On the universal unfolding of vector fields in one variable: A proof of Kostov's theorem

In this note we present variants of Kostov's theorem on a versal deformation of a parabolic point of a complex analytic $1$-dimensional vector field. First we provide a self-contained proof of Kostov's theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $C^\infty$ case, where we show that only versality is possible.


Introduction
Let us consider a singular point of a germ of analytic vector field X on (C, 0). If the singular point is simple, then the germ of vector field is analytically linearizable. If the singular point is multiple, also called parabolic, then the vector field is analytically conjugate to any one of the following normal forms: where µ = Res x=0 X −1 is the residue of the dual form. The first normal form is more frequent in the older works of the Russian school. The second one is easier to manipulate; for instance, the rectifying coordinate (time coordinate) X −1 is simple to calculate.
The next natural question is to consider normal forms for unfoldings of germs of analytic vector fields at a singular point. When the singular point is simple the normal form of an unfolding is linear, and hence unique. When the singular point is parabolic, Kostov proved that the following standard deformation of (1.1) is versal [Kos84]: (1.3) X 1 (x; y) = x k+1 + y k−1 x k−1 + . . . + y 1 x + y 0 − (µ + y 2k+1 )x 2k+1 ∂ ∂x .
The proof uses that (1.3) is an infinitesimal deformation of (1.1), and then calls for the machinery of Martinet's Reduction Lemma (see for instance [Arn85]). The philosophy behind this normal form is two-fold. First, a parabolic point of codimension k is the merging of k + 1 simple singular points, each having its own eigenvalue, which is an analytic invariant. Hence it is natural that a full unfolding would involve k + 1 parameters. Second, the geometry of an unfolding of a parabolic point is simple, hence the convergence to the normal form.
Kostov's normal form is very important for many bifurcations problems. For instance, when one studies the unfolding of a parabolic point of a germ of 1diffeomorphism of (C, 0) (i.e. a multiple fixed point), then a formal normal form is given by the time one map of a vector field of the form (1.3). The change of coordinate to this normal form diverges and the obstruction to the convergence is the classifying object of the unfoldings (see for instance [MRR04], [Rou15], and [Rib08a,Rib08b]). The same normal form is used to classify germs of unfoldings of 2-dimensional vector fields in (C 2 , 0) with either a saddle-node or a resonant saddle point: indeed the vector fields are orbitally analytically equivalent if and only if the holonomy map of the separatrices (the strong separatrices in the case of a saddle-node) are conjugate (see [RoC07] and [RoT08]).
Very soon, other normal forms equivalent to (1.3) appeared in the literature without proof: and they are all called Kostov's theorem. In practice, most authors use the normal form (1.5), which is much more suitable for computations.
The paper [RoT08] indirectly suggests that the normal form (1.5) is universal by showing that the normal form (1.5) associated to a generic k-parameter unfolding of a parabolic point of codimension k is unique up to the action of the group Z/kZ of rotations of order k. This uniqueness property is extremely important in all classification problems of unfoldings under conjugacy or analytic equivalence: it shows that the parameters of the normal forms are essentially unique and hence analytic invariants of the unfoldings. Hence, to show that two unfoldings are analytically equivalent, the first step is to change to the canonical parameters and it then suffices to study the equivalence problem for fixed values of the parameters.
In this paper, we provide self-contained proofs that the three normal forms (1.3), (1.4) and (1.5) are unique up to the action of the group Z/kZ, and universal. These self-contained proofs are useful for further generalizations, for instance when the vector field has some symmetry or reversibility property, and also for the formal case, the C ∞ -case, and mixed cases where the variable is analytic and the dependence on the parameters is only real-analytic: this mixed case occurs when one considers bifurcations of antiholomorphic parabolic fixed points (i.e. f (x) =x ±x k+1 + o(x k+1 )).
As a second part of the paper, we briefly address the real analytic, formal, and smooth cases. In the first two cases, each of the corresponding unfoldings is universal. In the smooth case, we give an explicit example showing that the unfolding is only versal and cannot be universal, namely the two vector fields X(x; λ) = (x 2 + λ 2 ) ∂ ∂x , and X ′ (x; λ) = (x 2 + (λ + ω(λ)) 2 ) ∂ ∂x are C ∞ -conjugate when ω(λ) is infinitely flat at λ = 0. Let us explain one difference with the analytic case. In the latter case the eigenvalues at the singular points are complex C 1 -invariants and for a given set of k +1 eigenvalues at the singular points, there are only a finite number of solutions for the y j in any of the normal forms (1.3), (1.4) and (1.5) with the prescribed eigenvalues. In the smooth case, only the eigenvalues at the real singular points are C 1 -invariants and we can smoothly glue anything at the complex singular points. The two systems of our counterexample have purely imaginary singular points. An open question is to know if we have universal unfoldings in the smooth case when all the singular points are real.
The original articles [Kos84,Kos91] of Kostov cover a much more general case of deformations of differential forms of real power α. However, our goal is not to redo what has been done well by Kostov, but to provide an elementary and selfcontained proof in the case of vector fields, that is power α = −1, which is why we do not discuss the other cases. Nevertheless, we believe that our proof of the uniqueness in the formal/analytic case, which is missing in Kostov's article, could be well adapted to general α.

