ALGEBRAIC APPROXIMATION IN CR GEOMETRY NORDINE MIR

Keywords: 2000 Mathematics Subject Classification. 32H02, 32V20, 32V40, 32C07, 14P05, 14P20 Algebraic map, CR manifold, CR orbits

Introduction

A well-known theorem in algebraic geometry going back to Artin [A69] states that given any system of polynomial equations P (x, y) = 0, x = (x 1 , . . . , x n ), y = (y 1 , . . . , y m ) over the field of real numbers (resp. complex numbers) and given any germ of an analytic solution y(x) of the above system at a given point p ∈ R n (resp. C n ), there exists a sequence of germs at p of real-algebraic (resp. complex-algebraic) solutions of the system that converge to the given solution in the Krull topology.

In this paper, we provide a Cauchy-Riemann version of this approximation theorem. Given a real-algebraic CR submanifold M ⊂ C N , let us say that M has the Nash-Artin approximation property if for every point p ∈ M , every real-algebraic subset S ⊂ C N +N and every positive integer , if f : (C N , p) → C N is a germ of a holomorphic map such that Graph f ∩ (M × C N ) ⊂ S (as germs at (p, f (p))), then there exists a germ at p of complex-algebraic map f : (C N , p) → C N that agrees with f at p up to order with Graph f ∩ (M × C N ) ⊂ S . Observe that finding a sequence (f ) with the above properties consists of finding a sequence of germs of real-algebraic mappings from M into C N R 2N whose graphs are contained in S ⊂ C N +N R 2(N +N ) but which, furthermore, must satisfy the tangential Cauchy-Riemann equations (on M ). Hence, deciding whether M has the Nash-Artin approximation property is indeed equivalent to asking for a CR (or more generally PDE) version of Artin's theorem mentioned above.

We provide a positive solution to this approximation problem by making use of the CR geometry of the manifold M . Let T c M ⊂ T M denote the complex tangent bundle of M . Recall that for every p ∈ M , there exists a unique germ of a real-algebraic CR submanifold O p through p with the property that every point q ∈ O p can be reached from p by following a piecewise differentiable curve in M whose tangent vectors are in T c M (see [START_REF] Baouendi | Algebraicity of holomorphic mappings between real algebraic sets in C n[END_REF]). We call this germ the CR orbit of M at p and say that M is minimal at p if this CR orbit is a neighbhorhood of p in M . We also say that M is of constant orbit dimension if the CR orbits of M have all the same dimension. Our main result is the following: Theorem 1.1. Let M ⊂ C N be a real-algebraic CR submanifold of constant orbit dimension. Then M has the Nash-Artin approximation property.

The study of algebraicity properties of holomorphic mappings sending real-algebraic sets into each other has been extensively studied in the past years (see e.g. [W77, H94, HJ98, BER96, M98, CMS99, Z99, BRZ01b, BMR02, MMZ03b, LM09]). Most of the mentioned results are concerned with the automatic algebraic extension of all holomorphic mappings between two given real-algebraic sets and hold under some geometric nondegeneracy conditions on these sets. The situation considered in Theorem 1.1 goes beyond this setting and deals with the general case where M is allowed to be mapped to an arbitrary real-algebraic set by a non-algebraic holomorphic map. In such a situation, the only known sufficient condition implying that a (connected) real-algebraic CR submanifold possesses the Nash-Artin approximation property is that of minimality and goes back to the work of Meylan, Zaitsev and the author [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF][START_REF] Meylan | On some rigidity properties of mappings between CRsubmanifolds in complex space[END_REF]. Theorem 1.1 provides the same conclusion under a much weaker sufficient condition, that, in addition, is generically satisfied on every connected real-algebraic CR manifold (see e.g. [START_REF] Baouendi | Real Submanifolds in Complex Space and Their Mappings[END_REF]). Hence even the following immediate consequence of Theorem 1.1 is new.

Corollary 1.2. For every connected real-algebraic CR submanifold M ⊂ C N , there exists a closed proper real-algebraic subvariety Σ M of M such that M \ Σ M has the Nash-Artin approximation property.

In the case of real hypersurfaces, we can deduce from Theorem 1.1 the following stronger result.

Theorem 1.3. Any (smooth) real-algebraic hypersurface of C N has the Nash-Artin approximation property.

Indeed, any connected component of any real-algebraic (smooth) hypersurface of C N is either Levi-flat, in which case it is of constant orbit dimension and Theorem 1.1 applies, or is somewhere minimal, in which case it has the Nash-Artin approximation property in view of [START_REF] Meylan | On some rigidity properties of mappings between CRsubmanifolds in complex space[END_REF]Theorem 3.5].

One noteworthy and typical application of Theorem 1.1 deals with the problem of deciding whether the holomorphic equivalence of two germs of real-algebraic CR manifolds of the same dimension implies their algebraic equivalence (see the works [START_REF] Baouendi | Points in general position in real-analytic submanifolds in C N and applications[END_REF][START_REF] Baouendi | Reflection ideals and mappings between generic submanifolds in complex space[END_REF][START_REF] Lamel | Holomorphic versus algebraic equivalence for deformations of realalgebraic CR manifolds[END_REF]). Specializing Theorem 1.1 to this situation, we have: Corollary 1.4. Let M, M ⊂ C N be two real-algebraic CR submanifolds of constant orbit dimension. Then for all points p ∈ M and p ∈ M , the holomorphic equivalence of the germs (M, p) and (M , p ) implies their algebraic equivalence.

As for Theorem 1.1, Corollary 1.4 was known only when M and M are (connected and somewhere) minimal and goes back in this case from the work of Baouendi, Rothschild and the author [START_REF] Baouendi | Reflection ideals and mappings between generic submanifolds in complex space[END_REF]. For nowhere minimal real-algebraic CR manifolds, partial results towards Corollary 1.4 were previously established by Baouendi, Rothschild and Zaitsev [START_REF] Baouendi | Points in general position in real-analytic submanifolds in C N and applications[END_REF] and more recently by Lamel and the author in [START_REF] Lamel | Holomorphic versus algebraic equivalence for deformations of realalgebraic CR manifolds[END_REF]. In these two works, the conclusion given by Corollary 1.4 is obtained for all points p in a certain Zariski open subset of M . The proofs of [START_REF] Baouendi | Points in general position in real-analytic submanifolds in C N and applications[END_REF][START_REF] Lamel | Holomorphic versus algebraic equivalence for deformations of realalgebraic CR manifolds[END_REF] require to exclude from M a thin set of points corresponding to the locus of the degeneracy set of a certain holomorphic foliation. Corollary 1.4 answers one of the main questions left open from [START_REF] Baouendi | Points in general position in real-analytic submanifolds in C N and applications[END_REF][START_REF] Lamel | Holomorphic versus algebraic equivalence for deformations of realalgebraic CR manifolds[END_REF] which was to decide whether one could get rid off this locus.

Let us now discuss briefly the proof of Theorem 1.1. If M is as in Theorem 1.1 and p ∈ M , we assume that p = 0 and we may view the germ of M at 0 as a (small) algebraic deformation of its CR orbits. Namely, there exists an integer c ∈ {0, . . . , N } and a real-algebraic submersion S : (M, 0) → (R c , 0) such that S -1 (S(q)) = O q for all q ∈ M near 0. The level sets of S, i.e. the CR orbits, therefore foliate M near the origin by minimal real-algebraic CR submanifolds. We may hence identify the germ of M at 0 with an algebraic deformation (M t ) for t ∈ R c close to 0, where M t ⊂ C N -c is a germ at 0 of a minimal real-algebraic CR submanifold (see e.g. Lemma 2.7). If f :

(C N -c z × C c u , 0) → C N is
a germ of a holomorphic map such that Graph f ∩(M ×C N ) ⊂ S for some real-algebraic subset S ⊂ C N ×C N , then for every t ∈ R c sufficiently small, the holomorphic map

f t : (C N -c z , 0) z → f (z, t) ∈ C N satisfies Graph f t ∩ (M t × C N ) ⊂ S t where S t := {(z, z ) ∈ C N -c × C N : (z, t, z ) ∈ S }.
Since each submanifold M t is minimal, the conclusion of Theorem 1.1 boils down to providing a deformation version of the approximation theorem of [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF][START_REF] Meylan | On some rigidity properties of mappings between CRsubmanifolds in complex space[END_REF] associated to the deformation (M t ) t∈R c , the analytic family of holomorphic maps (f t ) t∈R c and the family of realalgebraic subsets (S t ) t∈R c .

To this end, we devote one part of the paper, namely Section 4, to build a system of polynomial equations with real-algebraic coefficients fulfilled by the restrictions to R N and R c of f and J(u) := (∂ α f (0, u); |α| ≤ k) for some suitable integer k. The system is constructed in such a way that it reduces our approximation problem to an application of a version of Artin's approximation theorem due to Popescu [P86] (see [START_REF] Lamel | Holomorphic versus algebraic equivalence for deformations of realalgebraic CR manifolds[END_REF] where the use of [P86] appeared for the first time in the subject). To construct the desired system, we introduce a suitable field of partially algebraic power series, denoted by S f , depending on the mapping f and its derivatives. An appropriate and careful study of the field extension generated by the field S f and the components of the mapping f leads to the desired system of real-algebraic polynomial equations.

At this point, we should mention that the above strategy depends on a key technical proposition, Proposition 3.2, which is an algebraic dependence result for certain power series which might be of independent interest. We devote Section 3 to the proof of this result : it requires to introduce algebraic power series rings defined over a field that is itself a field of fractions of partially algebraic power series and depending on the fixed mapping f . During the proof of Proposition 3.2, we adapt several tools concerning ratios of power series developed in [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF] and also make use of Artin's approximation theorem over the above defined field of partially algebraic power series. We should point out that, under the additional assumption that the source manifold M in Theorem 1.1 is connected and contains minimal points, most of the tools developed in this paper are unecessary and the proof can be simplified and reduced to that of [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF][START_REF] Meylan | On some rigidity properties of mappings between CRsubmanifolds in complex space[END_REF].

The paper is organized as follows. In Section 2, we introduce the notation and preliminary notions that will be used throughout the article. Section 3 is devoted to the proof of one key algebraicity property for certain ratios of power series that is used in Section 4 while building the system of real-algebraic equations associated to a given mapping f . The proof of Theorem 1.1 is completed in Section 5.

Preliminaries

2.1. Algebraic power series and two approximation theorems for polynomial systems. Throughout the paper, given a field K, and indeterminates x = (x 1 , . . . , x r ), r ≥ 1, we denote by K[[x]] the ring of formal power series with coefficients in the ring K. For T (x) ∈ K [[x]] and an integer m, we write j m T or (j m T )(x) for the collection of all partial derivatives of T up to order m. If the indeterminates x split as x = (x , x ), we write (j m T )(0, x ) for the power series mapping (j m T ) evaluated at x = 0. We will also be using the notation j m

x T for the collection of all partial derivatives with respect to x up to order m.

We shall frequently make use of the ring of algebraic power series over K : it is the subring

A K {x} of K[[x]] consisting of those T (x) ∈ K[[x]]
for which there exist a positive integer m and, for j = 0, . . . , m, polynomials P j ∈ K[x] with P m = 0 such that m j=0 P j (x)T (x) j = 0.

Note that when K = C, A C {x} is the usual ring of (complex-)algebraic (or Nash) functions over C, which is contained in the ring of convergent power series denoted in this paper by C{x}. Given any convergent power series η = η(x) ∈ C{x}, we will also denote throughout the paper by η = η(x) the convergent power series obtained from η by taking complex conjugates of its coefficients.

In this work, we will make use of two approximation theorems for polynomial systems of equations with algebraic coefficients. The first one is due to Artin [A69].

Theorem 2.1. (Artin [A69]) Let K be a field, P (x, y) = (P 1 (x, y), . . . , P m (x, y)) be m polynomials in the ring A K {x}[y] with x = (x 1 , . . . , x r ), y = (y 1 , . . . , y q ). Suppose that Y (x) = (Y 1 (x), . . . , Y q (x)) ∈ (K[[x]]) q is a formal solution of the system (2.1) P (x, Y (x)) = 0.

Then, for every integer , there exists Y (x) ∈ (A K {x}) q satisfying the system (2.1) such that Y (x) agrees with Y (x) up to order .

The second one is a more precise version of Theorem 2.1 and corresponds to Popescu's solution of the approximation problem on nested subrings.

Theorem 2.2. (Popescu [P86]) Let K be a field, P (x, y) = (P 1 (x, y), . . . , P m (x, y)) be m polynomials in the ring A K {x}[y] with x = (x 1 , . . . , x r ), y = (y 1 , . . . , y q ). Suppose that

Y (x) = (Y 1 (x), . . . , Y q (x)) ∈ (K[[x]]) q is a formal solution of the system (2.2) P (x, Y (x)) = 0.
Suppose furthermore that for every j = 1, . . . , q, there exists s j ∈ {1, . . . , r} such that

Y j (x) ∈ K[[x 1 , . . . , x s j ]].
Then, for every integer , there exists Y (x) ∈ (K{x}) q satisfying the system (2.2), such that Y (x) agrees with Y (x) up to order and each Y j (x) ∈ A K {x 1 , . . . , x s j }, j = 1, . . . , q.

