Center conditions: Rigidity of logarithmic differential equations ^{1}

Hossein Movasati

Abstract

In this paper we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko’s result on Hamiltonian differential equations. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A’Campo’s theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss-Manin connection on the so called Brieskorn lattice/Petrov module of the polynomial. Some applications on the cyclicity of cycles and the Bautin ideals will be given.

0 Introduction

Let us be given the 1-form ω = P (x, y)dy − Q(x, y)dx (1)

where P and Q are two relatively prime polynomials in ℂ^{2}. The degree of ω is the maximum of deg(P) and deg(Q). The space of degree d ω’s up to multiplication by a constant, namely F(d), is a Zariski open subset of the projective space associated to the coefficient space of polynomial 1-forms (1) with deg(P), deg(Q) ≤ d. We denote the elements of F(d) by F(ω) or F if there is no confusion about the underlying 1-form ω in the text. Any F(ω) induces a holomorphic foliation F on ℂ^{2} i.e., the restrictions of ω to the leaves of F are identically zero. Therefore we name an element of F(d) a (holomorphic) foliation of degree d. But note that in the Introduction we do not need the geometric meaning of this word.

The points in Sing(F(ω)) = {P = 0, Q = 0} are called the singularities of F. A singularity p ∈ ℂ^{2} of F(ω) is called reduced if D(P, Q)(p) ≠ 0. A reduced singularity p is called a center singularity or center for simplicity if there is a holomorphic coordinates system (x̃, ỹ) around p with x̃(p) = 0, ỹ(p) = 0 such that in this coordinate system ω ∧ d(x̃^{2} + ỹ^{2}) = 0. The leaves of F around the center p are given by x̃^{2} + ỹ^{2} = const.. The subset of F(d) containing F(ω)’s with at least one center, namely M(d), form an algebraic subset of F(d) (see Appendix A). Now the problem of identifying the irreducible components of M(d) arises. This problem is also known by the name ”Center conditions” in the context of real polynomial differential equations.

H. Dulac in [Du] proves that M(2) has exactly 9 irreducible components (see also [CL] p. 601). In this case any foliation in M(2) has a Liouvillian first integral. Since this problem finds applications on the number of limit cycles in the context of real differential equations, this classification problem is very active. It is recommended to the reader to do a search with the title center/centre conditions in mathematical review to obtain many recent papers in this problem (We find also a similar problem for Abel equations y′ = p(x)y^{2} +q(x)y^{3}, p, q polynomials in x) In the context of holomorphic foliations we can refer to [CL],[Mu] and [Mo]. One of the main objectives in this paper is to introduce some new methods in this problem using an elementary algebraic geometry. We have borrowed many notions like Brieskorn lattice, Picard-Lefschetz theory and so on from the literature of singularities of holomorphic functions (see [AGV]) therefore a basic knowledge of this area of mathematics will be useful in understanding this work.

1Keywords: Holomorphic foliations, Picard-Lefschetz theory Math. classification: 32L30-14D05 Supported by Max Planck Institut Für Mathematik-Germany

Let us be given the polynomials f_{i}, deg(f_{i}) = d_{i}, 1 ≤ i ≤ s and non-zero complex numbers λ_{i} ∈ ℂ^{∗}, 1 ≤ i ≤ s. The foliation

F = F(f_{1} ⋯ f_{s} ∑_{i=1}^{s} λ_{i} \frac{df_{i}}{f_{i}}) (2)

is of degree d = ∑_{i=1}^{s} d_{i} − 1 and has the logarithmic first integral f_{1}^{λ_{1}} ⋯ f_{s}^{λ_{s}}. Let L(d_{1}, …, d_{s}) be the set of foliations (2). Since L(d_{1}, …, d_{s}) is parameterized by λ_{i}’s and the coefficients of f_{i}’s, it is irreducible. The main theorem of this paper is:

Theorem 0.1. \overline{L(d_{1}, …, d_{s})} is an irreducible component of M(d), where d = ∑_{i=1}^{s} d_{i} −1 and \overline{L(d_{1}, …, d_{s})} is the closure of L(d_{1}, …, d_{s}) in F(d).

