BRIOT–BOUQUET’S THEOREM IN HIGH DIMENSION

Let X be a germ of holomorphic vector field at 0 ∈ Cn and let E be a linear subspace of Cn which is invariant for the linear part of X at 0. We give a sufficient condition that imply the existence of a non-singular invariant manifold tangent to E at 0. It generalizes to higher dimensions the conditions in the classical Briot–Bouquet’s Theorem: roughly speaking, we impose that the convex hull of the eigenvalues μi corresponding to E does not contain 0 and there are no resonances between the μi and the complementary eigenvalues. As an application, we propose an elementary proof of the analyticity of the local stable and unstable manifolds of a real analytic vector field at a singular point. 2010 Mathematics Subject Classification: Primary: 34C45; Secondary: 37D10, 34A20, 37F75, 32S65.


Introduction
This work is framed in the local study of a complex holomorphic vector field X at a singular point 0 ∈ C n .More precisely, in the problem of determining whether there exists a formal or analytic (germ of a) variety at 0 which is invariant for X.
In dimension n = 2, the problem is completely solved after Camacho and Sad's Separatrix Theorem [4] that asserts that at any singular point of a holomorphic planar vector field there is an analytic invariant curve (called a separatrix).The situation is much more complicated in higher dimension.For example, we can mention several works by Luengo et al. [8,12], where some examples of singularities of holomorphic vector fields in dimension greater than two with no invariant analytic curve are exhibited.
By contrast, we have reasonable results concerning existence of invariant varieties if the singularity is assumed to be "non degenerated" (usually, that the linear part D 0 X of the vector field at 0 is non-nilpotent) and "sufficiently generic" (that the eigenvalues of that linear part satisfy certain generic condition).For instance, returning to the planar case, if the eigenvalues λ, µ of the linear part D 0 X satisfy the non-resonance condition λ = kµ for any natural number k ≥ 2, it is easy to see, using indeterminate coefficients, that there exists a unique formal invariant curve Γ µ at the origin which is non-singular and tangent to the eigendirection corresponding to µ.Moreover, we have a classical and well known theorem due to Briot and Bouquet [3] that asserts that if µ = 0 then Γ µ is convergent (and thus gives rise to a non-singular analytic separatrix).It is worth to mention that Briot-Bouquet's Theorem is an essential ingredient in the proof of Camacho-Sad's Separatrix Theorem: after a process of reduction of singularities of a planar vector field X by blowingups, due to Seidenberg [15], one finds a non-degenerate singular point with the hypothesis of Briot-Bouquet's Theorem in such a way that the non-singular separatrix assured by this theorem is not destroyed by the process and produces a separatrix at the initial singular point 0 ∈ C 2 , which a priori is singular.
Different ways of generalizing Briot-Bouquet's Theorem to higher dimension are conceivable.This work is devoted to state and prove one of such generalizations, although we are not intended for the moment to use it for more general results about existence of invariant analytic varieties of holomorphic vector fields.The result is motivated by the fact that, in the planar case, the two hypothesis λ = kµ for k ≥ 2 and µ = 0, are equivalent to the condition inf Our main theorem is the following (see below for a more precise statement).
Main Theorem.Let X be a holomorphic vector field at 0 ∈ C n , E a linear r-dimensional subspace of C n invariant for the linear part D 0 X, and put Spec(D 0 X| E ) = {µ 1 , . . ., µ r } and Spec(D 0 X) = {µ 1 , . . ., µ r , λ 1 , . . ., λ n−r } where the µ i or the λ j need not be all different.Assume that there is α > 0 such that for j = 1, . . ., n − r and non-negative integers q i with q Then there exists a unique germ of analytic invariant r-manifold W E for X at 0 which is tangent to E at the origin.
The hypothesis (1) implies in particular the non-resonance condition (2) for any j and non-negative integers q i with q 1 + • • • + q r ≥ 2. This last condition is the only needed condition for having just a formal r-dimensional manifold W E tangent to the linear space E at 0 and invariant for X, but it is not sufficient for analyticity.We recall this well known result below (Theorem 3.1).On the other hand, it is worth to mention that condition (1) is equivalent to the two conditions: (a) the non-resonance condition ( 2) and (b) the convex hull of {µ 1 , . . ., µ r } in C does not contain the origin.Expressed in these terms, our condition ( 1) is an analogous counterpart for the existence of analytic invariant manifolds to the condition assumed for the Linearization Poincaré's Theorem (see for instance [1], [2] or [7] for a proof): X is analytically equivalent to D 0 X at the origin if the convex hull of the eigenvalues of D 0 X does not contain the origin and they have no resonances (condition (2) holds when we take the µ's as the totality of the eigenvalues).