Observe that by a noetherianity argument, Theorems 2.1 and 2.2 also hold for a polynomial system with infinitely many equations.

2.2. K-algebraicity of ratios of formal power series. In Section 3, we will have to study certain types of ratios of formal power series defined over a ground field that is not the usual field of complex numbers, but, rather a field extension over C. In order to prove a key property of these ratios, we shall use and modify some concepts about ratios of formal power series introduced in [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF] for the case of C[[x]] to treat ratios of power series over K [[x]] where C → K is a field extension.

Let C → K be a field extension. Given two power series N (x), D(x) ∈ K[[x]], x = (x 1 , . . . , x r ), we write (N : D) for a pair of formal power series where we both allow N and D to be zero. When D ≡ 0, we can think of (N : D) as being the usual ratio N/D. In the following, we say that two ratios (N 1 : D 1 ) and (N 2 : D 2 ) of formal power series in K [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF], it will be useful to introduce the following notion.

[[x]] are equivalent if N 1 D 2 -N 2 D 1 = 0. As in
Definition 2.3. Given two ratios (N 1 : D 1 ) and (N 2 : D 2 ) of formal power series in K[[x]], a complex linear subspace E ⊂ C r and a nonnegative integer q, we say that (N 1 : D 1 ) and (N 2 : D 2 ) are q-similar along

E if (j q (N 1 D 2 -N 2 D 1 ))| E = 0.
We have the following (weak) transitivity property of the notion of q-similarity, whose simple proof is completely analogous to that of [MMZ03a, Lemmas 3.2 and 3.3] and is left to the reader.

Lemma 2.4. Let (N 1 : D 1 ), (N 2 : D 2 ) and (N 3 : D 3 ) be three ratios of formal power series in K [[x]]. Suppose that we have a splitting of the indeterminates of the form x = (u 1 , u 2 , u 3 ) and let E be the linear subspace of C r given by E = {u 1 = 0, u 2 = 0}. Suppose that there exist an integer and integers q 1 , q 2 ≥ such that

(j (N 2 , D 2 ))| E = 0, (j q 1 u 1 j q 2 u 2 (N 1 D 2 -N 2 D 1 ))| E = 0, (j q 1 u 1 j q 2 u 2 (N 3 D 2 -N 2 D 3 ))| E = 0. Then (j q 1 - u 1 j q 2 - u 2 (N 3 D 1 -N 1 D 3 ))| E = 0.
We now define the notion of K-algebraicity for (formal) power series along a certain complex subspace that will be extremely useful in Section 3. Definition 2.5. Given a ratio (N : D) of formal power series in K[[x]] and a complex linear subspace E ⊂ C r , we say that (N : D) is K-algebraic along E if there exists an integer and for every integer q, formal power series N q , D q ∈ A K {x} such that (N : D) and (N q : D q ) are q-similar along E and such that (j (N q , D q ))| E ≡ 0.

Remark 2.6. Using Lemma 2.4, it is easy to see that if two ratios of formal power series in K[[x]] are equivalent and if one of them is nontrivial and K-algebraic along a complex subspace E ⊂ C r then the other ratio has the same property.

2.3. Generic real-algebraic submanifolds and local CR orbits. Let M ⊂ C N be a real-algebraic CR submanifold with N ≥ 2 and T 0,1 M its CR bundle. For every point p ∈ M , we denote by G M (p) the Lie algebra evaluated at p generated by the sections of T 0,1 M and its conjugate T 1,0 M . By a theorem of Nagano (see e.g. [START_REF] Baouendi | Real Submanifolds in Complex Space and Their Mappings[END_REF][START_REF] Berhanu | An introduction to involutive structures[END_REF]), for every point p ∈ M , there is a well-defined unique germ at p of a real-analytic submanifold V p satisfying CT q V p = G M (q) for all q ∈ V p . This unique submanifold is necessarily CR and is called the CR orbit of M at p. In fact, by [BER96, Corollary 2.2.5], this CR orbit is even a real-algebraic CR submanifold contained in M . It is not difficult to see that if M is connected, the dimension of the local CR orbits is constant (and of maximal dimension) except possibly on a proper real-algebraic subvariety Σ M of M (see e.g. [START_REF] Baouendi | Equivalences of real submanifolds in complex space[END_REF][START_REF] Baouendi | Points in general position in real-analytic submanifolds in C N and applications[END_REF]). If dim V p = dim M for some point p ∈ M , we say that M is of minimal (or also of finite type) at p.

Assume in what follows that M is a (connected) real-algebraic generic submanifold of C N of CR dimension n and codimension d. If p is a point in M \ Σ M , where Σ M is defined as above, then one may choose holomorphic (algebraic) coordinates such that M is described near p through the following well-known lemma.

Lemma 2.7. ([BRZ01a, Proposition 3.4] and [BER96, Lemma 3.4.1]) Let M ⊂ C N be a connected generic real-algebraic submanifold through a point p ∈ M whose CR orbit at p is of maximal dimension and let c ∈ {0, . . . , d} be the codimension of this CR orbit in M . Then there exists normal algebraic coordinates

Z = (z, η) ∈ C n × C d , η = (w, u) ∈ C d-c × C c ,
such that M is given near the origin by an equation of the form

(2.3) η = (w, u) = Θ(z, z, η) := (Q(z, z, w, ū), ū),
where Q is a C d-c -valued complex-algebraic map near 0 ∈ C n+N . Furthermore, there exist neighborhoods U, V of the origin in R c and C N -c respectively such that for every u ∈ U , the real-algebraic submanifold given by

(2.4) M u := {(z, w) ∈ V : w = Q(z, z, w, u)}
is generic in C N -c and minimal at 0.

As a real-analytic submanifold, M can be complexified and gives rise to its socalled complexification, which we denote by M. This complexification M is the germ at 0 of the complex-algebraic submanifold of C 2N given by (2.5)

{(Z, ζ) ∈ (C N × C N , 0) : σ = Θ(χ, z, η)}, where Z = (z, η) ∈ C n × C d and ζ = (χ, σ) ∈ C n × C d .
Recall also that the above choice of normal coordinates imply that the following two identities hold

(2.6) Θ(z, 0, σ) = Θ(0, χ, σ) = σ, Θ(z, χ, Θ(χ, z, η)) = η.
In what follows, we pick a point p ∈ M \ Σ M and choose normal coordinates for M at p as given by Lemma 2.7. In Section 3, we will need the iterated Segre mappings attached to such a germ of submanifold (see e.g. [START_REF] Baouendi | Equivalences of real submanifolds in complex space[END_REF]). These mappings are defined in the following way. For any nonnegative integer j, we denote by t j a variable lying in C n and also introduce the variable t [j] := (t 1 , . . . , t j ) ∈ C nj . We also use the notation s for a variable lying in the euclidean space C d-c . Then we first set v 0 (s, u) := (0, s, u) for (s, u) ∈ C d sufficiently close to 0 and define the map

v j : (C nj × C d-c × C c , 0) → C N for j ≥ 1 inductively as follows: (2.7) v j (t [j]
, s, u) := (t j , U j (t [j] , s, u)), where U j (t [j] , s, u) := Θ(t j , vj-1 (t [j-1] , s, u)).

From the construction we clearly see that each iterated Segre mapping v j defines an algebraic map in a neighbhorhood of 0 in C nj+d . In fact, for every point (s, u) ∈ C d sufficiently close to the origin, the map v j (•, s, u) parametrizes the usual Segre set of order j attached to the point (0, s, u) (see e.g. [START_REF] Baouendi | Algebraicity of holomorphic mappings between real algebraic sets in C n[END_REF][START_REF] Baouendi | Real Submanifolds in Complex Space and Their Mappings[END_REF]). Notice also that, thanks to (2.6), one has the following useful identities (2.8) v j (0, s, u) = (0, s, u), v j+2 (t [j+2] , s, u)| t j+2 =t j = v j (t [j] , s, u), j ≥ 0, and that for every j ≥ 0, the germ at 0 of the holomorphic map (v j , vj-1 ) takes its values in M, where M is the complexification of M as defined by (2.5).

3. An algebraicity property for certain ratios of power series on real-algebraic generic submanifolds

Throughout this section, we assume that M is a germ of a (connected) realalgebraic generic submanifold through the origin in C N with N ≥ 2 and that f : (C N , 0) → C N is a germ of a holomorphic mapping. We also assume that the CR orbit of M at the origin is of maximal dimension and that normal coordinates Z = (z, w, u) for (M, 0) have been chosen (and fixed) as in Lemma 2.7. In such a setting, we are going to prove a crucial algebraicity property for certain ratios of (convergent) power series constructed from the mapping f and whose restriction on M is CR. In order to state the algebraic criterion given by Proposition 3.2 below, we need to define the following ring of power series that depend on the above fixed choice of normal coordinates Z = (z, w, u). Definition 3.1. Let f and Z = (z, w, u) be as above. Let B f be the subring of C{u}[z, w] consisting of those power series T (z, w, u) for which there exists an integer k and

S ∈ C[z, w, u, Λ k , Γ k ] such that T (z, w, u) = S(z, w, u, (j k f )(0, u), (j k f )(0, u)).
The field of fractions of B f will be denoted by S f . In what follows, we say that a holomorphic vector X defined near 0

∈ C N Z × C N ζ is a (0, 1) vector field if it annihilates the natural projection C N ×C N (Z, ζ) → Z ∈ C N .
The goal of this section is to prove the following: Proposition 3.2. Let M , f and Z = (z, w, u) be as above. Let ϕ 1 , ϕ 2 ∈ C{Z, ζ} with ϕ 2 | M ≡ 0, where M is the complexification of M as given by (2.5). Assume that there exist an integer k, and power series

Φ 1 , Φ 2 ∈ A C {Z, ζ}[Λ k , Γ k , Υ k ] such that (3.1) ϕ j (Z, ζ) = Φ j (Z, ζ, (j k f )(0, u), (j k f )(0, u), (j k f )(ζ))), j = 1, 2.
Assume furthermore that the ratio ϕ 1 /ϕ 2 is annihilated by any (0, 1) holomorphic vector field tangent to M near 0. Then ϕ 1 /ϕ 2 is algebraic over the field S f (as introduced in Definition 3.1).

Other (different) types of CR ratios of (formal) power series on real-analytic generic submanifolds have been studied in [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF]. Though the technique used in [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF] deals only minimal generic submanifolds, we will use part of this technique to prove Proposition 3.2.

3.2. Proof of Proposition 3.2. We assume in what follows that we are in the setting of Proposition 3.2. In order to prove Proposition 3.2, we need to work in appropriate rings of formal power series. All these formal power series will have the same ground field that is defined as follows.

Definition 3.3. Let D f u be the subring of C{u} consisting of those (convergent) power series T (u) for which there exists an integer k and

S ∈ A C {u}[Λ k , Γ k ] such that T (u) = S(u, (j k f )(0, u), (j k f )(0, u)).
The field of fractions of the ring D f u will be denoted by K f u . The first step in the proof of Proposition 3.2 relies on the following lemma.

Lemma 3.4. In the setting of Proposition 3.2, there exists power series

∆ 1 (Z), ∆ 2 (Z) ∈ C{Z} ∩ K f u [[z, w]
] such that ∆ 2 ≡ 0 and such that the ratios (ϕ 1 : ϕ 2 ) and (∆ 1 : ∆ 2 ) are equivalent. In addition, the ratio (∆ 1 : ∆ 2 ) (viewed as a ratio of power series in

K f u [[z, w]]
) is K f u -algebraic along the (n + c)-dimensional complex subspace {w = 0} (as explained in Definition 2.5).

Proof.

By assumption, we know that

(3.2) ∂ ∂χ ϕ 1 (Z, χ, Θ(χ, Z)) ϕ 2 (Z, χ, Θ(χ, Z)) = 0.
Since ϕ 2 | M = 0, we can choose a multiindex β ∈ N n of minimal length such that

∂ |β| χ ϕ 2 (Z, χ, Θ(χ, Z)) χ=0 ≡ 0. Setting ∆ 1 (Z) := ∂ |β| χ ϕ 1 (Z, χ, Θ(χ, Z)) χ=0 , ∆ 2 (Z) := ∂ |β| χ ϕ 2 (Z, χ, Θ(χ, Z)) χ=0
, it follows from (3.2) that (∆ 1 : ∆ 2 ) is a ratio of convergent power series that is equivalent to (ϕ 1 : ϕ 2 ). In addition, it follows from the chain rule and from (3.1) that there exists power series

Φ j ∈ A C {Z}[Λ k , Γ k , Υ k+|β| ], j = 1, 2, such that (3.3) ∆ j (Z) = Φ j (Z, (j k f )(0, u), (j k f )(0, u), (j k+|β| f )(0, w, u))), j = 1, 2.
Then (3.3) readily implies that both power series ∆ 1 , ∆ 2 belong to

K f u [[z, w]].
In fact, one may even notice that for every integer q, if ∆ 1,q and ∆ 2,q denote the truncated power series

∆ 1,q (z, w, u) = ν∈N d-c |ν|≤q ∂ ν ∆ 1 ∂w ν (z, 0, u) w ν ν! , ∆ 2,q (z, w, u) = ν∈N d-c |ν|≤q ∂ ν ∆ 2 ∂w ν (z, 0, u) w ν ν! , then ∆ 1,q , ∆ 2,q ∈ A K f
u {z, w}, and (∆ 1,q : ∆ 2,q ) and (∆ 1 : ∆ 2 ) are q-similar along the subspace L := {w = 0}. Furthermore, since there exists an integer n 0 such that (j n 0 (∆ 1 , ∆ 2 )) L ≡ 0, it follows that for all q ≥ n 0 , we have (j n 0 (∆ 1,q , ∆ 2,q )) L ≡ 0, which shows that the ratio (∆ 1 , ∆ 2 ) is K f u -algebraic along the complex subspace L. The proof of Lemma 3.4 is complete.

