞ in A around R j,n (x) cannot be contained in R j,n (Ω j,n ). Finally we obtain the sought contradiction since then A can be split into a countable union of closed subset with empty interior :

A = n∈N j∈N q∈N R j,n (K j,n,q ) ∩ A ,
which is impossible since A is a Banach thus Baire space.

Analyticity of Martinet-Ramis invariants and proof of the main theorem

The tool that we need is a map which to a saddle-node equation, written in the most general form (0.1), associates its invariant of orbital classification. This is the goal of this section, as well as that this map is actually analytic.

Firstly we need to put the general equation (0.1) in a prepared form; this is done using Dulac's prenormalization procedure in Section 3.2. We will restrict our construction to those equations whose first topological invariant equals 1. Geometrically speaking this invariant is the order of tangency between the foliation defined by the equation and the separatrix tangent to the eigenspace associated to the eigenvalue λ 2 = 0. This defines the stratum E 1 , studied in Section 3.3. After applying Dulac's procedure D we deal with equations in the form Theorem. (Martinet-Ramis, [START_REF] Martinet | Classification analytique des équations différentielles non linéaires résonnantes du premier ordre[END_REF]) There exists a map M : E 1 → M such that two equations E and Ẽ of E 1 are orbitally conjugate if, and only if, M (E) = M Ẽ . Moreover this map is onto and if t ∈ (C n , 0) → E t ∈ E 1 is an analytic family of equations written in Dulac's form then t → M (E t ) is an analytic family too (that is, for all t one can choose a representant of M (E t ) such that this family is analytic).

In other words, once written in Dulac's form the germ-component of Martinet-Ramis' map, which we will write φ MR , is strongly analytic with respect to R. The aim of this section is to provide a proof for the :

Theorem 13. The complete Martinet-Ramis map (Ady -Bdx) D -→ x 2 dy -(y + R) dx φMR -→ φ ∈ Dif f (C, 0) is a strongly analytic association.
Thus all that remains is to show that D is strongly analytic. Before investigating this result we begin with giving the proof of Theorem 1 in the upcoming section.

3.1. Proof of the existence of countably many non-algebraizable saddlenode singularities (Theorem 1). First we show that there there exists at least one such non-algebraizable equation. Suppose on the contrary that any saddle-node singularity is algebraizable. We define P d to be the set of all equations in E 1 with polynomial coefficients of maximum degree d which, as we will see in Corollary 15, is an analytic space. Then, according to Theorem13 the restriction of φ MR • D to the space of polynomials must be onto : therefore, we would obtain the decomposition

Dif f (C, 0) ≃ C {h} = n∈N φ MR • D (P n )
which would be a countable union of analytical sets. This is impossible in view of the analytic Baire property of C {h} (Theorem 6). Hence, there exists at least one saddle-node equation which is not algebraizable.

Obviously the same argument works for countably many equations as a point of C {h} is a compact with empty interior. Notice that the latter change of variable depends rationally on the coefficients of the linear part of A and B. In all the sequel the only changes of variables we allow will be required to preserve this diagonal form. The existence of a unique analytic solution x = s (y) (a separatrix of X A,B ) tangent to the eigenspace {x = 0} at (0, 0) is well known (see [START_REF] Camacho | Invariant varieties through singularities of holomorphic vector fields[END_REF] for example). The other separatrix y = ŝ (x), tangent to {y = 0}, only exists a priori at a formal level (and generically this series, though unique, is divergent). The reader will find in [START_REF] Dulac | Recherches sur les points singuliers des équations différentielles[END_REF] the material needed to carry out the complete prenormalization procedure. What we retain from it is the following steps :

• Applying the change of coordinates (x, y) → (x + s (y) , y) transforms X A,B into a vector field X A1,B1 where

A 1 (x, y) ∈ xC {x, y} .
• It is possible to further orbitally normalize A 1 to obtain a new vector field X AD ,BD such that

A D (x, y) = x k+1 B D (x, y) = y + r (x) + yR (x, y) (3.1)
with r (0) = r ′ (0) = R (0, 0) = 0. The integer k ∈ N >0 is a topological invariant (but not a complete topological invariant).