We cannot take for granted that our result is completely new and unknown.Surely many specialists in dynamical systems could be aware of it or of near formulations of it.However, we did not find any reference where the result is stated and proved in the way we do here.
On the other hand, we give an interesting application of the Main Theorem in the last section of the paper.We show that the stable and unstable manifolds (the so called strong invariant manifolds) of a real analytic vector field at a singular point are analytic.This result is quite well known for all specialists in dynamical systems or vector fields.It appears mentioned in many general references (see for instance [1] or [14]) and proved in several particular cases (see for instance Hadamard's work [9] or [13], which uses the same arguments).A complete recent proof can be deduced from a more general result established in [10].
In all these references, the proof consists in constructing the real strong manifold as the real trace of a complex manifold obtained as a graph of a uniform limit of a sequence of holomorphic maps.The proof that we present here, as a consequence of the Main Theorem, is more direct and (maybe) more elemental: we just prove that the (real) formal stable or unstable invariant manifold, which exists and it is unique by the general theory of existence of formal invariant manifolds, is convergent.

Some properties of formal power series
In this section we fix notations and establish very basic results about formal power series that will be used in the proof of the main result, Theorem 3.3 below.If x = (x 1 , . . ., x r ) is an r-tuple of variables and q = (q 1 , . . ., q r ) ∈ Z r ≥0 , we denote, as usual, ] of the ring of complex formal power series in r variables, written as a = q∈Z r ≥0 a q x q , will be also written as the sum of its homogeneous components a = ∞ k=0 a (k) , where a (k) = |q|=k a q x q .
For each positive integer k, consider P k the C-vector space of homogeneous polynomials of total degree k in the variables x 1 , . . ., x r endowed with the norm k defined by Notice that we have the inequality To any a ∈ C[[x 1 , . . ., x r ]], we associate the real series in a single variable called the majorant of a.It is clear that a is a convergent series if and only if its majorant a is a convergent series.
On the other hand, the operation of taking the majorant does not give a homomorphism between the rings of formal power series.It has, however, useful properties concerning domination of series.To be more precise, given series In fact, we will need a more general result about domination of majorant series obtained by composition: . ., x r , y 1 , . . ., y s ]] the series obtained from g replacing any of its coefficients by its corresponding norm and consider the composition G(x 1 , . . ., x r ) = g(x 1 , . . ., x r , h 1 , . . ., h s ).Then we have G |g|(t, . . ., t, h 1 , . . ., h s ).
Proof: Write g = q∈Z r ≥0 , J∈Z s ≥0 g q,J x q y J , where x = (x 1 , . . ., x r ), y = (y 1 , . . ., y s ).For each J = (j 1 , . . ., j s ) ∈ Z s ≥0 and i = 1, . . ., s, write the j i -th power of the series h i as the sum of its homogeneous components Note that each term in this equation has degree Taking norms and using (3), we have On the other hand, if we write the j i -th power of the majorant h i as Comparing ( 5) and ( 6), the result follows once we prove that, for any j i and for any integer k ≥ j i , we have In order to prove (7), put for simplicity h = h i , j = j i and write h = k≥1 h (k) , the sum of homogeneous components of h.The homogeneous component of degree Equation ( 7) follows from this last equation together with Together with Proposition 2.1 we will use the following result about domination of the majorant of the partial derivatives of a series.Proposition 2.2.Let a ∈ C[[x]] be a formal power series where x = (x 1 , . . ., x r ).For i = 1, . . ., r, we have Proof: Write a = q∈Z r ≥0 a q x q = k≥0 a (k) with the same notations introduced above.We have that, for any k ≥ 1, ∂a (k)  ∂xi is the homogeneous component of ∂a ∂xi of order k − 1. Therefore ( 8) On the other hand, we have ∂a ∂xi = q=(q1,...,qr)∈Z r ≥0 |q|=k q i a q x q−ei , where e i is the i-th vector of the standard basis of C r and |q i a q | ≤ |q||a q | for any q ∈ Z r ≥0 .Thus and the proof follows from (8) and the definition of the majorant series (4).

Invariant manifolds of holomorphic vector fields
During the rest of the paper we are interested in the following situation.Let X be a germ of a holomorphic vector field at 0 ∈ C n so that X(0) = 0 and denote by D 0 X its linear part at the origin.Suppose that we have a linear subspace E of dimension r of C n , with 0 < r < n, which is invariant for D 0 X.The question is whether there exists a non-singular r-dimensional analytic manifold W E through the origin, whose tangent space at 0 is equal to E and invariant for the vector field X.
It is well known that such an invariant manifold W E does not exists in general, so that several conditions must be imposed.If we are only interested in formal invariant manifolds, there is a well known and satisfactory result which we want to recall here.