For the second step of the proof of Proposition 3.2, we shall use the iterated Segre maps introduced in section 2.3. For any integer j, recall that v j is the mapping given by (2.7). Note that since each map v j is algebraic and since the power series ∆ 1 = ∆ 1 (z, w, u) and ∆ 2 = ∆ 2 (z, w, u) given by Lemma 3.4 belong to the ring K f u [[z, w]], it follows from the first identity of (2.8) that the power series ∆ 1

•v j = (∆ 1 •v j )(t [j] , s, u) and ∆ 2 • v j = (∆ 2 • v j )(t [j] , s, u) both belong to the ring K f u [[t [j] , s]].
In what follows, when writing any iterated map v j , we shall sometimes omit to write the variables for sake of brevity.

Lemma 3.5. In the setting of Proposition 3.2, let j be a positive integer and assume that the ratio

(∆ 1 • v j : ∆ 2 • v j ) of power series in K f u [[t [j] , s]] is K f u -algebraic along the complex subspace E j := {s = 0} ⊂ C nj+d-c . Then the ratio (∆ 1 • v j+2 : ∆ 2 • v j+2 ) of power series in K f u [[t [j+2] , s]] is K f u -algebraic along the complex subspace E j+2 := {s = 0} ⊂ C n(j+2)+d-c .
In order to prove Lemma 3.5, we need the following preliminary result.

Lemma 3.6. Under the assumptions of Lemma 3.5, the ratio

(∆ 1 • v j+2 : ∆ 2 • v j+2 ) is K f u -algebraic along the complex subspace E j+2 := {s = 0, t j+2 = t j } ⊂ C n(j+2)+d-c .
Proof of Lemma 3.6. By assumption, there exist an integer 0 and for every integer q, (formal) power series N q , D q ∈ A K f u {t [j] , s} such that (3.4) (j 0 (N q , D q )) E j ≡ 0, (j q ((∆

2 • v j )N q -(∆ 1 • v j )D q )) E j = 0.
Consider the algebraic map I : (C nj+d , 0) (t [j] , s, u) → C nj+d given by (3.5) I(t [j] , s, u) := (t [j] , U j (t j , t j-1 , . . . , t 1 , s, u)) for j odd, (t [j] , U j (t j , t j-1 , . . . , t 1 , s, u)) for j even, where U j is the mapping given by (2.7). The reader can check that, by using the second identity of (2.6), the map I satisfies the following properties

(3.6) v j (I(t [j] , s, u)) = (t j , s, u), I(t [j-1] , v j (t [j] , s, u)) = (t [j] , s, u).
Consider the algebraic map J(t [j+2] , s, u)

:= I(t [j-1] , v j+2 (t [j+2]
, s, u)) and, for every integer q, define N q := N q • J and D q := D q • J. Since J(0, 0, u) = (0, 0, u) and since

J ∈ A C {t [j+2]
, s, u}, the power series N q and D q belong to the ring j+2] , s]]. In fact, noticing that the last C c -valued component of J(t [j+2] , s, u) is equal to u and using the fact that N q , D q ∈ A K f u {t [j] , s} and the second identity of (3.6), we see that N q , D q ∈ A K f u {t [j+2] , s}. Furthermore, it follows from (3.6) that v j+2 = v j • J and therefore we have

K f u [[t [
(3.7) (∆ 2 • v j+2 ) N q -(∆ 1 • v j+2 ) D q = ((∆ 2 • v j )N q -(∆ 1 • v j )D q ) • J.
Noticing furthermore that (3.6) and (2.8) imply that J( E j+2 ) ⊂ E j , it follows from (3.4) and (3.7) that for every integer q, the ratios (∆ 1 • v j+2 : ∆ 2 • v j+2 ) and ( N q : D q ) are q-similar along E j+2 . In addition, the second identity in (3.6) implies N q = N q t j+2 =t j and D q = D q t j+2 =t j which shows, in view of the first identity in (3.4) that (j 0 ( N q , D q )) E j+2 ≡ 0. This proves Lemma 3.6.

Proof of Lemma 3.5. We first note that in view of (3.1), there exists power series Φ 1,j+2 and Φ 2,j+2 both in the ring

A C {t [j+2] , s, u}[Λ k , Γ k , Υ k ] such that for r = 1, 2 one has (3.8) ϕ r,j+2 (t [j+2] , s, u) := ϕ r (v j+2 , vj+1 ) = Φ r,j+2 (t [j+2] , s, u, (j k f )(0, u), (j k f )(0, u), (j k f ) • vj+1 )).
Note in addition that the (convergent) power series ϕ r,j+2 , r = 1, 2, both belong to the ring K f u [[t [j+2] , s]]. Using the fact that ϕ 2 M ≡ 0 and that the mapping (v j+2 , vj+1 ) is clearly submersive from C n(j+2)+d into C N , we get that there exists an integer 1 such that (3.9) (j 1 (ϕ 1,j+2 , ϕ 2,j+2 )) E j+2 ≡ 0.

Furthermore, since the ratios (ϕ 1 : ϕ 2 ) and (∆ 1 : ∆ 2 ) are equivalent by Lemma 3.4, it follows that the ratios (ϕ 1,j+2 : ϕ 2,j+2 ) and (∆ 1 • v j+2 : ∆ 2 • v j+2 ) are also equivalent. Hence from Lemma 3.6 and Remark 2.6, we get that the ratio (ϕ 1,j+2 : ϕ 2,j+2 ) is

K f u -algebraic along E j+2 .
We are now going to follow the strategy of the proof of [MMZ03a, Lemma 3.8]. There exists an integer 2 and for every integer q, power series A q , B q ∈ A K f u {t j+2 , s} such that the ratios (A q : B q ) and (ϕ 1,j+2 : ϕ 2,j+2 ) are q-similar along E j+2 with (3.10) (j 2 (A q , B q )) E j+2 ≡ 0.

Furthermore, we may assume without loss of generality that 2 ≥ 1 . For the rest of the proof, we fix an arbitrary integer m ≥ 2 and set Y (t [j+1] , s, u) := (j k f ) • vj+1 .

Then for every integer q ≥ m + 2 , we have

(3.11) (j q-m t j+2 j m s (A q ϕ 2,j+2 -ϕ 1,j+2 B q )) E j+2 = 0.
Since A q and B q are both in the ring A

K f u {t [j+2]
, s} and in view of (3.8), we may rewrite (3.11) as follows (3.12) S q,m (t [j+1] , u, (j m s Y )(t [j+1] , 0, u)) = 0, ∀q ≥ m + 2 , where S q,m is a polynomial in its last argument with coefficients in the ring A

K f u {t [j+1] }.
By construction, the power series mapping (j m s Y )(t [j+1] , 0, u) belongs to the ring

K f u [[t j+1 ]
] and is a formal solution of the polynomial system of equations with coefficients in A K f u {t [j+1] } provided by (3.12). By Theorem 2.1, for every integer r, we may find a power series mapping Y r (t

[j+1] , u) ∈ A K f u {t [j+1] } such that (3.13) (j r t [j+1] Y r ) t [j+1] =0 = j r t [j+1] (j m s Y ) s=0 t [j+1] =0 and satisfying (3.14) S q,m (t [j+1] , u, Y r (t [j+1] , u)) = 0, ∀q ≥ m + 2 . Choose T r ∈ A K f u {t [j+1] }[s]
such that (j m s T r ) s=0 = Y r and set for e = 1, 2 (3.15)

ϕ r e,j+2 (t [j+2] , s, u) := Φ e,j+2 (t [j+2] , s, u, (j k f )(0, u), (j k f )(0, u), T r (t [j+1] , s, u))). In view of (3.9), (3.13) and the fact that m ≥ 2 ≥ 1 , we may choose r such that (3.16) (j 1 (ϕ r 1,j+2 , ϕ r 2,j+2 )) E j+2 ≡ 0, and keep such a choice of r for the remainder of this proof. From the construction of the power series mapping T r , it follows that ϕ r 1,j+2 , ϕ r 2,j+2 ∈ A

K f u {t [j+2]
} and that the following identity holds

(3.17) (j q-m t j+2 j m s (A q ϕ r 2,j+2 -ϕ r 1,j+2 B q )) E j+2 = 0, ∀q ≥ m + 2 .
Lemma 2.4 together with (3.11), (3.17) and (3.10) implies

(3.18) (j q-m-2 t j+2 j m-2 s (ϕ 1,j+2 ϕ r 2,j+2 -ϕ r 1,j+2 ϕ 2,j+2 )) E j+2 = 0, ∀q ≥ m + 2 .
Hence, (3.18) obviously implies

(3.19) (j m-2 s (ϕ 1,j+2 ϕ r 2,j+2 -ϕ r 1,j+2 ϕ 2,j+2
)) E j+2 = 0, which shows that the ratios (ϕ 1,j+2 : ϕ 2,j+2 ) and (ϕ r 1,j+2 : ϕ r 2,j+2 ) are (m -2 )-similar along E j+2 . Since (3.16) immediately implies that (j 1 (ϕ r 1,j+2 , ϕ r 2,j+2 )) E j+2 ≡ 0, we obtain that the ratio (ϕ 1,j+2 :

ϕ 2,j+2 ) is K f u -algebraic along E j+2 . Since the ratio (∆ 1 • v j+2 : ∆ 2 • v j+2 ) is equivalent to (ϕ 1,j+2 : ϕ 2,j+2
), the conclusion of Lemma 3.5 follows from Remark 2.6.

Completion of the proof of Proposition 3.2. In view of Lemma 3.4, we have to show that ∆ 1 /∆ 2 is algebraic over the field S f . We start by noticing that Lemma 3.4 provides the fact the ratio (∆

1 • v 1 : ∆ 2 • v 1 ) of power series in K f u [[t 1 , s]
] is K f ualgebraic along the subspace E 1 := {s = 0}. Applying Lemma 3.5, we get that for every odd integer, the ratio (∆ 1 •v j : ∆ 2 •v j ) (of power series in the ring K f u [[t [j] , s]]) is K f u -algebraic along the complex subspace E j := {s = 0} ⊂ C nj+d-c . Choose j = 2d+3 for the remainder of this proof, where we recall that d is the codimension of M in C N . There exists an integer 3 and for every integer q, power series N q , D q ∈ A K f u {t [2d+3] , s} such that the ratios (N q : D q ) and (Θ • v 2d+3 : ∆ • v 2d+3 ) are q-similar along the subspace {s = 0} with (3.20) (j 3 (N q , D q )) s=0 ≡ 0.