In the case s = 1 we can assume that λ_{1} = 1 and so L(d+1) is the space of foliations of the type F(df), where f is a polynomial of degree d + 1. This case is proved by Ilyashenko in [Il]. The similar result for foliations with a first integral of the type \frac{F^{p}}{G^{q}} , \frac{deg(F)}{deg(G)} = \frac{q}{p}, g.c.d.(p, q) = 1 is obtained in [Mo]. Some basic tools of this kind of generalizations for Lefschetz pencils on a manifold is worked in [Mu].

Let us reformulate our main theorem as follows: Let F ∈ L(d_{1}, …, d_{s}) be given by (2), p one of the center singularities of F and F_{ϵ} a holomorphic deformation of F in F(d) such that its unique singularity p_{ϵ} near p is still a center.

Theorem 0.2. In the above situation, there exists an open dense subset U of L(d_{1}, …, d_{s}), such that for all F ∈ U, F_{ϵ} admits a logarithmic first integral. More precisely, there exist polynomials f_{iϵ}, deg(f_{iϵ}) = d_{i}, i = 1, …, s and non-zero complex numbers λ_{iϵ} such that F_{ϵ} is given by

f_{1ϵ} ⋯ f_{sϵ} ∑_{i=1}^{s} λ_{iϵ} \frac{df_{iϵ}}{f_{iϵ}} = 0

f_{iϵ} and λ_{iϵ} are holomorphic in ϵ and f_{i0} = f_{i}, λ_{i0} = λ_{i}, i = 1, …, s.

In the above theorem we can put U the complement of L(d_{1}, …, d_{s}) ∩ sing(M(d)) in L(d_{1}, …, d_{s}). One may not be satisfied with this U and try to find explicit conditions like as: A foliation F(f_{1} ⋯ f_{s} ∑_{i=1}^{s} λ_{i} \frac{df_{i}}{f_{i}}) in U satisfies 1. \{f_{i} = 0\} intersects \{f_{j} = 0\} transversally 2. f_{1}^{λ_{1}} ⋯ f_{s}^{λ_{s}} has nondegenerated critical points in ℂ^{2} − ∪_{i=1}^{s}{f_{i} = 0} and so on. In general one may be interested to identify the set L(d_{1}, …, d_{s}) ∩ sing(M(d)). In any case these questions are not in the focus of this paper.

The above theorem also says that the persistence of one center implies the persistence of all other centers and dicritical singularities {f_{i} = 0} ∩ {f_{j} = 0}, i, j = 1, …, s.

Since this paper is inspired by Ilyashenko’s paper [Il] let us give a sketch of the proof in the case L(d + 1): Let f be a degree d + 1 polynomial with the following condition

• f has d^{2} non-degenerate critical points with distinct images.

This condition is generic, i.e. in the space of polynomials of degree d + 1 there is an open dense subset such that for all f in this subset the condition is satisfied. Let df +ϵω +h.o.t. be a deformation of df such that the singularity near a center singularity of df , namely p_{1}, persists in being center. Then ∫_{δ} ω = 0 for all vanishing cycles δ around p_{1}. The action of the monodromy on a single vanishing cycle δ ∈ H_{1}(f^{−1}(b), ℤ), where b is a regular value of f, generates the whole homology (the most significant part of the proof). Therefore our integral is zero in all cycles in the fibers of f and so it is relatively exact. Knowing the fact deg(ω) ≤ d we conclude that ω = dP, where P is a polynomial of degree less than d + 1. Since M(d) is an algebraic set, the hypothesis on df is generic and the tangent vector ω of any deformation of df in M(d) is tangent also to L(d + 1), we conclude that L(d + 1) is an irreducible component of F(d).

Essentially the main steps of our proof is as the above proof. The main difference is that we have to work with a non-generic polynomial (Example 1.3) and in this case the associated foliation may lie in many irreducible components of M(d) (all L(d_{1}, …, d_{s})). Therefore our strategy will be to identify as far as possible these irreducible components. Let us explain the structure of this paper:

In § 1 we introduce the Brieskorn lattice/Petrov module H associated to a polynomial f in ℂ^{2} and the action of Gauss-Manin connection ∇ on it. The general objective is to obtain some information about H and ∇ when f has some bad critical fibers. Using a theorem of Mattei-Cerveau we prove Corollary 1.1 which is enough for our needs in this paper. This corollary classifies all ω ∈ H with ∇^{n}ω = 0 for a given natural number n. In § 2 we analyze the action of the monodromy on a Lefschetz vanishing cycle in Example 1.3. Using the well-known Theorem 2.2 and Gusein-Zade/A’Campo’s Theorem 2.1 we prove Theorem 2.3 and then Theorem 2.4. In § 3 we give the precise form of the tangent vector of a deformation of Example 1.3 with a persistent center. § 4 is devoted to the proof of Theorem 0.1. In § 5 we discuss our result in the context of real differential equations and its connection with concepts like cyclicity and Bautin Ideals. In § 6 we introduce some other problems concerning center conditions. In Appendix A we prove that M(d) is an algebraic set. The last Appendix is devoted to the proof of a lemma which does not fit in the paper. Following the arguments of this paper with Examples 1.1, 1.2 and specially 1.3 is recommended to the reader.