A proof of this result can be derived from the corresponding one in [6, Theorem 3.7, p. 417].However, we outline here the main arguments for two reasons.On one hand, below we will explicitly use several notations appearing in the proof.On the other hand, our hypothesis are slightly weaker than the ones stated in the aforementioned reference (there, it is assumed that there exists another linear subspace F , invariant for D 0 X and complementary to E, but one can see that this last condition is inessential).
Outline of the proof of Theorem 3.1: Choose analytic coordinates (x, y) = (x 1 , . . ., x r , y 1 , . . ., y s ) at 0 ∈ C n such that E is tangent to {y = 0}.We look for W E given by the graph of a formal map (10) where, for any j, h j (0) = 0 and h j (0) = 0 (tangent to E), and such that, if the local expression of X is where a = (a 1 , . . ., a r ) and b = (b 1 , . . ., b s ) are vectors of convergent power series, then h = (h 1 , . . ., h s ) is a formal solution of the the system of partial differential equations written in matricial notation as Since E is invariant for D 0 X we may write the linear part as a block-triangular matrix where A is the matrix of the restriction D 0 X| E .Thus equation ( 11) becomes where f = (f 1 , . . ., f r ) and g = (g 1 , . . ., g s ) are vectors of convergent power series in the variables (x, y) of order greater or equal than two.Write each h j as sum of its homogeneous components and let F (k) and G (k) be the homogeneous components of degree k of f (x, h(x)) and g(x, h(x)), respectively.Notice that F (1) = G (1) = 0.
Recall also that we are looking for a solution h for which h (1) = 0. Therefore, for k ≥ 2, comparing the homogeneous components of degree k of both sides of equation ( 12), we obtain Denote by P k the vector space over C of homogeneous polynomials of degree k.Notice that, due to the fact that f and g have order greater or equal than two as series in the variables (x, y), in the right-hand side of (13) only the homogeneous components h (2) , . . ., h (k−1) are involved, but not h (k) .On the other hand, we write the left-hand side of equation (13) as L k,A,B (h (k) ), where L k,A,B : (P k ) s → (P k ) s is the linear operator defined as The spectrum of L k,A,B is precisely the set q, µ − λ j | q ∈ Z r ≥0 , |q| = k, j = 1, . . ., s (see the proof in [6, Lemma 2.5, p. 409]).The hypothesis of nonresonance (9) implies then that L k,A,B is non-singular and equation ( 13) can be solved recursively and uniquely for h (k) by means of the formula ( 14) This finishes the proof of Theorem 3.1.Notice that if A, B are real matrices then L k,A,B preserves the set of real elements in (P k ) s .This remark will we used in Section 4.
If all the components h j of h in (10) are convergent power series, then their sum will give rise to a non-singular analytic manifold W E , tangent to E at the origin and which is invariant for X in virtue of (11).Moreover, its germ at the origin is the unique germ of analytic r-manifold which is invariant for X and tangent to E, by uniqueness in Theorem 3.1.We will simply say that the formal invariant manifold W E is convergent.
It is well known that W E does not need to be convergent.A typical example is Euler's equation, associated to the planar vector field for which E is the line {y = x}.The eigenvalue, µ, associated to E is equal to 0; in particular we have the non-resonance conditions ( 9) and Theorem 3.1 applies.But the corresponding formal invariant curve W E is not convergent: one checks that W E is the graph of the series h A classical result due to Briot and Bouquet asserts that the annulation of the eigenvalue µ is the only obstruction to convergence in the planar situation.
Theorem 3.2 (Briot-Bouquet).Let X be a germ of holomorphic vector field at 0 ∈ C 2 .Let Spec(D 0 X) = {µ, λ} be the spectrum of the linear part and let E be an invariant line of D 0 X associated to the eigenvalue µ.Assume that conditions (9) hold.If µ = 0 then the formal curve W E is convergent.
A modern proof of Briot and Bouquet's Theorem can be found in [5].It is worth to mention that the situation considered in that reference is slightly more restrictive than the one stated here: the hypothesis about the eigenvalues is that µ = 0 and λ/µ is not a positive rational number (X is said to have a simple singularity at the origin).However, their proof works as well with the weaker hypothesis (9); that is, λ = kµ for any integer k ≥ 2. In fact, this last condition is the one stated in the original Briot and Bouquet's paper [3,Théorème XVIII,p. 168].
Notice that in the notations of Briot-Bouquet's Theorem, the nonresonance conditions ( 9), together with µ = 0, are equivalent to the condition This property motivates our main result, a generalization of Theorem 3.2 in higher dimension.
(In particular the non-resonance conditions (9) hold.)Then the formal invariant manifold W E given by Theorem 3.1 is convergent.
The idea of the proof is to reduce the problem to the planar situation and apply Briot-Bouquet's Theorem (Theorem 3.2).We divide the proof in several steps.