Take q = 3 and choose

β 0 ∈ N (2d+3)n+d-c with |β 0 | ≤ 3 of minimal length such that (∂ β 0 N 3 , ∂ β 0 D 3 ) s=0 ≡ 0.
Then we have

(3.21) (∆ 2 • v 2j+3 ) ∂ β 0 N 3 -(∆ 1 • v 2j+3 ) ∂ β 0 D 3 s=0 = 0.
Since the manifold M 0 given by Lemma 2.7 is minimal at the origin, the minimality criterion given in [START_REF] Baouendi | Algebraicity of holomorphic mappings between real algebraic sets in C n[END_REF][START_REF] Baouendi | Real Submanifolds in Complex Space and Their Mappings[END_REF] shows that the mapping 2d+3] , 0, u) ∈ C N is of generic rank N . Hence (3.21) implies that (∂ β 0 D 3 ) s=0 ≡ 0 and that the following identity holds in the quotient field of

C (2d+3)n+c (t [2d+3 ], u) → v 2d+3 (t [
K f u [[t 2d+3 ]] (3.22) (∆ 1 • v 2j+3 )(t [2d+3] , 0, u) (∆ 2 • v 2j+3 )(t [2d+3] , 0, u) = (∂ β 0 N 3 )(t [2d+3] , 0, u) (∂ β 0 D 3 )(t [2d+3] , 0, u) . Since N 3 , D 3 ∈ A K f u {t [2d+3]
, s}, it follows from (3.22) that the germ of the meromorphic function given by the left hand side of (3.22) is also algebraic over the quotient field of the ring K f u [t [2d+3] ]. Therefore there exist two positive integers a, b and, for µ = 0, . . . , a, polynomials

P µ = P µ (u, t [2d+3] , Λ b , Γ b ) ∈ A C {u}[t [2d+3] , Λ b , Γ b ] such that P a (u, t [2d+3] , (j b f )(0, u), (j b f )(0, u)) ≡ 0
and such that the following identity holds (3.23)

a µ=0 P µ u, t [2d+3] , (j b f )(0, u), (j b f )(0, u) (∆ 1 • v 2j+3 )(t [2d+3] , 0, u) (∆ 2 • v 2j+3 )(t [2d+3] , 0, u) µ = 0.
In fact, the above mentioned minimality criterion even provides points arbitrarily close to 0 in C 2d+3 such that the (algebraic) map t [2d+3] → v 2d+3 (t [2d+3] , 0, 0) sends those points to the origin and has rank N -c there. Pick a point T 0 with this property such that the left hand side of (3.22) is meromorphic in an open neighborhood V ×W of the origin in C (2d+3)n × C c with T 0 ∈ V . From the rank theorem, we may find an algebraic map λ : (C N , 0) → (C (2d+3)n , T 0 ) such that v 2d+3 (λ(z, w, u), 0, u) = (z, w, u). Composing (3.23) with the algebraic mapping λ, we get (3.24)

a µ=0 P µ u, λ(z, w, u), (j b f )(0, u), (j b f )(0, u) ∆ 1 (z, w, u) ∆ 2 (z, w, u) µ = 0,
for (z, w, u) ∈ C N sufficiently close to the origin. It is easy to see that one may choose the mapping λ so that P a u, λ(z, w, u), (j b f )(0, u), (j b f )(0, u) ≡ 0, which proves that ∆ 1 /∆ 2 is algebraic over the quotient field of A K f u {z, w}. Since this latter field is algebraic over the quotient field of K f u [z, w], which is itself algebraic over the field S f , we see that the proof of Proposition 3.2 is complete.

3.

3. An application of the algebraicity property. We shall now provide an application of Proposition 3.2 in the spirit of [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF]Proposition 4.3]. This result will be one of the key point of the proof of Theorem 1.1. Proposition 3.7. Let f : (C N , 0) → C N be a germ of a holomorphic mapping and M be a connected real-algebraic generic submanifold through the origin. Assume that the CR orbit of M at 0 is of maximal dimension and choose normal coordinates Z = (z, w, u) for M near 0 as in Lemma 2.7. Let M be the complexification of M as defined in (2.5) and assume the mapping f splits as follows f = (g, h) ∈ C e × C N -e for some integer e ∈ {1, . . . , N }. Assume also that there exists an integer k 0 and a

polynomial R ∈ A C {Z, ζ}[Λ k 0 , Γ k 0 ][ξ , ] where (ξ , ) ∈ C e × C e such that: (i) R(Z, ζ, (j k 0 f )(0, u), (j k 0 f )(0, u), ξ , ) ≡ 0 (Z, ζ, ξ , ) ∈ M × C e × C e near 0, (ii) R(Z, ζ, (j k 0 f )(0, u), (j k 0 f )(0, u), g(Z), ḡ(ζ)) = 0 for (Z, ζ) ∈ M near 0.
Then the components of the mapping g are algebraically dependent over the field S f (as introduced in Definition 3.1).

Proof. Let E be the set of all polynomials B ∈ C{Z, ζ}[ξ ] such that there exist an integer m and

D ∈ A C {Z, ζ}[Λ m , Γ m , Υ m ][ξ ] such that B(Z, ζ, ξ ) = D(Z, ζ, (j m f )(0, u), (j m f )(0, u), (j m f )(ζ), ξ ), with B(Z, ζ, ξ ) ≡ 0 for (Z, ζ, ξ ) ∈ M × C N near 0 and such that B(Z, ζ, g(Z)) = 0 for (Z, ζ) ∈ M near 0.
We claim that E is not empty. Indeed, let R as by the assumption of the lemma. There are two cases to consider. On one hand, if

R(Z, ζ, (j k 0 f )(0, u), (j k 0 f )(0, u), ξ , ḡ(ζ)) ≡ 0, (Z, ζ, ξ ) ∈ M × C e near 0,
then we are clearly done. On the other hand, if for (Z, ζ, ξ

) ∈ M × C e near 0 R(Z, ζ, (j k 0 f )(0, u), (j k 0 f )(0, u), ξ , ḡ(ζ)) = 0, then we also have R(ζ, Z, (j k 0 f )(0, u), (j k 0 f )(0, u), , g(Z)) = 0 for (Z, ζ, ) ∈ M × C e . Writing R(ζ, Z, Γ k 0 , Λ k 0 , , ξ ) = α∈N N C α (ζ, Z, Γ k 0 , Λ k 0 , ξ )( ) α ,
where the above sum is finite and each

C α ∈ A C {Z, ζ}[Γ k 0 , Λ k 0 ][ξ ], assumption (i) implies that there exists β ∈ N N such that C β (ζ, Z, (j k 0 f )(0, u), (j k 0 f )(0, u), ξ ) ≡ 0 for (Z, ζ, ξ ) ∈ M × C e near 0. Since C β (ζ, Z, (j k 0 f )(0, u), (j k 0 f )(0, u), g(Z)) = 0 for (Z, ζ) ∈ M near 0, the claim is proved.
Since E is not empty, we may choose B 0 ∈ E with the minimum number r 0 of monomials in ξ . If r 0 = 1, we see that one component of the mapping g necessarily vanishes identically, which proves the proposition in this case. Otherwise, let m ∈ Z + such that we may write

B 0 (Z, ζ, ξ ) = D 0 (Z, ζ, (j m f )(0, u), (j m f )(0, u), (j m f )(ζ), ξ ) for some D 0 ∈ A C {Z, ζ}[Λ m , Γ m , Υ m ][ξ ].
We expand D 0 in monomials in ξ in the following way

D 0 (Z, ζ, Λ m , Γ m , Υ m , ξ ) = r 0 j=1 D 0,j (Z, ζ, Λ m , Γ m , Υ m ) E j (ξ ).
We also set for j = 1 . . . , r 0 , (3.25)

T j (Z, ζ) := D 0,j (Z, ζ, (j m f )(0, u), (j m f )(0, u), (j m f )(ζ)),
and note that from the definition of r 0 , it follows that T j (Z, ζ) ≡ 0 for (Z, ζ) ∈ M near 0 and for every j = 1, . . . , r 0 . Let L 1 , . . . , L n be a local basis of (0, 1) holomorphic vector fields tangent to M near 0, where we recall that n is the CR dimension of M . Since M is real-algebraic, we may assume, without loss of generality, that these vector fields have (complex-)algebraic coefficients. Since

(3.26) B 0 (Z, ζ, g(Z)) = r 0 j=1 T j (Z, ζ) E j (g(Z)) = 0, (Z, ζ) ∈ M near 0,
applying each vector L ν to this identity yields, (3.27)

r 0 -1 j=1 L ν T j (Z, ζ) T r 0 (Z, ζ) E j (g(Z)) = 0,
for ν = 1, . . . , n and (Z, ζ) ∈ M near 0. For every ν ∈ {1, . . . , n} and j = 1 . . . , r 0 -1, we note that the ratio of convergent power series L ν T j (Z,ζ) ≡ 0 for all ν = 1, . . . , r 0 -1. We may therefore apply Proposition 3.2 to conclude that each ratio T j /T r 0 is algebraic over the field S f (as given in Definition 3.1). In other words, we may say that in the quotient field of C{Z, ζ}, the subfield F generated by S f and all the ratios T j /T r 0 , j = 1 . . . , r 0 -1, is an algebraic extension of S f . Since (3.26) tells us that the components of the mapping g are algebraically dependent over F, it follows that they are also algebraically dependent over S f . The proof is therefore complete. 4. A polynomial system of equations with real-algebraic coefficients associated to a given holomorphic map

Tr 0 (Z,ζ) may be written in the form T j,ν (Z, ζ)/(T r 0 (Z, ζ)) 2 where T j,ν (Z, ζ) = G j,ν (Z, ζ, (j m+1 f )(0, u), (j m+1 f )(0, u), (j m+1 f )(ζ)) for some G j,ν ∈ A C {Z, ζ}[Λ m+1 , Γ m+1 , Υ m+1 ].
In this section, we assume that we are in the same situation as in the previous section. Namely, we assume that M is a germ of a (connected) real-algebraic generic submanifold through the origin in C N with N ≥ 2 and that f : (C N , 0) → C N is a germ of a holomorphic mapping. We also assume that the CR orbit of M at 0 is of maximal dimension and that normal coordinates Z = (z, w, u) ∈ C n × C d-c × C c for (M, 0) have been chosen (and fixed) as in Lemma 2.7.

Our goal in this section is to construct a system of real-algebraic equations over R N associated to the mapping f such that f | R N and (j k f )| {0}×R c are solutions of such a system for a suitable integer k. The constructed system will possess further additional properties (see Proposition 4.3 for the precise statement) that will be crucial in order to prove Theorem 1.1. 4.1. Field extension associated to a given holomorphic map. We refer the reader to [START_REF] Zariski | Commutative Algebra[END_REF] for the basic notions of field theory used in the remainder of this paper and keep using the notation introduced in previous sections.

Let S f (f ) denote the field generated by S f and all the components of the mapping f . Then denoting M{Z} the field of fractions of C{Z}, we have the following field extensions S f → S f (f ) → M{Z}. Let e = e(f ) ∈ {0, . . . , N } be te the transcendence degree of the field extension S f → S f (f ). We may therefore choose e components of the mapping f , denoted by g in the rest of this paper, such that g forms a transcendence basis of S f (f ) over S f . We shall assume in the rest of Section 4 that e < N .

If we write f = (g, h) ∈ C e ξ × C N -e ω
, then the components of g are all algebraically independent over S f and any other component of h is algebraically dependent over S f (g). Writing h = (h 1 , . . . , h N -e ) and ω = (ω 1 , . . . , ω N -e ), we may find an integer k and, for every j ∈ {1, . . . , N -e}, a polynomial

P j ∈ C[Z, Λ k , Γ k , ξ ][ω j ]
such that for all Z = (z, w, u) ∈ C N sufficiently close to the origin (4.1) P j (Z, (j k f )(0, u), (j k f )(0, u), g(Z), h j (Z)) = 0.

Furthermore, writing P j (Z, Λ k , Γ k , ξ , ω j ) = m j ν=0 P j,ν (Z, Λ k , Γ k , ξ )(ω j ) ν , we may assume that (4.2) P j,m j Z, (j k f )(0, u), (j k f )(0, u), g(Z) ≡ 0, j = 1, . . . , N -e.

We now come to a useful lemma that is a more refined version of [MMZ03a, Lemma 6.2] (or of [LM09, Lemma 4.3]) but whose proof is completely analogous. For sake of completeness, we provide the proof of this statement. In what follows, we write Z = (ξ , ω ) ∈ C e × C N -e and will denote by another variable lying in C e and θ = (θ 1 , . . . , θ N -e ) ∈ C N -e . We also denote k = card {α ∈ N N : |α| ≤ k}.

Lemma 4.1. In the above setting, for every real-valued polynomial ρ(Z, Z, Z , Z ) in C N × C N , there exists a nontrivial polynomial

Q ρ ∈ C[Z, ζ, Λ k , Γ k , ξ , ][X]
such that the following holds: for all germs of holomorphic maps H : (C N , 0) → C N -e , G : (C N , 0) → C e , T : (C c , 0) → C k satisfying P j (Z, T (u), T (u), G(Z), H j (Z)) ≡ 0, j = 1, . . . , N -e, and P j,m j Z, T (u), T (u), G(Z) ≡ 0,

then Q ρ (Z, ζ, T (u), T (u), G(Z), Ḡ(ζ), ρ(H(Z), G(Z), H(ζ), Ḡ(ζ))) ≡ 0,
for all (Z, ζ) ∈ C 2N near 0. Furthermore, we may write

Q ρ (Z, ζ, Λ k , Γ k , ξ , , X) = δ ν=0 Q ρ ν (Z, ζ, Λ k , Γ k , ξ , )X ν ,
with δ independent of ρ and with

Q ρ δ (Z, ζ, T (u), T (u), G(Z), Ḡ(ζ)) ≡ 0 for (Z, ζ) ∈ M near the origin.
Proof. For j = 1, . . . , N -e, consider

P j (Z, Λ k , Γ k , ξ , ω j ) = m j ν=0 P j,ν (Z, Λ k , Γ k , ξ )(ω j ) ν , V j (ζ, Λ k , Γ k , , θ j ) := m j ν=0 Pj,ν (ζ, Γ k , Λ k , )(θ j ) ν .
For every j ∈ {1, . . . , N -e}, let A j (resp. B j ) be the complex-algebraic variety in C N +2 k +e given by the zero set of the polynomial P j,m j (resp. Pj,m j ) and set

E := ∪ N -e j=1 A j ∪ ∪ N -e j=1 B j . For (Z, Λ k , Γ k , ξ ) ∈ C N +2 k +e \ E (resp. (ζ, Γ k , Λ k , ) ∈ C N +2 k +e
\E) and for every j ∈ {1, . . . , N -e}, denote by σ

(1)

j (Z, Λ k , Γ k , ξ ), . . . , σ (m j ) j (Z, Λ k , Γ k , ξ ) (resp. ς (1) j (ζ, Γ k , Λ k , ), . . . , ς (m j ) j (ζ, Γ k , Λ k , )
) the m j roots of the polynomial P j (resp. V j ).