It is an interesting fact that we can restate center conditions for arbitrary algebraically closed field (see Appendix A). This arises the following question: Is Theorem 0.1 is true in an arbitrary algebraically closed field? Is there a completely algebraic proof for Theorem 0.1? The main obstacle in answering these questions is the Picard-Lefschetz theory which we use in this paper. One of the main source of this theory is the celebrated book of Lefschetz [Le], where Lefschetz’s first, second and hard theorems is proved by topological methods. All of these theorems have latter completely algebraic proofs in algebraic geometry and this was the reason for me to put these questions in this paper.

Acknowledgments: Here I would like to thank Max-Planck Institut for Mathematics in Bonn for financial supporting and for the exceptional ambient of research. I thank Alcides Lins Neto for putting the paper [Il] in the first steps of my mathematical carrier. Thanks go also to Claus Hertling and Susumu Tanabé for many useful conversations. I thank also Konstanz university for hospitality during a visiting.

1 Brieskorn lattices/Petrov Modules

Let f be a non-composite polynomial of degree d + 1 in ℂ^{2}, i.e. f cannot be composed as p ∘ g, where p is a polynomial of degree greater than one in ℂ and g is a polynomial in ℂ^{2}. It is easy to see that this condition is equivalent to the fact that for all b ∈ ℂ except a finite number the fiber f^{−1}(b) is irreducible. Let Ω^{i}, i = 0, 1, 2 be the set of polynomial differential i-forms in ℂ^{2} and ℂ[t] be the ring of polynomials in t. Ω^{i} is a ℂ[t]-module in the following way

p(t).ω = p(f)ω, p ∈ ℂ[t], ω ∈ Ω^{i}

The Brieskorn lattice/Petrov module

H = \frac{Ω^{1}}{dΩ^{0} + Ω^{0}df}

is a ℂ[t]-module. Also we define

V = \frac{Ω^{2}}{df ∧ Ω^{1}} ≅ \frac{ℂ[x, y]}{< f_{x}, f_{y} >}

Multiplying by f defines a linear operator on V which is denoted by A.

Lemma 1.1. If f has isolated singularities then the followings are true:

1. V is a ℂ-vector space of dimension μ, where μ is the sum of local Milnor numbers of f;

2. Eigenvalues of A are exactly the critical values of f.

This lemma is a well-known fact, for instance see [Ko] but I did not find its proof in the literature and so I have proved it in Appendix B. In fact the canonical restriction V → ⊕_{p} \frac{O_{ℂ^{2},p}}{<f_{x},f_{y}>}, where O_{ℂ^{2},p} is the ring of germs of holomorphic functions in a neighborhood of p in ℂ^{2} and the sum runs through all critical points of f, is an isomorphism of vector spaces.

From now on we assume that f has isolated singularities and we denote the corresponding critical values by c_{1}, c_{2}, …, c_{r}. We denote by p(t) the minimal polynomial of A. The Gauss-Manin connection

∇ : H → H/ < p >

where H/ < p > is the localization of H by the multiplicative subgroup < p >= {p^{i} | i = 1, 2, …} of ℂ[t], is defined as follows: For an ω ∈ H we have p(f)dω = 0 in V. Therefore there is a polynomial 1-form η in ℂ^{2} such that

p(f)dω = df ∧ η (3)

we define ∇ω = η/p(t). Of course we must check that this operator is well-defined. If η_{1} and η_{2} are two polynomial 1-forms satisfying (3) then (η_{1} − η_{2}) ∧ df = 0 and so by de Rham lemma η_{1} − η_{2} = P df (= 0 in H), for a P polynomial in ℂ^{2}. Also if ω = dP + Qdf, P and Q two polynomials in ℂ^{2}, then dω = dQ ∧ df and so ∇ω = dQ (= 0 in H).