• We define the map

D : (A, B) ∈ E → B D -y ∈ C {x, y} .
At this stage this map may not be well defined. We will give a canonical way of obtaining D (A, B) from the original vector field without ambiguity. We do this in Section (3.4).

3.3.

The stratum E 1 . Denote by E 1 the stratum of E consisting of equations that can be put under the previous form (3.1) with k = 1.

Proposition 14. The stratum E 1 is constructable: it is the complementary of a dimension 1 affine subspace of E.

Proof. First we apply the change of coordinates (x, y) → (x + s (y) , y) which brings X A,B to X x Ã, B . In this situation the separatrix is straightened to {x = 0}. Write à (x, y) = ax + by + o (||x, y||); we claim that X A,B belongs to E 1 if, and only if, a = 0. On the one hand suppose that there exists a local analytic change of coordinates Ψ (x, y) = (αx + C (x, y) , βy + D (x, y)), with C and D in C {x, y} 1 , defining a conjugacy between X x Ã, B and some U X x 2 , B with η := U (0, 0) = 0. Then :

U (x + C, y + D) (αx + C) 2 = x à α + ∂C ∂x + B ∂C ∂y . (3.2)
Written for the term of least homogeneous degree this equation becomes, since B (x, y) = y + o (||x, y||) :

ηα 2 x 2 = αx (ax + by) + y (δx + γy) where δ = ∂ 2 C ∂x∂y (0, 0) and γ = 1 2 ∂ 2 C
∂y 2 (0, 0). Hence αη = a, meaning a = 0 as requested. On the other hand we use Dulac's result : we know that there exists such a Ψ between X x Ã, B and some U X x k+1 , B . If a = 0 then necessarily k = 1, as can be seen for the analog of (3.2) (the term B -y ∂C ∂y is indeed of homogeneous degree strictly greater than 2 and thus cannot cancel αax 2 out). To complete the proof we only have to mention that the condition a = 0 is equivalent to A 2,0 = 0. But this is obviously the case : we even have A 2,0 = a according to

A (x + s (y) , y) = x à (x, y) + B (x, y) s ′ (y) with s ′ (0) = s (0) = 0. Hence E 1 = E\ {A 2,0 = 0} is constructable.
Corollary 15. Let C [x, y] ≤d be the space of all polynomials of degree at most d and define

P d := E 1 ∩ C [x, y] ≤d × C [x, y] ≤d .
Then P d is a constructable set.

Particularly P d is a finite union of smooth analytical sets.

3.4.

Strong analyticity of Dulac's procedure. We first begin with building the map D in a canonical way.

(1) As already stated, there exists a unique germ s (y) such that {x = s (y)} is a separatrix of X A,B .

(2) Applying the change of coordinates (x, y) → (x -s (y) , y) transforms X A,B into X A1,B1 where To prove that the map D is strongly analytic, it is enough to prove that each step of the above construction shares this property: it should be obvious for the second and third steps. It remains to check that it is also the case for the first step.

Lemma 9 .

 9 Let S be a closed set in C {z} for the norm ||•||. Then S ∩ A is closed in A for the norm ||•|| ∞ .Proof. Let (f n ) be a sequence in S ∩ A which tends to f when n tends to infinity for the norm ||•|| ∞ . Then f belongs to A since it is closed. Moreover, as||f n -f || ≤ e ||f n -f || ∞ ,the sequence is convergent in C {z} for the norm ||.||. Since S is closed f must belong to S too. Lemma 10. A family f 1 , . . . , f n ∈ C {z} is free over C if, and only if, there exists p ∈ N such that their p-jets are free over C.

x 2

 2 dy = (y + R (x, y)) dx . Define the space M := C × C × Dif f (C, 0) the equivalence relation on M by (µ, τ, φ) ∼ μ, τ , φ if, and only if, µ = μ and there exists c ∈ C =0 such that φ (ch) = φ f (h) and τ = cτ .

3. 2 .