Next for (Z, Λ k , Γ k , ξ ) and (ζ, Γ k , Λ k , ) as above, consider the following polynomial W in X

(4.3) W (Z, ζ, Λ k , Γ k , ξ , , X) := m 1 i 1 =1 . . . m N -e i N -e =1 m 1 n 1 =1 . . . m N -e n N -e =1 X -ρ(Z, ζ, ξ , σ (i 1 ) 1 , . . . , σ (i N -e )
N -e , , ς

(n 1 ) 1 , . . . , ς (n N -e ) N -e ) ,
where, for r = 1, . . . , N -e, we have written σ (ζ, Γ k , Λ k , ). It follows from Newton's theorem on symmetric polynomials that Then a convergent power series mapping T as in Lemma 4.2 satisfies (4.8) if and only if for every α, β ∈ N N -e , every ν ∈ {0, . . . , p ρ -1}, we have (4.9) Q ρ ν,α,β (Z, ζ, T (u), T (u)) ≡ 0, (Z, ζ) ∈ M, close to 0. Using Lemma 2.7, we may choose a parametrization (R 2N -d-c × R c , 0) (y, u) → ϕ(y, u) ∈ C N of M near 0 such that ϕ is real-algebraic and such that for each fixed u ∈ R c sufficiently close to the origin, the mapping (R 2N -d-c , 0) y → ϕ(y, u) parametrizes (the germ at 0 of) the manifold M u as defined in (2.4). Hence for all u ∈ C c sufficiently close to the origin and for every γ ∈ N 2N -d-c and every α, β, ν, we may set (4.10)

S ρ ν,α,β,γ (u, Λ k , Γ k ) := ∂ γ ∂y γ Q ρ ν,α,β (ϕ(y, u), φ(y, u), Λ k , Γ k ) y=0 .
From our construction, each

S ρ ν,α,β,γ ∈ A C {u}[Λ k , Γ k ].
Furthermore a convergent power series mapping T (u) as in Lemma 4.2 satisfies (4.9) if and only if for every γ ∈ N 2N -d-c and every α, β, ν as above S ρ ν,α,β,γ (u, T (u), T (u)) ≡ 0 for all u ∈ R c sufficiently close to the origin. By a noetherian argument, we may extract a finite collection of polynomials in the family (S ρ ν,α,β,γ ) providing the conclusion of the lemma. The proof is therefore complete.

We are now in a position to prove the main technical result of this section.

Proposition 4.3. Let M and f be above. Let e be the transcendence degree of the field extension S f → S f (f ) and assume that e < N . For j ∈ {1, . . . , N -e} and q ∈ {1, . . . , b}, let

P j ∈ C[Z, Λ k , Γ k , ξ ][ω j ] and S q ∈ A C {u}[Λ k , Γ k ]
given by (4.1) and Lemma 4.2 respectively. For x ∈ R n+d-c , u ∈ R c , consider the following (complexvalued) polynomial system (4.11) P j (x, u, Λ k , Λ k , ξ , ω j ) = 0, S q (u, Λ k , Λ k ) = 0, j ∈ {1, . . . , N -e}, q ∈ {1, . . . , b}.

Then Λ k = (j k f )(0, u), ξ = g(x, u), ω = (ω 1 , . . . , ω N -e ) = h(x, u) is a complexvalued real-analytic solution of (4.11). Furthermore, the system (4.11) has the following property: for every real-algebraic set Σ ⊂ C N × C N and every sequence Λ k = T (u), ξ = g (x, u), ω = h (x, u) of germs at 0 ∈ R N of real-analytic mappings converging as → ∞ in the Krull topology to (j k f )(0, u), g(x, u) and h(x, u) respectively and satisfying (4.11), if Graph f ∩ (M × C N ) ⊂ Σ (as germs at (0, f (0))), then for large enough, we also have Graph f ∩ (M × C N ) ⊂ Σ (as germs at (0, f (0))) where f : (C N , 0) → C N is the holomorphic map obtained by complexifying the realanalytic mapping (g , h

) : (R N , 0) → C N .
Proof. The first part of the conclusion of the proposition follows immediately the construction of the system (4.11) (e.g. Lemma 4.2 and (4.7) and (4.1)). Let us the prove the second part of the desired conclusion. To this end, let Σ be a fixed realalgebraic subset of C N × C N (that we may assume to be, without loss of generality, different from C N × C N ) and assume that Graph f ∩ (M × C N ) ⊂ Σ (as germs at (0, f (0))). Let also (T (u)) , (g (x, u)) , (h (x, u)) be a sequence of germs at 0 ∈ R N of real-analytic mappings converging as → ∞ in the Krull topology to (j k f )(0, u), g(x, u) and h(x, u) respectively, and, satisfying for every integer the system (4.11). We may assume that Σ = C N ×C N since otherwise the conclusion of the proposition is obvious. We prove the proposition by contradiction. Assume therefore that there exists a subsequence (f ηr ) r such that Graph f ηr ∩ (M × C N ) ⊂ Σ for every integer r. Choose a finite number of nontrivial real-valued polynomials ρ 1 , . . . , ρ m defined over C N ×C N such that Σ = ∩ m j=1 Σ j where Σ j is the zero set of the polynomial ρ j . Then, by the pigeonhole principle, we may assume that there exists a subsequence (f ηr ) r of (f ηr ) r such that Graph f ηr ∩ (M × C N ) ⊂ Σ 1 for every integer r. Without loss of generality, we may assume that this subsequence is the whole sequence (f ) . Furthermore, in what follows, when complexifying germs at 0 ∈ R N of real-analytic mappings, we will keep the same notation for the obtained germs at 0 ∈ C N of holomorphic maps. For every integer , we thus have (4.12)

ρ 1 (Z, ζ, f (Z), f (ζ)) ≡ 0, (Z, ζ) ∈ (M, 0).
Since for every integer , the mapping Λ k = T (u), ξ = g (x, u), ω = h (x, u) satisfies the system of complex-valued real-algebraic equations given by (4.11), we have, in view of Lemma 4.2 that for every ν = 0, . . . , p ρ 1 -1 that (4.13)

Q ρ 1 ν (Z, ζ, T (u), T (u), ξ , ) M×C 2(N -e) = 0, near 0,
where Q ρ 1 is the polynomial associated to ρ 1 and given by Lemma 4.1 and written as in (4.6). Note also that we also have for j = 1, . . . , N -e and for Z ∈ C N sufficiently close to the origin P j (Z, T (u), T (u), g (Z), h j (Z)) = 0.

Observe that the sequence of germs of holomorphic mappings (f (Z)) and (T (u)) converge in the Krull topology as → ∞ to the holomorphic maps f (Z) = (g(Z), h(Z)) and (j k f )(0, u) respectively. This implies in particular that, in view of (4.2), that for large enough, we have P j,m j Z, T (u), T (u), g (Z) ≡ 0, j = 0, . . . , N -e.

Hence by Lemma 4.1, for large enough and for all (Z, ζ) ∈ C 2N sufficiently to the origin, we have

(4.14) Q ρ 1 (Z, ζ, T (u), T (u), g (Z), g (ζ), ρ 1 (Z, ζ, f (Z), f (ζ))) = 0.
Then (4.14) and (4.13) imply that, for (Z, ζ) ∈ M near 0, we have (4.15)

δ ν=pρ 1 Q ρ 1 ν (Z, ζ, T (u), T (u), g (Z), g (ζ)) ρ 1 (Z, ζ, f (Z), f (ζ)) ν = 0.
Using (4.12), we may therefore write for (Z, ζ) ∈ M near 0, (4.16)

Q ρ 1 pρ 1 (Z, ζ, T (u), T (u), g (Z), g (ζ)) = δ ν=1+pρ 1 Q ρ 1 ν (Z, ζ, T (u), T (u), g (Z), g (ζ)) ρ 1 (Z, ζ, f (Z), f (ζ)) ν .
Since Graph f ∩(M ×C N ) ⊂ Σ (as germs at (0, f (0))), we have ρ 1 (Z, ζ, f (Z), f (ζ)) ≡ 0 for (Z, ζ) ∈ M near 0. Hence the restriction to M of the right hand side of (4.16) converges in the Krull topology, as → ∞, to 0, whereas the left hand side of (4.16) converges to Q ρ 1 pρ 1 (Z, ζ, (j k f )(0, u), (j k f )(0, u), g(Z), ḡ(ζ)) M . This implies that for (Z, ζ) ∈ M sufficiently close to 0, we have (4.17)

Q ρ 1 pρ 1 (Z, ζ, (j k f )(0, u), (j k f )(0, u), g(Z), ḡ(ζ)) = 0. By the definition of p ρ 1 (see (4.7)), we have

Q ρ 1
pρ 1 (Z, ζ, (j k f )(0, u), (j k f )(0, u), ξ , ) M×C 2e ≡ 0, near 0. Proposition 3.7 implies that the components of g are algebraically dependent over the field S f . This contradicts the definition of the mapping g. The proof of Proposition 4.3 is complete.

Proof of Theorem 1.1

Consider first the case N = 1. Then M is a real-algebraic curve in the complex plane. Fix a point p ∈ M . Since we can algebraically flatten M near p, we may assume without loss of generality that M is a piece of the real line and that p = 0. Then the conclusion of the theorem follows easily from Theorem 2.1 with K = R.

Assume in the remainder of this section that N ≥ 2. We first prove Theorem 1.1 in the case where M is generic in C N . Pick a point p ∈ M and choose normal coordinates Z = (z, w, u) ∈ C n × C d-c × C c vanishing at p as in Lemma 2.7. As in Section 4, let e be the transcendence degree of the field extension S f → S f (f ) where S f is the quotient field of the ring B f as given in Definition 3.1. We use in what follows the notation introduced in the previous section. Let S be the real-algebraic subset of C N × C N containing the germ at (0, f (0)) of Graph f ∩ (M × C N ).

First case. If e = N , we claim that M × C N ⊂ S (as germs at (0, f (0))). Indeed, choose a finite collection of real-valued polynomials r 1 , . . . , r m over C N × C N such that S = {r 1 = 0, . . . , r m = 0}. Suppose that there exists j ∈ {1, . . . , m} such that r j (Z, Z, Z , Z ) M ×C N ≡ 0 near the origin. Since r j (Z, ζ, f (Z), f (ζ)) ≡ 0 for (Z, ζ) ∈ (M, 0), Proposition 3.7 implies that the components of the mapping f are algebraically dependent over S f , which contradicts the fact that e = N . Hence for every integer j ∈ {1, . . . , m}, r j (Z, Z, Z , Z ) ≡ 0 for (Z, Z ) ∈ (M × C N , 0), which proves the claim. Therefore, we see that, if for every integer , f denotes the Taylor polynomial of the mapping f up to order , then the sequence of polynomial mappings (f ) satisfies the required conclusion.

Second case. Assume that e < N . In view of Proposition 4.3, we see that it is enough to find for every integer , germs at 0 ∈ R N of complex-valued realanalytic mappings Λ k = T (u), ξ = g (x, u), ω = h (x, u) that solve the system (4.11) with the following additional two properties: the sequences (T (u)) , (g (x, u)) , (h (x, u)) converge as → ∞ in the Krull topology to (j k f )(0, u), g(x, u) and h(x, u) respectively and the real and imaginary part of every component of these mappings must belong to the ring A R {x, u}. But the existence of these three sequences of germs of real-analytic mappings with the desired properties follows by a direct application of Theorem 2.2 (with K = R) to the system of real-valued real-algebraic equations associated to the system (4.11). The proof of Theorem 1.1 is therefore complete in case M is generic in C N .

To conclude the proof of Theorem 1.1, we will show as in [START_REF] Meylan | Approximation and convergence of formal CR-mappings[END_REF] that the case where M is not generic follows from the generic case treated above. Indeed, if M is not generic and p ∈ M , then the germ (M, p) is equivalent (through a local complex-algebraic biholomorphism) to a germ of real-algebraic submanifold of the form (M 1 × {0}, 0) ⊂ C N -r t × C r s where r ∈ {1, . . . , N -1}, M 1 is a real-algebraic generic submanifold in C N -r of constant orbit dimension near 0(see e.g. [START_REF] Baouendi | Real Submanifolds in Complex Space and Their Mappings[END_REF]). We may therefore assume that (M, p) = (M 1 × {0}, 0) and apply the generic case treated above to the holomorphic map (C N -r , 0) t → f (t, 0) ∈ C N . Hence, for every integer , there exists a (germ of a) complex-algebraic mapping ψ : (C N -r , 0) → C N such that Graph ψ ∩ (M 1 × C N ) ⊂ S and that agrees with the power series mapping f (t, 0) up to order at 0. Denoting, for every integer , by ϕ the Taylor polynomial of order of the mapping (t, s) → f (t, s) -f (t, 0) and setting f := ψ + ϕ , we get the desired result.