Let H̃ be the localization of H by polynomials in t vanishing only on c_{i}’s. We can extend ∇ as a function from H̃ to H̃ by the rule

∇(ω/q) = (q∇ω − q′ω)/q^{2}

Let f_{1} = 0, f_{2} = 0, …, f_{k} = 0 be irreducible components of all critical fibers of f

D = ∑_{i=1}^{k}{f_{i} = 0}

and Ω̃^{i} be the set of meromorphic i-forms in ℂ^{2} with poles in D. We define H̃′ = \frac{Ω̃^{1}}{dΩ̃^{0}+Ω̃^{0}df}

and in a similar way as for H a connection ∇′ on H̃′.

Lemma 1.2. (H̃, ∇) is isomorphic to (H̃′, ∇′).

Proof. Every meromorphic 1-form in ℂ^{2} with poles in D determines a unique element of H̃ as follows: if ω = _{\frac{ω̃}{f_{i_{1}}^{r_{1}}}} and f − c_{1} = f_{i_{1}}^{k_{1}} ⋯ f_{i_{l}}^{k_{l}} is the decomposition of f − c_{1} to

irreducible factors then we multiply both ω̃ and f_{i_{1}}^{r_{1}} by f_{i_{1}}^{k_{1}m−r_{1}}f_{2}^{k_{2}m} ⋯ f_{i_{l}}^{k_{l}m}, where m is an integer number satisfying m − 1 < \frac{r_{1}}{k_{1}} ≤ m, and we obtain ω = \frac{˜ω̃}{(t−c_{1})^{m}}. Repeating this process by ˜ ω̃ we obtain an element of H̃. If ω_{1} = dP + Qdf, where P and Q are two meromorphic functions on ℂ^{2} with poles along D, then by applying the above method on P, Q we can see that ω is associated to zero in H̃. Therefore we obtain a map H̃′ → H̃ which is the inverse of the canonical map H̃ → H̃′ and so it is an isomorphism. Since ∇ and ∇′ coincide on H ⊂ H̃, H̃′, the mentioned isomorphism sends ∇′ to ∇.

Let b be a regular value of f and {δ_{t}}_{t∈(ℂ,b)}, δ_{t} ∈ f^{−1}(t) be a continuous family of cycles in the fibers of f. For an ω ∈ H̃ the integral ∫_{δ} ω is well-defined and

\frac{∂}{∂t} ∫_{δ_{t}} ω = ∫_{δ_{t}} ∇ω (4)

(see [AGV]). In fact this formula is a bridge between topology and algebra in this paper.

Our objective in this section is to analyze the action of ∇ on H. For this purpose let us state a classical theorem. Let ω be a meromorphic 1-form in ℂ^{2} and ∪_{i=1}^{k}{f_{i} = 0} be the pole locus of ω. Suppose that the multiplicity of ω along {f_{i} = 0} is r_{i}.

Theorem 1.1. ( [CM]) If ω is closed, i.e. dω = 0 then there are λ_{1}, …, λ_{k} ∈ ℂ and a polynomial g such that

1. If r_{i} = 1 then λ_{i} ≠ 0;

2. If r_{i} > 1 then f_{i} does not divide g;

3. ω can be written

ω = (∑_{i=1}^{k} λ_{i} \frac{df_{i}}{f_{i}}) + d(\frac{g}{f_{1}^{r_{1}−1} … f_{k}^{r_{k}−1}} )

Note that if ω has a pole of order r_{∞} at the line at infinity then the degree of g is ∑_{i=1}^{k} d_{j}(r_{i} − 1) + r_{∞} − 1 and r_{∞} + ∑_{i=1}^{k} λ_{i}d_{i} = 0.

Let L be the subset of H̃ containing the 1-forms of the type ∑_{i=1}^{k} λ_{i} \frac{df_{i}}{f_{i}}, λ_{i} ∈ ℂ. We

have the decomposition L = L_{1} ⊕ ⋯ ⊕ L_{r}, where L_{i} contains all the terms λ_{j} \frac{df_{j}}{f_{j}}, f_{j} = 0 being an irreducible component of the fiber f^{−1}(c_{i}). Note that if f^{−1}(c_{i}) is irreducible then L_{i} = 0.

Corollary 1.1. For the pair (H̃, ∇) and a positive integer number n 1.