 2 Dulac's procedure. Let E be the set of couples (A, B) ∈ C {x, y} × C {x, y} such that the matrix ∂A ∂x ∂A ∂y ∂B ∂x ∂B ∂y has exactly one non-vanishing eigenvalue. One can assume that, up to a linear change of variables, the linear part of X A,B = -B ∂ ∂x + A ∂ ∂y is diagonal : A (x, y) = o (||x, y||) B (x, y) = y + o (||x, y||) .

B 1 ( 1 xα ( 1 +

 111 x, y) := B (x -s (y) , y) A 1 (x, y) := A (x -s (y) , y) -B 1 (x, y) s ′ (y) =: x (a 0 (y) + αxA 2 (x, y))with A 2 (0, 0) = 1 and α = 0.(3) There exists a unique holomorphic function y → C (y) such that C (0) = 0 and (x, y)→ x α (1 + C (y)) , y transforms X A1,B1 into U X x 2 ,BDwhereU (x, y) := a 0 (y) B 1 (0, y) B 1 (0, y) -B 1 (x, y) αx + A 2 (x, y) B D (x, y) := B C (y)) , y U (x, y) .This function C is the unique holomorphic solution to the (regular) linear differential equation with C (0) = 0 :B 1 (0, y) C ′ (y) = (1 + C (y)) a 0 (y) ,whose solution is given by C (y) = e ´y 0 a 0 (u)B 1 (0,u) du -1. We have U ∈ C {x, y} * since U (0, 0) = A 2 (0, 0) = 1.Definition 16. We define Dulac's map asD (A, B) := B D -y ∈ C {x, y} 1 .

Lemma 17 .

 17 The correspondence (A, B) ∈ E 1 → s (y) ∈ C {z} is strongly analytic.Proof. It is enough to prove that one can control the disc of convergence s in terms of parameters depending on A and B . Let (A ǫ , B ǫ ) be some analytic family in E 1 with ǫ ∈ (C p , 0) The lemma is deduced from the following formal computation. Let us write s(y) = j≥0 s j y j , s 0 = s 1 = 0 .Then for all n ∈ N :s n (y) = for p ≥ j we have S p,,j = 0. WriteA ǫ (x, y) = n,m a ǫ n,m x n y m , a ǫ 0,0 = a ǫ 1,0 = a ǫ 0,1 = 0 so that A ǫ (s ǫ (y) , y) = n,ma ǫ n,m s ǫ (y) n y m =

  Since the series f is convergent ||f || is well defined and is a norm on the space C {z}. Notice that the space (C {z} , ||.||) is not complete since the sequence The map F is said to be analytic if for any point x in Ω, there exists a linear map L : C p → C {z} such that

	f → f (0) is continuous as well as the evaluation at 0 of any derivative. Hence, the family of projectors J N which associates to f is N -jet is continuous.
	1.2. Analytical functions from C p to C {z}.
	Definition 2. Let Ω be a domain of C p for p ∈ N =0 . (1) A map F : Ω → C {z} is said to be strongly analytic if the map (x, z) → F (x) (z) is analytic with respect to the n + p complex variables x 1 , • • • , x p and z 1 , • • • , z n on a neighbourhood of Ω × {0}. (2)
	|J|≤n in the space of convergent series. It is not even a Baire space. The evaluation √ J!z J has the Cauchy property but is not convergent n∈N

The equation defining s ǫ , namely A ǫ (s ǫ (y) , y) = B ǫ (s ǫ (y) , y) s ′ ǫ (y), thus becomes with a similar notation for

After identifying the coefficients in y p we derive

Hence, in a standard fashion, for any p we have s ǫ p ≤ sp , where sp satisfies the same recurrence equation as s ǫ p except that we set a m,n = b m,n = M ρ m+n , where M is a constant and ρ a lower bound for the radius of convergence of A ǫ and B ǫ . Thus s ǫ p is less or equal than the coefficient sp of s satisfying

1-ρs(y) -1 -ρy -ρs (y) Since this equation admits a convergent solution with s (0) = 0, its radius of convergence is a lower bound for the radius of convergence of the family s ǫ .