The analytic theory
The following definitions are classical: see for instance [Arn83].
(1) Two germs of analytic (resp. real analytic, formal, C ∞ ) parametric families of vector fields X(x; λ), X ′ (x ′ ; λ) depending on a same parameter λ are conjugate if there exists an analytic (resp. real analytic, formal, C ∞ ) invertible change of coordinate changing one family to the other. We write X = φ * X ′ as a pullback of X ′ . (2) Let λ → λ ′ = ψ(λ) be a germ of analytic (resp. real analytic, formal, C ∞ ) map (not necessarily invertible), then X(x, λ) = X ′ (x, ψ(λ)) is a family induced from X ′ . (3) A parametric family of vector fields X(x; λ) is a deformation of X(x; 0). Two deformations X(x; λ), X ′ (x, λ) of the same initial vector field X(x; 0) = X ′ (x; 0) with the same parameter λ are equivalent (as deformations) if the two families are conjugate by means of an invertible transformation (2.
In this section we provide a self-contained proof of the following theorem: Theorem 2.2. In the analytic case, for k ≥ 1, the deformation (1.5) of (1.2) is universal.
As explained in the introduction, the proof of the versality is due to Kostov (for (1.3) see [Kos84], while for (1.5) it has been often stated in literature without explicit proof, see e.g. [Arn93, p.116]), and the uniqueness comes from [RoT08]. Theorem 2.2 can be rephrased in more precise terms as the following theorem of which it is a direct consequence.
for some l ∈ Z k and some analytic germ t(λ).
The first step in proving Theorem 2.4 is the following "prenormal form", which can be also found for example in [Rib08a, Proposition 5.13].
The following lemma is classical (see for example [Tey04, Proposition 2.2] to which it is essentially equivalent).
Lemma 2.6. Let X 0 , X 1 be two germs of analytic families of vector fields vanishing at the origin, and assume there exists an analytic germ α(x, λ) such that 1+X0.α . Then the flow map of the vector field Y (x, t; λ) = ∂ ∂t − αX0 , conjugates X 1 with X 0 = φ * 1 X 1 . The statement is also true if α is meromorphic such that X 0 .α and αX 0 are analytic and vanish for (x, λ) = 0 (so that the flow of Y is defined for all t ∈ [0, 1]).
Proof. Let X t = P (x,y) Q(x,t;u,µ) ∂ ∂x be as above (2.5). We want to construct a family of transformations depending analytically on t ∈ [0, 1] between X 0 and X t , defined by a flow of a vector field Y of the form for some ξ j and H, such that [Y , X t ] = 0, that is where U (x; u) = u 0 + . . . + u k−1 x k−1 and Ξ(x; ξ) = ξ 0 + . . . + ξ k−1 x k−1 , which is equivalent to (2.6) H ∂ ∂x P − P ∂ ∂x H + QΞ = U P. We see that we can choose H as a polynomial in x: H = h 0 (t, y; u, µ) + . . . + h k (t, y; u, µ)x k .