3. 1 .

 1 Statement of the algebraicity property. Given any integer k, we denote by J k 0 (C N , C N ) the jet space of order k at the origin of holomorphic maps from C N to C N . Throughout the paper, Λ k , Γ k and Υ k will denote coordinates in J k 0 (C N , C N ) and we write Λ k = (Λ k α ) |α|≤k where Λ k α ∈ C N for α ∈ N N (and analogously for Γ k and Υ k ).

  Hence, it follows from (3.27) and the minimality of r 0 that L ν T j (Z,ζ) Tr 0 (Z,ζ)

  , Λ k , Γ k , ξ ) and ς (nr) r for ς (nr) r

The author was partially supported by the French National Agency for Research (ANR), projects ANR-10-BLAN-0102 and ANR-09-BLAN-0422. 1

W can be rewritten in the following form (4.4)

where δ is independent of ρ and where for each γ = 0, . . . , δ -1,

for some polynomial D γ of its arguments (depending only on ρ) and where 1 ≤ j ≤ N -e and 1 ≤ ν j ≤ m j . Setting

we see that for a suitable integer r ρ ,

. We leave it to the reader to check that the construction of the polynomial Q ρ provides all the desired properties of the lemma. The proof is therefore complete.

4.2. Construction of the polynomial system attached to the mapping f . For every real polynomial ρ(Z, Z, Z , Z ) defined over C N × C N , let Q ρ be the polynomial given by Lemma 4.1. As in Lemma 4.1, we write

and define (4.7) p ρ := inf {ν ∈ {0, . . . , δ} : Q ρ ν (Z, ζ, (j k f )(0, u), (j k f )(0, u), ξ , ) M×C 2e ≡ 0, near 0.}. It follows from (4.1) and (4.2) and Lemma 4.1 that each p ρ is well defined.

We need the following lemma.

Lemma 4.2. In the above setting, there exists a finite collection of polynomials S 1 , . . . , S b ∈ A C {u}[Λ k , Γ k ] such that for every T ∈ (C{u}) k , T satisfies the system of equations

u), T (u), ξ , ) M×C 2(N -e) ≡ 0, near 0, for all ν ∈ {0, . . . , p ρ -1} and every real-valued polynomial ρ defined over C N × C N if and only if S q (u, T (u), T (u)) ≡ 0, near 0 ∈ R c , q = 1, . . . , b.

Proof. For every real-valued polynomial ρ defined over C N × C N and every 0 ≤ ν ≤ p ρ -1, we write


Christophe O Laux

Laurent Pierrot

Richard J Gessman

State-to-state modeling of a recombining nitrogen plasma experiment

An atmospheric pressure nitrogen/argon plasma flows at high velocity through a water-cooled test-section in which it is forced to recombine within 250 s. At the test-section inlet, the plasma is in Local Thermodynamic Equilibrium (LTE) at about 7200 K. Because recombination rates are finite, the plasma reaches a state of chemical nonequilibrium at the exit of the test-section, at a temperature of about 4715 K. At the test-section exit, the radiation emitted by the plasma, measured by emission spectroscopy, shows significant departures from equilibrium in the populations of the excited electronic states of nitrogen (N 2 B 3  g , N 2 C 3  u ) and of the nitrogen ion (N 2 + B 2  u  ). This experiment is thus proposed as a test-case to validate collisional-radiative (CR) models. A vibrationally specific CR model is then used to predict the emission of these states. The rate coefficients of the model are calculated with the Weighted Rate Coefficient method based on elementary cross-section data. These rates depend explicitly on the kinetic temperatures of electrons (T e ) and heavy species (T g ). The predictions of the CR model are in good agreement with the measured vibrational population distribution in the N 2 B state. A method is then proposed to determine ground state nitrogen atom densities based on the measured vibrational population distribution of the N 2 B state.

Introduction

The chemistry of nonequilibrium molecular plasmas is important to predict nonequilibrium radiation in atmospheric re-entry and to interpret optical diagnostics in applications ranging from material processing or biodecontamination to plasma-assisted combution. Simulating the radiation of recombining plasmas or reentry flows requires to predict the populations of the emitting internal levels of atoms and molecules under nonequilibrium conditions. Various forms of nonequilibrium can exist. We distinguish thermal nonequilibrium, where the kinetic temperature T g of heavy species is different from the kinetic temperature T e of the electrons, and chemical nonequilibrium, where chemical reactions are not equilibrated. Local Thermodynamic Equilibrium (LTE) corresponds to the case of both thermal and chemical equilibrium.

The nonequilibrium populations of internal energy states can be determined by solving the so-called master equation, which is a system of coupled state-specific rate equations (collisionalradiative model, or CR). While for atomic species it is usually sufficient to consider 10-100 electronic states, for molecular species a much larger number of rovibronic levels must be taken into account. Many CR models [1][START_REF] Park | Nonequilibrium Air Radiation (NEQAIR) Program: User's Manual[END_REF][START_REF] Sarrette | [END_REF][4][5] use the simplifying assumption that the rotational and vibrational energy levels of molecules are populated according to Boltzmann distributions at temperatures T r and T v , respectively. In these models, the master equation is solved for electronic levels only, with global rates corresponding to averages over the rovibrational states of each electronic level. Thus these approaches implicitly assume that departures from Boltzmann distributions in the populations of rotational and vibrational levels are small. Yet, departures from Boltzmann distributions, at least in the vibrational levels, are fairly common. Experiments conducted at Stanford University [START_REF] Gessman | Experimental study of kinetic mechanisms of recombining atmospheric pressure air plasmas[END_REF] with a nitrogen/argon plasma forced to recombine within 250 s from a state of equilibrium at a temperature of 7200 K to a state of chemical nonequilibrium at 4715 K have shown that the populations of vibrational levels in certain electronic states, in particular the B 3  g of N 2 , can strongly depart from a Boltzmann distribution when the density of atomic nitrogen is higher than its LTE value. To predict nonequilibrium populations, a vibrational state-specific collisional-radiative (CR) model must be used. Most existing vibrational state-specific CR models [START_REF] Cacciatore | [END_REF][START_REF] Colonna | State to State Electron and Vibrational Kinetics in Supersonic Nozzle Expansion: an Improved Model[END_REF][START_REF] Loureiro | [END_REF] usually assume that the rotational levels follow a Boltzmann distribution owing to fast rotational relaxation, especially at atmospheric pressure. This assumption is also made in the present work.

In this paper, we first give an overview of the vibrational state-to-state CR model that we have developed for nitrogen plasmas [START_REF] Pierrot | Vibrationally-Specific Collisional-Radiative Model for Nonequilibrium Nitrogen Plasmas[END_REF][START_REF] Pierrot | Collisional-radiative modeling of nonequilibrium effects in nitrogen plasmas[END_REF][START_REF] Pierrot | Chemical Kinerics and Vibrationally Specific Collisional-Radiative Models for Nonequilibrium Nitrogen Plasmas[END_REF]. The second part of the paper presents the recombining nitrogen plasma experiment [START_REF] Gessman | Experimental study of kinetic mechanisms of recombining atmospheric pressure air plasmas[END_REF] which is proposed as a test-case to validate nitrogen CR models. The third part compares the vibrational population distributions measured in that experiment with the simulations obtained with the nitrogen CR model. In the last part of the paper, we propose a method to determine absolute ground state N atom densities from the measured nonequilibrium emission of the N 2 B state.

Nitrogen CR model

Details about the nitrogen CR model and derivation of the vibrational state-specific rate coefficients are given by Pierrot et al. [START_REF] Pierrot | Vibrationally-Specific Collisional-Radiative Model for Nonequilibrium Nitrogen Plasmas[END_REF][START_REF] Pierrot | Collisional-radiative modeling of nonequilibrium effects in nitrogen plasmas[END_REF][START_REF] Pierrot | Chemical Kinerics and Vibrationally Specific Collisional-Radiative Models for Nonequilibrium Nitrogen Plasmas[END_REF]. This model was subsequently extended to air plasmas [START_REF] Chauveau | Vibrationally Specific Collisional-Radiative Model for Nonequilibrium Air Plasmas[END_REF], but these extensions will not presented because the focus of the present paper is on modeling a recombining nitrogen plasma. We give below an overview of the nitrogen CR model and of its capabilities.

Species and energy levels considered

The species considered in the nitrogen CR model are N, N + , N 2 , N 2 + , and electrons. The internal structure of N is accounted for by using the 22 grouped electronic levels of Park [START_REF] Park | Nonequilibrium Hypersonic Aerothermodynamics[END_REF].

For N + , only the ground state is taken into account. The X, A, B, W, B, and C electronic states of N 2 and the X, A, and B electronic states of N 2 + are included, and all rovibrational levels belonging to these states are considered up to the dissociation limit. As listed in [Insert Table 1, the model takes into account 192 vibronic levels of N 2 and 142 levels of N 2 + . Note that the model for the ground state of N 2 contains fewer vibrational levels that the more accurate model recently proposed by Le Roy et al [START_REF] Le Roy | [END_REF] (48 levels for present model vs. 61 levels for Le Roy et al), but this difference has little impact on the excited state populations presented here.

[Insert Table 1.]

Reactions considered

The nitrogen CR model considers approximately 46,000 reactions that are summarized in [insert Table 2. These reactions include electron-impact ionization, excitation, and dissociation of atoms and molecules, heavy-particle impact dissociation of molecules, as well as electronimpact vibrational excitation of ground state nitrogen (VE transfer), vibrational-translational (VT) and vibrational-vibrational (VV) transfer, charge exchange, dissociative recombination, radiation, and predissociation.

[insert Table 2]

Example of rate coefficient calculation: N 2 ground state ionization by electron impact

To illustrate the principles of the vibrationally specific CR model, we present the method of calculation of the rate coefficients for electron-impact ionization of ground state N 2 . We begin by considering the elementary ionization reaction between rovibronic level (X,v",J") of N 2 to level (A,v',J') of N 2 + .

N 2 (X,v",J") + e  N 2 + (A,v',J') + e + e

We first determine the rate coefficient of this elementary reaction, and then we average the elementary rate coefficients over all initial (J") and final (J') rotational levels of (X,v") and (A,v') in order to obtain the vibrational state-specific rate coefficients.

Elementary rate coefficients

The elementary rate coefficient of electron-impact ionization of N 2 (X,v",J") into N 2 + (A,v',J'), can be written as:

      ++ 22 22 N Av'J' N Av'J' N Xv"J" N Xv"J" 3/ 2 0 81 exp 2 e e e e K T d kT m kT             (1)
where

  + 2 2 N Av'J' N Xv"J"
 is the cross-section of the elementary reaction, m e the electron mass, and  the electron energy. In Eqn. (1), we have implicitly assumed that electrons follow a Maxwellian energy distribution at temperature T e . In passing, we note that all rate coefficients of the CR model assume Maxwellian energy distributions for electrons and heavy species at T e and T g , respectively. Although this assumption is not justified in general for electrons, we will see later that it is valid for the conditions of the recombination experiment presented in this paper.

Following Park [START_REF] Park | Nonequilibrium Hypersonic Aerothermodynamics[END_REF], we consider that the Franck-Condon principle holds for transitions in molecules caused by collisions with electrons, and that no change in rotational quantum number occurs as a result of such collisions. Accordingly, we write the elementary cross-section of ionization from rovibronic level N 2 (X,v",J") to rovibronic level N 2 + (A',v',J') as:

  + 2 2 N Av'J' Av' ' Av'J' N Xv"J" Xv'J' Xv" , J J qE       (2)
where Av'

Xv"

q designates the Franck-Condon factor, Av'J' Xv"J" E  the ionization energy, and ' " J J

 is equal to 1 if J" = J' and to 0 otherwise. The reduced cross-section

A'v'J' Xv"J" ( , ) E   
is calculated with the Binary-Encounter-Bethe (BEB) model of Hwang et al [17], which provides ionization crosssections within 15% of experimental values for a wide variety of atoms and molecules. The BEB expression is:

  2 Av'J' 2 Xv'J' 0 Av'J' 2 Xv'J' 1 ln 1 1 ln , 4 1 1 1 2 1 R t t E a N E t u t t t                       (3)
where R is the Rydberg constant (13.61 eV), a 0 is the Bohr radius (0.5292 Å), N is the electron occupation number in the shell from which ionization takes place, t stands for / Av'J' Xv"J" E  where  is the kinetic energy of the electron, and u stands for U/ Av'J' Xv"J"

E



where U is the average kinetic energy of a bound electron in the shell. The values of N and U are given by Hwang et al [17] for several molecular orbitals of N 2 and O 2 .

Vibrational state-specific rate coefficients

To obtain the vibrational state-specific rate coefficients, we sum the elementary rate coefficients over all final rotational levels, and we average over the distribution of initial rotational levels. We assume that the initial rotational levels follow a Boltzmann distribution at rotational temperature T r , which is a reasonable approximation for many molecules of interest (except perhaps the hydrides) owing to fast collisional relaxation at atmospheric pressure. The vibrational state specific rate coefficient is then: " FJ is the rotational term energy of level J". The reverse rates are determined by detailed balance.