Kernel(∇^{n}) = {ω ∈ H̃ | ω = ∑_{j=0}^{n−1} α_{j}t^{j}, α_{j} ∈ L}

2.

Kernel(∇^{n}) ∩ H = {ω ∈ H | ω = ∑_{j=1}^{n−1} ∑_{i=1}^{r} α_{ij}(t^{j} − c_{i}^{j}), α_{ij} ∈ L_{i}}

where ∇^{n} = ∇ ∘ ⋯ ∘ ∇ n-times.

Proof. Let us prove the first part by induction on n. We use the isomorphism in Lemma 1.2. If for ω ∈ H̃, ∇ω = 0 then dω = dP ∧ df, where P is a meromorphic function in ℂ^{2} with poles along D. Now d(ω − P df) = 0 and by Theorem 1.1 we have ω ∈ L. This proves the first part n = 1 of the induction. Now if ∇^{n+1}ω = 0 then by induction ∇ω = ∑_{j=0}^{n−1} α_{j}t^{j} = ∇ ∑_{j=0}^{n−1} \frac{α_{j}}{j+1}t^{j+1} or equivalently ∇(ω − ∑_{j=0}^{n−1} \frac{α_{j}}{j+1}t^{j+1}) = 0. Using the case n = 1 we finish the proof of the first part. Now let us prove the second part. Let ω = ∑_{j=1}^{n−1} α_{j}t^{j} + α_{0} ∈ H and α_{j} = ∑_{i=1}^{r} α_{ij} be the decomposition of α_{j}. We write

ω = ∑_{j=1}^{n−1} ∑_{i=1}^{r} α_{ij}t^{j} + α_{0} =

∑_{j=1}^{n−1} ∑_{i=1}^{r} α_{ij}(t^{j} − c_{i}^{j}) + ∑_{j=1}^{n−1} ∑_{i=1}^{r} α_{ij}c_{i}^{j} + α_{0}

The first summand in the above belongs to H and hence the second one belongs to L ∩ H and so the second summand must be zero.

Before we go to the next section let us give three simple but important examples. The last one has a very special role in this paper. Corollary 1.1 with n = 2 will be used in the next section. Therefore we explain it with these examples.

Example 1.1. f = (x^{2} + y^{2} − 1)x. Since xy = 0, 3x^{2}x^{i} = x^{i}, y^{2}y^{i} = y^{i}, i ≥ 1 in V, 1, x, y, x^{2} form a basis for the vector space V. f has four critical points p_{1} = (0, 1), p_{2} = (0, −1), p_{3} = (\sqrt{1/3}, 0), p_{4} = (−\sqrt{1/3}, 0) with three critical values c_{1} = c_{2} = 0, c_{3} = −2/3\sqrt{1/3}, c_{4} = 2/3\sqrt{1/3}. We define ω_{1} = (x^{2} + y^{2} − 1)dx. We have

∇ω_{1} = −\frac{xd(x^{2} + y^{2} − 1)}{t} = \frac{ω_{1}}{t} , ∇^{2}ω_{1} = 0

Since f^{−1}(0) is the only reducible fiber of f, by Corollary 1.1, 2 we know that any other 1-form in H with the property ∇^{2}ω_{1} = 0 is some multiply of ω_{1} by a constant.

Example 1.2. f = xy(x + y − 1). There are four critical points p_{1} = (0, 0), p_{2} = (1, 0), p_{3} = (0, 1), p_{4} = (1/3, 1/3) with two critical values c_{1} = c_{2} = c_{3} = 0, c_{4} = −1/27. Knowing Corollary 1.1, we can see that the 1-forms ω_{1} = x(x+y−1)dy, ω_{2} = y(x+y−1)dx form a basis for the vector space {ω ∈ H | ∇^{2}ω = 0}.