Step 1. Choosing coordinates.Put s = n − r.Take coordinates (x, y) at 0 where x = (x 1 , . . ., x r ) and y = (y 1 , . . ., y s ) such that E is tangent to {y = 0}.In these coordinates we may write where A is a square r-dimensional matrix, the matrix of the restriction D 0 X| E in the x coordinates.
Step 2. The equations of W E .As we have done in the proof of Theorem 3.1, if the local expression of X in the chosen coordinates is then the formal invariant manifold W E is given by the graph y = h(x) of the vector of formal power series h ) s which is a solution of the system (12) of PDEs.We have to show that any component h i of h is a convergent series in the variables x.We write the system (12) in a different way.First put Write f = (f 1 , . . ., f r ), g = (g 1 , . . ., g s ) the components of f and g, where f j , g j ∈ C{x, y} and write also C = (c lm ) 1≤l≤r 1≤m≤s . Then ( 12) is equivalent to for i = 1, . . ., s.
Step 3. Formulas for the homogeneous components of u i .By the especial form of the matrices A and B we have Writing explicitly the homogeneous components i,q x q , we can write where e i is the i-th vector of the standard basis of C r .With the convention that h (k) i,q−ej−1+ej = 0 if q j−1 = 0, and changing the index q by q+e j−1 −e j in the first summand in (17), the homogeneous component of degree k of u i is given by (18) u Step 4. Relation between the norms of u (k) i and h (k) j .For each q in Z r ≥0 with |q| = k, we consider the triangle inequality Summing up over all q with |q| = k and using the hypothesis (15), we obtain, from (18), Notice that if q j = k then q j−1 = 0 and so h (k) i,q−ej−1+ej = 0 by the convention we have established.Thus, |q|=k r j=2 σ(q j + 1) h Since we have chosen σ with σr < α 2 we have finally Step 5. Using majorants.In order to prove that the series ] is convergent, we will prove that its majorant It is enough to show that the series We look for a differential inequality for H.In one hand, adding the inequalities (20) for any i = 1, . . ., s, we have is a convergent series.
To finish the proof, we are going to show that any coefficient z n of the series z(t) is non-negative and that, if we write H = ∞ n=2 H n t n , then H n ≤ z n for any n ≥ 2.
We proceed by induction on n.Looking more carefully how the series F (t, z) and G(t, z) are constructed (23), we can write where, for each n ≥ 2, F n and G n are polynomials in n − 2 variables with non-negative coefficients (F 2 and G 2 are non-negative constants).Consequently, we have also

Inequality (24) and equation (25) give for
Then H 2 ≤ z 2 .Suppose we have shown that H k ≤ z k for k ≤ n − 1 where n ≥ 3. Since the coefficients of F k and G k are non-negative, using as was to be proved.

Application to real analytic vector fields
In this final section we give an application of our main result, Theorem 3.3, to (germs of) real analytic vector fields X at 0 ∈ R n .
For such a vector field X, denote by D 0 X its linear part at the origin and consider the decomposition into the linear subspaces associated to the eigenvalues of D 0 X with positive, zero and negative real parts, respectively.Put also E cs = E s ⊕ E c , E cu = E u ⊕ E c .A well known result in the theory of invariant manifolds (see for instance [11] or [7]) asserts that, for any k ∈ Z ≥1 , there are smooth embedded manifolds W s k , W cs k , W c k , W cu k , W u k of class C k in a neighborhood of the origin which are invariant by X and such that T 0 W k = E with = s, cs, c, cu, u.They are called local stable, center-stable, center, center-unstable and unstable manifold of class C k of X at 0, respectively.Moreover, the stable and unstable manifolds are uniquely determined by dynamical properties, so that they do not depend on k (thus they are of class C ∞ ).Both are called strong invariant manifolds.In contrast, the weak invariant manifolds W cs k , W c k , W cu k , are not unique, may depend on k and may not be even of class C ∞ at the origin.
The existence (and uniqueness) of such invariant manifolds in the formal setting is easier, however much less treated in the literature.Also, the strong manifolds are (real) analytic at the origin.We summarize these results in the following statement, which can be proved as a corollary of Theorem 3.1 and our main result, Theorem 3.3.invariant manifold W E s given by Theorem 3.1 and, being Re(µ i ) < 0 for any eigenvalue µ i corresponding to E s and Re(λ j ) ≥ 0 for any eigenvalue corresponding to the complementary space E cu , the condition (15) in Theorem 3.3 holds and thus W s is convergent.
in a single variable t, we say that g dominates f and we write f g if |f n | ≤ |g n | for any n, where | • | denotes the usual norm on complex numbers.Then, using inequality (3), if a, b ∈ C[[x 1 , . . ., x r ]], we can assert a + b a + b and a • b a • b.