          ++ 2 22 22 2 N Av' N Av'J' N Xv" N Xv" N Xv"J" N Xv" " J' rot " 1 , 2 " 1 exp e r e J rr FJ K T T J K T Q T kT         (4)
We compare in [insert Table 3 the total vibrationally specific ionization cross-sections of N 2 (X 1  g + ,v) with the values presented by Kosarim et al [26] in Figure 4 of their paper. As can be seen in [insert Table 3, the present model predicts a slight increase of the cross-sections with the electron impact energy, whereas the model of Kosarim et al [26] predicts the opposite. Nevertheless, reasonable agreement is obtained overall between the two models since the ionization cross-sections agree within 80% for v=0, and within 30% for v=20. It should be noted that the model of Kosarim et al [26] includes ionization to three states of N 2 + , namely the X 2  g + , A 2  u , and B 2  u + states whereas the present model is limited to ionization into states X 2  g + and A 2  u . However, according to Van Zyl and Pendleton [27], the branching ratios of the ionization cross-sections from N 2 X to N 2 + X, N 2 + A, and N 2 + B are respectively equal to 0.320±01.47, 0.535±0.112, and 0.145±0.019 for an electron impact energy of 100 eV. Thus, even at these high impact energies, the cross-section for ionization of N 2 + B represents less than 15% of the total ionization cross-section. Therefore, we do not expect ionization from the ground state of N 2 to the B state of N 2 + to greatly affect the overall ionization.

[insert Table 3]

Predissociation rates

As will be seen in the analysis of the recombination experiments presented in Sections 3 and 4, predissociation plays a key role on the vibrational population distribution of the B state of N 2 . Therefore, we describe here the method by which these rates are computed in the CR model.

The predissociation rate constants (v,J) of the B state of N 2 are taken from Geisen et al [24]:

(v,J) =  1 (v) +  2 (v)J(J+1) (5)
where the second term on the right hand side of Eqn. ( 5) is a corrective term corresponding to indirect predissociation of the 3  0 f component via rotational coupling with the 3  1 ef components (the 3  0 f component would otherwise remain unpredissociated). This corrective term cannot be larger than the value  2 max corresponding to the 3  1 predissociation rate constant.

In our CR model, the total predissociation rate constant of a given level v is calculated as a function of T r by averaging the J-dependent values given by Eqn. ( 5) as follows:

        ) , ( ) ( exp 1 2 1 min 2 ) , ( J v kT J F J T Q v r Bv J J r v B N rot        (6)
where J min is the minimum rotational quantum number for which the vibrational level considered predissociates. According to Lofthus and Krupenie [28], these values are J min = 32 for v=12 and J min =0 for v ≥ 13. The summation over J in Eqn. ( 6) is truncated at the limiting value  2 max . The values of  1 (v),  2 (v), and  2 max are listed in [insert Table 4. When rate constants for a particular component of a given vibrational level are missing, they are taken as the average rate constants of that level. The predissociation rate constants of v=12, not given by Geisen et al, are assumed to be equal to those of v=13.

[insert Table 4]

In summary, we have presented a vibrationally specific CR model for nitrogen plasmas. This model will be applied later in this paper to the analysis of the recombining nitrogen plasma experiment presented next.

Nonequilibrium recombining nitrogen plasma experiment

Experiments were conducted at Stanford University [START_REF] Gessman | Experimental study of kinetic mechanisms of recombining atmospheric pressure air plasmas[END_REF] with a recombining atmospheric pressure nitrogen plasma. The experimental set-up is shown in [insert Figure 1. In these experiments, a mixture of ~100 slpm N 2 , ~50 slpm argon, and ~2.3 slpm H 2 was inductively heated with a 50 kW RF plasma torch. Argon was added to stabilize the torch operating conditions, and H 2 was added in order to measure the electron number density from the Starkbroadened H  line. The plasma produced by the torch was accelerated to a velocity of ~1 km/s by passing through a 1 cm diameter torch exit nozzle. From the nozzle, the plasma was then rapidly cooled (within ~250 s) by flowing through a 15 cm-long water-cooled brass testsection. A shorter test-section, 10 cm in length, was also used to obtain intermediate results at a distance of 10 cm from the nozzle exit. Calorimetric measurements were taken on the water flow used to cool the 10 and 15 cm test-sections. The total heat flux removed by the testsections was 2.4 and 4.5 kW (0.1 kW) for the 10-and 15-cm test-sections, respectively.

[insert Figure 1.] Emission spectroscopy measurements were made using a SPEX model 750M, 0.75-meter, scanning monochromator fitted with a Hamamatsu model R1104 photomultiplier tube. Two 1200 lines/mm gratings blazed at 200 and 500 nm were used to cover the spectral range of the present study. The entrance and exit slits of the monochromator were set to 200 and 400 m, respectively, yielding a trapezoidal slit function of base 0.66 nm and top 0.22 nm. The spectra measured along the diameter of the plasma at the nozzle and test-section exits were calibrated in absolute intensity using two NIST traceable radiance standards: an Optronics model OL550 standard for wavelengths between 300 and 800 nm and a 1-kW Argon Mini-Arc from Arc Applications Research between 200 and 400 nm.

At the exit of the nozzle (position termed 0 cm), several temperature profiles were obtained by Abel-inverting absolute intensity profiles of various emission lines of N, Ar, and H [START_REF] Gessman | Experimental study of kinetic mechanisms of recombining atmospheric pressure air plasmas[END_REF]. As can be seen from Fig. 2a, the good agreement among the various temperature profiles indicates that the plasma is close to LTE at the nozzle exit (which is the inlet of the test-section). The maximum temperature at the centerline is approximately 7200 K. A similar technique was applied at the exit of the 10 cm nozzle (Fig. 2b). At the exit of the 15-cm test-section, a temperature profile was measured by Abel-inverting lateral profiles of the (0,0) band of the N 2 + first negative system, as described also in Ref. [START_REF] Gessman | Experimental study of kinetic mechanisms of recombining atmospheric pressure air plasmas[END_REF]. The centerline temperature was found to be 4715±100 K (Fig. 2c). Subsequent high-resolution spectroscopy measurements [29] using resolved rotational lines of the N 2 + first negative system confirmed these measurements with a measured centerline temperature of 4850±100 K.

[insert Figure 2]

At 7200 K, the LTE composition of the plasma is 55% N, 19% N 2 , with a balance of argon and hydrogen. The electron mole fraction is 5.5x10 -4 . At 4715 K, the LTE composition is 63% N 2 , 1.3% N, with a balance of argon and hydrogen, and an electron mole fraction of 3.4x10 -7 . However, because the plasma residence time within the 15-cm test section, 250 s, is shorter than the characteristic time of atomic nitrogen recombination [START_REF] Gessman | An Experimental Investigation of the Effects of Chemical and Ionizational Nonequilibrium in Recombining Air Plasmas[END_REF], the atomic nitrogen population tends to freeze and hence is expected to be higher than its LTE value at the exit of the 15-cm test-section.

Electron number densities were also measured using the Stark broadened H  line at 486 nm, using the technique described in Ref. [START_REF] Laux | [END_REF]. The resulting values at 0, 10 and 15 cm are compared in [insert Table 5 with the corresponding LTE number densities at these locations. As can be seen from the table, the electron number density remains close to LTE until at least the exit of the 10 cm test-section. On the other hand, the measured electron number density at 15 cm is about two orders of magnitude larger than the LTE value, consistent with the existence of nonequilibrium conditions at that location.

[insert Table 5]

We now examine the emission spectra at the various locations. Absolute emission spectra between 280 and 800 nm were measured along the plasma diameter, both at the nozzle exit and at the test-section exits. These measured spectra were then compared with LTE spectra calculated with the line-by-line radiation code SPECAIR [START_REF] Laux | [END_REF] using the measured gas temperature profiles.

At the nozzle exit, the measured and LTE spectra are in close agreement (see [insert Figure 3), which supports the existence of LTE conditions at the nozzle exit. At the 10-cm test section exit, the comparison also indicates that the plasma remains close to LTE. At the 15-cm test-section exit, however, the measured spectrum is more intense than the computed LTE spectrum (see [insert Figure 4). In particular, the radiative emission of the first positive system of N 2 , second positive system of N 2 , and first negative system of N 2 + is higher than LTE values by factors of about 6, 30, and 10, respectively. These differences indicate that the populations of the emitting vibrational levels are higher than the corresponding LTE values, and thus that these levels follow nonequilibrium distributions. We will now extract the nonequilibrium vibrational distributions from the measured spectra in order to provide quantitative data to test the CR model.

Note that several bands of NH A-X and OH A-X, the latter coming from some entrainment of ambient air into the plasma plume, were observed in the spectra, and that these bands interfere with the bands of the second positive system of N 2 (C-B transition) and of the first positive system of N 2 + (B-X). For clarity, we have excluded from [insert Figure 4 the regions where these bands appear, but details of the analysis including these bands are given in Appendix A.

[insert Figure 3]

[insert Figure 4]
The nonequilibrium vibrational population distributions of the observed emitting states at the 15-cm test-section exit were examined. To this end, we multiplied the LTE populations of individual vibrational levels by nonequilibrium factors in SPECAIR, until the predicted spectrum (obtained by solving the radiative transport equation along the plasma diameter) matched the measured spectrum (see Figure 5). Note that this procedure assumes that the nonequilibrium factors are uniform along the plasma diameter. However, because the line-of-sight emission is strongly dominated by the emission from the central region of the plasma, the inferred nonequilibrium factors closely reflect the nonequilibrium population distribution at the center of the plasma. These nonequilibrium factors are defined by:

measured v v LTE v n n  (7)
where n v measured and n v LTE stand respectively for the measured and equilibrium (at the local gas temperature) number density in level v of the B state of N 2 . Hence, a nonequilibrium factor greater than unity implies that the vibrational level v is overpopulated with respect to equilibrium at the local gas temperature.

[insert Figure 5]

The measured distribution of nonequilibrium population factors of the N 2 B and C states and of the N 2 + B state are shown in [insert Figure 6. In this graph, a Boltzmann distribution at T v = 4715 K would be represented by the straight line  v =1. For the N 2 C and N 2 + B states, the inferred overpopulation factors do not vary much with the vibrational levels, with values of about 30 for the N 2 C state and 11 for the N 2 + B state. For the N 2 B state, the measured vibrational distribution is non-Boltzmann, with a marked peak at v=13. We will now compare these measurements with the predictions of the CR model for the B and C states of N 2 .

[insert Figure 6]

Analysis of the recombining nitrogen experiments

The nitrogen CR model is used in this section to analyze the recombining nitrogen plasma experiments. We are particularly interested in the nonequilibrium vibrational distribution of the B state of N 2 , which exhibits a peculiar dependence with a peak at v=13. Argon is not included in the CR model, and we make the assumption that it does not significantly affect the populations of the observed nitrogen states. We therefore consider the case of a pure nitrogen plasma in chemical nonequilibrium. The CR model calculations are performed by making use of the quasisteady-state (QSS) approximation, which assumes that all excited levels of atoms and molecules are in steady-state, and equations relative to ground levels are replaced with mass conservation equations.

The input parameters of the model are the temperature T = 4715 K, the pressure p = 1 atm, and the number densities of the different species (e, N 2 , N 2 + , N, and N + ) at the exit of the 15-cm test-section. The electron number density was measured from the Stark-broadened H  line and was found to be 7.3x10 13 cm -3 . The N atom overpopulation factor (recall that we expect an overpopulation of nitrogen atoms owing to the finite rate of nitrogen recombination) has been adjusted to  N = 8.1 in order to obtain the best possible agreement with the peak of the distribution at v=13. This value will be justified later in Section 4. Thus the N density is set to 1.5x10 17 cm -3 . The density of N 2 is assumed to stay close to its LTE value. Finally, the densities of N + and N 2 + are chosen in such a way that the plasma is electrically neutral and that the charge exchange reaction N 2 +N +  N 2 + +N is balanced. Note however that the densities of N + and N 2 + have little influence on the N 2 B state population. The final set of input parameters for the CR model computations at 15 cm is summarized in [insert Table 6.

In general, superelastic and electron-electron collisions can have a strong influence on the electron energy distribution function of nitrogen plasmas at low translational temperatures [32,33]. However, for the conditions of this experiment, the electron density distribution function can be considered to be Maxwellian, as shown by Capitelli et al [34]. This observation justifies our assumption in the CR model of a Maxwellian electron energy distribution for the calculation of electron-impact rate coefficients.