Example 1.3. The lines l_{p} = (d − p)x + py − p(d − p) = 0, p = 0, 1, …, d are in a general position in ℝ^{2} i.e., they are distinct and no three of them have a common intersection point (l_{p} is the line through (p, 0), (0, d − p)). The polynomial f = l_{0}l_{1} ⋯ l_{d} satisfies the properties

1) All the critical points of f in ℂ^{2} are real and non-degenerated;

2) The values of f at all saddle critical points equal zero; By a small perturbation of the lines l_{i} we also get the property

3) The values of f at center critical points are distinct. (See Appendix of [Mo2] for this kind of arguments). In a real coordinates system (x̃, ỹ) around a saddle (resp. center) critical point p the function f can be written as f(p)+x̃^{2}−ỹ^{2} (resp. f(p) + x̃^{2} + ỹ^{2}). f has a_{2} = \frac{d(d+1)}{2} saddle critical points, a_{1} = ∑_{i=2}^{d}[\frac{i−1}{2}] center critical points with negative value and a_{3} = \frac{d(d−1)}{2} − a_{1} center critical points with positive value, where [q] is the integer number satisfying [q] − 1 < q ≤ [q]. By Corollary 1.1 the set of ω ∈ H with ∇^{2}ω = 0 is a vector space generated by

l_{0}l_{1} ⋯ l_{p−1}l_{p+1} ⋯ l_{d}dl_{i}, p = 0, 1, …, d − 1

Note that ∑_{p=0}^{d} l_{0}l_{1} ⋯ l_{p−1}l_{p+1} ⋯ l_{d}dl_{i} = df = 0 in H.

In what follows when we refer to the polynomial f in Example 1.3 we mean the one with a small perturbation of the lines l_{p} and hence satisfying the property 3.

Remark 1.1. E. Brieskorn in [Br] introduced three O_{(ℂ,0)}-modules H, H′, H′′ in the context of singularity of holomorphic functions f : (ℂ^{n}, 0) → (ℂ, 0) (The H used in this paper is the equivalent of Brieskorn’s H′). After him these modules are called the Brieskorn lattices and recently the similar notions in a global context are introduced by many authors (see [Sa],[DS], [BD],[Mo]). In the context of differential equations H appears in the works of G.S. Petrov [Pe] on deformation of Hamiltonian equations of the type d(y^{2}+p(x)), where p is a polynomial in x and is named by L. Gavrilov in [Ga2] the Petrov module. For this reason I have used the both names Brieskorn lattice and Petrov module for H.

Restriction of an ω ∈ H to each fiber defines a global section of the cohomology fiber bundle of the function f and looking in this way ∇ is the usual Gauss-Manin connection in the literature. For this reason I have named ∇ again the Gauss-Manin connection. But of course we can name ∇ω the Gelfand-Leray form of ω (see [AGV]).

2 Action of the monodromy

Suppose that f is a polynomial function in ℝ^{2} with the properties 1,2,3 in Example 1.3. For a c ∈ ℂ we define L_{c} = f^{−1}(c) in ℂ^{2}. {δ_{t}}_{t∈(ℂ,c)} with δ_{t} ∈ H_{1}(L_{t}, ℤ) denotes a continuous family of cycles.

Choose a value b ∈ ℂ with Im(b) > 0 and fix a system of paths joining b with the critical values of f, subject to the condition that these paths lie in their entirety in the upper halfplane Im(z) > 0 except for the ends which coincide with the critical values. Now we can define a distinguished basis of vanishing cycles in H_{1}(L_{b}, ℤ) (see [AGV] for the definition). Critical points of f are in one to one correspondence with the self intersection points of the real curve f = 0, namely p_{j}, a_{1} + 1 ≤ j ≤ a_{1} + a_{2}, and relatively compact components of its complement, namely U_{i}^{0}, 1 ≤ i ≤ a_{1}, U_{k}^{2}, a_{1} + a_{2} + 1 ≤ k ≤ a = a_{1} + a_{2} + a_{3}. U_{i}^{0} contains a critical point of f with negative value and U_{k}^{2} contains a critical point with positive value. We denote by

δ_{i}^{0}, δ_{j}^{1}, δ_{k}^{2}, 1 ≤ i ≤ a_{1}, a_{1} + 1 ≤ j ≤ a_{1} + a_{2}, a_{1} + a_{2} + 1 ≤ k ≤ a (5)

the distinguished basis of vanishing cycles. δ_{i}^{0}, δ_{k}^{2} are called the center vanishing cycles and δ_{j}^{1} is called the saddle vanishing cycle.

Theorem 2.1. (S. Gusein-Zade, N. A’Campo) After choosing a proper orientation for the cycles δ_{i}^{σ} we have

• < δ_{i}^{σ}, δ_{j}^{σ} >= 0;

• < δ_{i}^{0}, δ_{j}^{1} > equal to the number of vertices of the polygon U_{i}^{0} coinciding with the point p_{j};

• < δ_{k}^{2}, δ_{j}^{1} > equal to the number of vertices of the polygon U_{k}^{2} coinciding with the point p_{j};

• < δ_{k}^{2}, δ_{i}^{0} > equal to the number of common edges of U_{k}^{2} and U_{i}^{0}.