Proof of Theorem 2.4.
(i) The existence of an analytic normalizing transformation to (2.3) when k ≥ 1 follows directly from Propositions 2.5 and 2.7. For k = 0 it follows from Lemma 2.6.
Suppose now that f ∈ I n λ . Developing the right side of the transformation equation The left side being a polynomial of order ≤ k − 1 in x, this means that both sides vanish modulo I n+1 λ . Therefore on the left side P = P ′ mod I n+1 λ , while the right side can be rewritten as Proof of Corollary 2.3. Consider the two families (1.3) and (1.5) By Theorem 2.4 we know that there exists a map x ′ = φ(x, y), y ′ j = ψ j (y), j = 0, . . . , k − 1, 2k + 1, such that X 1 (x, y) = φ * X 3 (x, ψ(y)), that is, such that We want to show that ψ is invertible. For y = 0 we have and (up to a pre-composition with a flow map of X 1 killing the term in x k+1 ) we can assume that φ(x, 0) = x + µ 2 1 k x 2k+1 + O(x 2k+2 ).

Real analytic, formal and smooth theory
Theorem 3.1 (Real analytic theory). The statement of Theorem 2.4 is also true in the real analytic setting, with the exception that (2.3) needs to be replaced by Consequently, the real analytic parametric family is a universal real analytic deformation for X 3,real (x; 0).
Proof. Assuming the initial vector fieldX is real analytic then so are all the transformations of Propositions 2.5 and 2.7 with the only exception: the leading coefficient ofX(x; 0) = cx k+1 + . . . ∂ ∂x can be brought to either c = ±1 if k is even, and c = 1 if k is odd.
Corollary 3.2. If we consider deformations which are symmetric (resp. antisymmetric (also called reversible)) with respect to the real axis, then their associated universal deformations y 0 , . . . y k−1 , cy 2k+1 ∈ R, cµ ∈ R, with c = (±1) k+1 (resp. c = i (±1) k+1 ), have the same property, and the conjugacy commutes with the symmetry. (1) The statement of Theorem 2.4 and therefore of Theorem 2.2 is also true in the formal setting, of formal parametric germs of vector fields and formal transformations (2.1), where by formal we mean a formal power series in (x, λ). In the formal real case (i.e. the series have real coefficients), then the normal form is given by (3.1).
(2) (Ribon [Rib08a, Proposition 6.1]) Two analytic germs of vector fields X, X ′ are formally conjugate if and only if they are analytically conjugate. Moreover, denotingÎ λ the ideal of formal series that vanish when λ = 0, if φ(x, λ) is a formal conjugating transformation, then for any n > 0 there exists an analytic conjugacy φ n (x, λ), φ * n X ′ = X, such that φ n =φ modÎ n λ . The second statement is an analogue of the Artin approximation theorem. Proof.
(1) The proof follows exactly the same lines. The key fact is that a formal flow map of a formal vector field which is analytic in t and the parameters (u, µ) is well defined: see Lemma 3.4 below.
(2) This is a consequence of the uniqueness of the normal form (2.3): each analytic germ of parametric vector field is analytically conjugate to a normal form (2.3), and two such normal forms are formally conjugate if and only if they are conjugate by a rotation x → e 2πi l k x, l ∈ Z k , which is analytic. Moreover, the formal conjugacy is a composition of the rotation and of a formal timet(λ)-flow map of the vector field. Replacingt(λ) with an analytic t n (λ) =t(λ) modÎ n λ does the trick.
Proof. For any n ∈ Z ≥0 , the n-jet with respect to the variable z ofŶ is an entire vector field j n zŶ (z, t) in C p × C with well defined flow z • exp(s j n zŶ ) fixing the origin in z. For any m ≤ n, the m-jet of this flow agrees with the m-jet of the one for m: j m z z • exp(s j n zŶ ) = j m z z • exp(s j m zŶ ) , meaning that they converge in the Krull topology as n → +∞ to a well-defined formal flow map z •ê xp(sŶ ). See also [IlY08, Theorem 3.9].
In real C ∞ -smooth setting, the deformation X 3,real (x; y) (3.2) is a versal deformation of the normal form vector field X 3,real (x; 0).
Proof. The only purely analytic tools used in the proof of the existence of a normalizing transformation were the Weierstrass preparation and division theorems (used in the proof of Proposition 2.7), which have their counterpart in the C ∞ -setting in the Malgrange preparation and division theorems [Mal64,GoG73].
The deformation (3.2) is not universal in the C ∞ -setting in general. The issue is the non-uniqueness in the Malgrange division and the lack of control over the potential non-real singularities in the family.
Problem 3.8. Can we expect uniqueness of the induced coefficients in (3.1) in the special case when the deformation is such that it has k +1 merging real singularities when counted with multiplicity?