[insert Table 6]

The computed vibrational overpopulation factors of the B state of N 2 are plotted in [insert Figure 7 together with the measured overpopulation factors. The curve labeled "CR model" was obtained using the baseline rate coefficients of the CR model. This calculation reproduces the general shape of the experimental curve, with a peak at v = 13, but predicts larger overpopulation factors at v's above and below 13. Thus we examined the sensitivity of the CR model predictions by varying selected rates. Curve (a) was calculated by multiplying the N 2 (X-A) and N 2 (X-B) electron-impact excitation and de-excitation rates by a factor of 4, and by dividing the predissociation rate of level N 2 (B,v = 12) by a factor of 5. Note that the value of the N 2 (B,v = 12) predissociation rate used in the baseline CR model is an estimate based on the value measured by Geisen et al. [24] for v = 13, so that the foregoing adjustment to the rate is reasonable. Similarly, the multiplicative factor of 4 for the X-A and X-B excitation rates is not unreasonable, given the uncertainty on cross-sections for these reactions [35]. With these modified rates, good agreement is obtained with the experimental curve for v=0-13.

Curve (b) presents another calculation in which the rates are modified as in curve (a) and predissociation of levels v 14  is neglected. Better agreement with the experimental curve is now obtained for the high-lying levels v 14

 . This result may suggest that the predissociation rates of levels v 14  are overestimated in the model, but it may also be due to uncertainties on other rates. The SSH-based VT transfer rate, for instance, may be somewhat inaccurate for the high lying levels, and multi-quantum jumps may have to be considered although typically they should be unimportant at the relatively low temperatures considered here [36,37]. Nevertheless, these discrepancies are not significant because the experimental uncertainties on overpopulation factors of high-lying levels v≥15 are quite large (see [insert Figure 6).

[insert Figure 7]

The CR model also predicts the populations of additional vibronic states. [insert Figure 8 shows the vibrational overpopulation factors of the X, A, B, W, B, and C states of N 2 as a function of the vibrational quantum number v, calculated with the same modified rates as curve (a) of [insert Figure 7. The predicted overpopulation factors for the low-lying levels of all states are in the range 2 to 4. The overpopulation factors of the X, A, and C states increase steadily with v to reach values close to 66, i.e. corresponding to dissociation equilibrium (by this, we mean that the last vibrational level below the dissociation limit of a given electronic state is nearly in equilibrium with the dissociation products). The measured overpopulation factors of the C state of N 2 are practically constant with v, whereas the CR model predicts a sharp increase with v. One possible explanation is that we may overestimate the degree of coupling between the last level (v=4) with nitrogen atoms. In the baseline model, we have simply considered that this level, which is known to be subject to predissociation [28], is fully coupled with the atoms. This assumption appears to be too strong. In addition, it seems important to incorporate in the model the rates of quenching by heavy-particle impact, as well as the rates of V-T and V-V transfer for the C (and B) states of N 2 . According to the measurements of Dilecce et al [38,39], these rates are relatively fast and thus could flatten the overpopulation distribution in the C state of N 2 . Finally, the W state overpopulation curve exhibits two inflexion points, and the B state overpopulation curve has a local maximum similar to that of the B state. These behaviors are caused by collisional coupling with the predissociating levels of the B state.

Method for measuring N atom densities by emission spectroscopy

Despite the remaining uncertainties on the shape of the vibrational population distribution, the results obtained with the baseline CR model reproduce the observed nonequilibrium distribution of the N 2 B state reasonably well. An interesting result of the analysis presented in the previous section is that vibrational level v=13 of the B state of N 2 appears to be fully coupled with N atoms by predissociation and inverse predissociation. This can be seen by writing the rate equation at steady-state for the population of N 2 , B, v=13:

  2 2 2 2 2 2
, ,13 13 14 13 12 , ,

',v' 0 NB pred e e N B M VT VT e ion e B Y Y X A B W C inv pr e e N N Y N B M VT N B M VT e rec Y e Y B N dn n k A n k k n k n k dt n k n A n n k n n k n n k n n k                          , , ', , Y X A B W C   13 v 13 13v" ,13 v" , , ', , 2 . . 14 13 12 13 2 v 13 , ',v' v'13 , ,14 , ,12 ,13 Y
The numerical values of the depleting terms that appear in the first bracket, as calculated with the CR model, are listed in [insert Table 7.

[insert Table 7]

It can be seen from [insert Table 7 that the predissociation term is about 30 times larger than the sum of the other depletion terms. Similarly, it can be shown that the largest term among the processes populating B,v=13 is the inverse predissociation term. It follows that, at steady-state, Eqn. ( 8) can be approximated by:

  2 2 . , ,13 v 13 v 13 pred inv pred N B N n k n k   (9)
Using superscript eq to designate equilibrium densities, we also have, from detailed balance, the following relation:

  2 2 . , ,13 v 13 v 13 eq pred eq inv pred N B N n k n k   (10)
Dividing Eqn. ( 9) by Eqn. [START_REF] Pierrot | Vibrationally-Specific Collisional-Radiative Model for Nonequilibrium Nitrogen Plasmas[END_REF], we obtain:

  22 2 , ,13 , ,13 eq eq N B N B N N n n n n  (11)
which, using nonequilibrium factor notation, can finally be written as:

2 2 , ,13 N B N   (12)
Equation ( 12) expresses a very interesting relation, namely that the overpopulation of the N 2 , B,v=13 state is approximately equal to the square of the overpopulation factor of atomic nitrogen in the ground state. In other words, the vibrational state v=13 is in partial equilibrium with nitrogen atoms in their ground state (in the same way as the last vibrational level in any electronic state tends to be in partial equilibrium with the dissociation products issued from this given state). This relation means that it is possible to use emission spectroscopy to infer ground state atomic nitrogen densities. For our experimental conditions, the overpopulation factor of v=13 is measured to be  N2,B, v=13  66±4. Thus the overpopulation of nitrogen atoms is approximately 8.1±0.3. At 4715±100 K, the equilibrium density of nitrogen atoms in the nitrogen/argon plasma mixture is 1.80.410 16 cm -3 . Multiplying this number by the measured overpopulation factor  N , we infer a nonequilibrium nitrogen atom density of 1.50.410 17 cm -3 , which corresponds to a mole faction of about 10% (whereas the LTE mole fraction of atomic nitrogen at 4715 K is only 1.3%). Note that relation [START_REF] Chauveau | Vibrationally Specific Collisional-Radiative Model for Nonequilibrium Air Plasmas[END_REF] is also valid at pressures lower than one atmosphere because the various depletion terms decrease with decreasing pressure and therefore become even more negligible relative to the rate of predissociation. Thus this technique provides a way to measure absolute densities of ground state nitrogen atoms using emission spectroscopy. It could be particularly useful to determine the density of nitrogen atoms in recombining plasmas such as those produced in arcjets.

Conclusions

Because the characterization of re-entry flow radiation by onboard detectors can only be conveniently performed using optical emission spectroscopy, accurate CR models are needed to infer species densities and temperatures from emission measurements. Collisional-radiative models have recently emerged that can predict certain nonequilibrium plasma conditions reasonably well. In this paper, we described a detailed vibrationally specific CR model and we showed how it can be used to understand and predict the nonequilibrium radiation emitted by a recombining nitrogen plasma at atmospheric pressure. One important result was that the nonequilibrium emission from the N 2 B state can be used to measure the density of ground state atomic nitrogen, hence the dissociation fraction of nitrogen in the flow-field. This method complements existing techniques such as VUV absorption and Two-photon Absorption Laser-Induced Fluorescence (TALIF). In addition, we have provided an experimental test-case with a plasma flow under well-characterized thermodynamic conditions and with detailed quantitative measurements of nonequilibrium vibrational population distributions in the B and C states of N 2 and in the B state of N 2 + . These data can be used to test advanced CR models.

Uncertainties remain on many of the state-specific reaction rates. Accurate cross-sections are needed, in particular for state-specific electron and heavy-particle impact dissociation and excitation rate coefficients. At low pressure, it is also important to take into account possible departures from a Maxwellian distribution for the free electrons. Finally, additional quantitative measurements in nonequilibrium flows are required to further test and validate the nonequilibrium CR models. *equilibrium is taken at the local gas temperature and 1 atm.

Table 13. Species densities used to simulate the recombining nitrogen plasma conditions at the exit of the 15-cm test-section. T=4715 K, P = 1 atm.

S

n S eq (cm -3 ) n S (cm

[insert Figure 8 ]

 8

Figure 9 .Figure 10 .Figure 11 .Figure 12 .

 9101112 Figure 9. Schematic cross-section of torch head with test-section.

Figure 13 .Figure 14 . 2 /Figure 15 . 2 (

 13142152 Figure[START_REF] Chauveau | Vibrationally Specific Collisional-Radiative Model for Nonequilibrium Air Plasmas[END_REF]. Measured and computed non-LTE (i.e. using the overpopulation factors shown in Figure6) spectra of the nitrogen/argon plasma at the exit of the 15-cm test-section. Note that the H  line at 656 nm is not modeled in SPECAIR.

Figure 16 .Figure A. 1 .

 161 Figure 16. Vibrational overpopulation factors of the X, A, B, W, B and C states of N 2 . Same conditions as curve (a) of [insert Figure 7.

Table 8 . Rovibronic states of N 2 and N 2 + in the nitrogen CR model. J max is the highest rotational level considered in the ground vibrational level.

 8

	N 2

Table 10 . Comparison of vibrationally specific cross-sections for the ionization of levels v=0 and v=20 of the ground state of N 2 , as a function of the impacting electron energy, E=20, 30, 50 and 100 eV.

 10

			 (E=20 eV)	 (E=30 eV)	 (E=50 eV)	 (E=100 eV)
			×10 -21 m 2	×10 -21 m 2	×10 -21 m 2	×10 -21 m 2
	v=0	Present work Kosarim et al [26]	3.2 1.74	10.9 6.1	17.9 10.9	20.4
	v=20	Present work Kosarim et al [26]	3.9 2.9	9.3 7.5	13.8 12.2	15.1

Table 11 . Predissociation rate constants for N 2 (B,v) (from Geisen et al [24])

 11

	v	 1 (s -1 )	 2 (s -1 )	 2	max (s -1 )
	12	2.6 x 10 8	1.2 x 10 5	5.1 x 10 7
	13	2.6 x 10 8	1.2 x 10 5	5.1 x 10 7
	14	1.6 x 10 8	5.4 x 10 4	2.5 x 10 7
	15	1.1 x 10 8	3.2 x 10 4	1.5 x 10 7
	16	2.5 x 10 7	8.9 x 10 3	4.3 x 10 6
	17	7 x 10 6	5.1 x 10 2	2.5 x 10 5
	18	4 x 10 6	1.5 x 10 3	7.5 x 10 5

Table 12 . Measured and equilibrium electron number densities (cm -3 ) for the N 2 /Ar plasma experiments

 12

		0 cm	10 cm	15 cm
	Measured	5.0  0.3  10 14	9.8  1.8  10 13	7.3  4.0  10 13
	Equilibrium *	5.4  0.4  10 14	6.4  2.0  10 13	5.3  2.7  10 11

Table 14 . Reaction rates (normalized by the population of n N2,B,v=-13 ) for the major depletion processes of N 2 (B,v=13) in the recombining nitrogen plasma (T=4720 K, P=1 atm,  e =180,  N =8.1).

 14

			-3 )	 S
	N	16 1.8 10 	17 1.5 10 	8.1
	N +	10 4.8 10 	13 3.8 10 	~800
	N 2	18 1.5 10 	18 1.5 10 	1
	N 2 +	11 3.5 10 	13 3.5 10 	100
	E	11 4.0 10 	13 7.3 10 	180

Acknowledgments

Financial support from the US Air Force Office of Scientific Research, NASA Ames Research Center, the European Space Agency, and the French Ministry of Research (Chaire d'Excellence) is gratefully acknowledged.

Appendix A

In this appendix, details about the fitting of the nonequilibrium spectra at the exit of the 15 cm test-section are presented. The radiative transitions considered in this analysis are the N 2 first (B-A) and second (C-B) positive systems, the N 2 + first negative system (B-X), the (A-X) transition of OH, and the (A-X) transition of NH. The OH and NH vibrational populations were arbitrarily adjusted to match the features due to these systems in the experimental spectra. ).

Reaction

Cross-section or rate coefficient data Remarks

Drawin [16] BEB [17] XX, XA, AA,

Park [START_REF] Park | Nonequilibrium Hypersonic Aerothermodynamics[END_REF] V-T and V-V transfer to the last vibrational level below dissociation limit Ground state only

Peterson [19] Ground state only Charge exchange N 2 (X,v")+N + (X)  N 2 + (X,v')+N(X)

Stebbings [20] Ground state only Electronic excitation N(Z")+e  N(Z')+e N 2 (X,v")+e

Park [START_REF] Park | Nonequilibrium Hypersonic Aerothermodynamics[END_REF] Cartwright [21] Seaton [START_REF] Seaton | The Theory of Excitation and Ionization by Electron Impact[END_REF], Drawin [16] Seaton [START_REF] Seaton | The Theory of Excitation and Ionization by Electron Impact[END_REF], Drawin [16] Y

Geisen [24] Equilibrium with N(X)

Park [START_REF] Park | Nonequilibrium Hypersonic Aerothermodynamics[END_REF] Laux [25] Laux [25] Table A

v (emitting level)

Bands of the N 2 C-B system Bands of the N 2 + B-X system 0 (0,1) -357.7 nm (0,2) -380.4 nm (0,3) -405.8 nm (0,0) -391.1 nm (0,1) -427.