This theorem, in an apparently local context, is proved by S. Gusein-Zade [Gu],[Gu1] and N. A’Campo [AC],[AC1] independently. However the proof in our case is the same. The above theorem gives us the Dynkin diagram of f (see [AGV]).

Now let us state another theorem which we will use in this paper:

Theorem 2.2. In the above situation, the vanishing cycles δ_{i}^{0}, δ_{j}^{1}, δ_{k}^{2} generate H_{1}(L_{b}, ℤ) freely.

The proof of the above theorems is classical and the reader can consult with [DN],[Mo2]. Also the main core of the proof can be found in [La].

Let us compactify ℂ^{2} in ℙ^{2} = {[x; y; z] | (x, y, z) ∈ ℂ^{3} −0}. Here ℂ^{2} = {[x; y; 1] | x, y ∈ ℂ} and 𝕃 = {[x; y; 0] | x, y ∈ ℂ} is the projective line at infinity. Let f = f_{0}+f_{1}+⋯+f_{d+1} be the decomposition of f to homogeneous parts. We look at f as a meromorphic function on ℙ^{2} by rewriting f as

f = f(x/z, y/z) = \frac{z^{d+1}f_{0}(x, y) + z^{d}f_{1}(x, y) + ⋯ + f_{d+1}(x, y)}{z^{d+1}}

The indeterminacy set of f is given by

R = {[x; y; 0] | f_{d+1}(x, y) = 0}

Now suppose that R has d+1 distinct points. For instance the polynomial in Example 1.3 has this property. This implies that the fibers of f intersect the line at infinity transversally. Doing just one blow-up in each point of R and using Ehresmann’s fibration theorem we conclude that the map f is a C^{∞} fibration on ℂ − C, where C = {c_{1}, …, c_{r}} is the set of critical values of f. In general case we must do more blow-ups to obtain this conclusion and the set of critical points of f may be a proper subset of C. Therefore we have the action of the monodromy on the first (co)homology group of L_{b}:

h : π_{1}(ℂ − C, b) × H_{1}(L_{b}, ℤ) → H_{1}(L_{b}, ℤ)

Recall the system of paths in the beginning of this section. When we say the ”Monodromy around a critical value ” we mean the monodromy associated to the path which gets out of b, goes along γ (the path connecting b to the critical value in this system of paths) , turns around the critical value counterclockwise and then comes back to b along γ. By Picard-Lefschetz formula (see [La]) the monodromy around zero is given by

δ → δ − ∑_{j=a_{1}+1}^{a_{2}} < δ, δ_{j}^{1} > δ_{j}^{1} (6)

and the monodromy around a center critical value c_{i} is given by

δ → δ− < δ, δ_{i} > δ_{i} (7)

For the regular value b ∈ ℂ − C, \overline{L_{b}} = L_{b} ∪ R is a compact Riemann surface. Let I be the subgroup of H_{1}(L_{b}, ℤ) generated by the cycles around the points of R in \overline{L_{b}}. We have

I = {δ ∈ H_{1}(L_{b}, ℤ) |< δ, δ′ >= 0, ∀δ′ ∈ H_{1}(L_{b}, ℤ)} (8)

Elements of I are fixed under the action of the monodromy.

Now we are in a position such that we can look at our objects in an abstract way: We have a union of curves C = f^{−1}(0) in ℝ^{2}. To C we associate an Abelian group G (= H_{1}(L_{b}, ℤ)) freely generated by the symbols (5). These symbols are in one to one correspondence with the self intersection points of C and the relatively compact components of the complement of C in ℝ^{2}. We have an antisymmetric pairing < . > given by the items of Theorem 2.1. Also the non-Abelian freely generated group π_{1}(ℂ − C, b) acts from left on G with the rules (6), (7). We have also the subgroup I of G defined by (8). These are all we are going to need. From now on we can think about (vanishing) cycles in this abstract context.

We will apply the above arguments for the Example 1.3. For the line l_{p} = 0, p = 0, 1, …, d we can associate the saddle critical points of f on l_{p} = 0 and the corresponding vanishing cycles. We rename these vanishing cycles by δ_{j}^{l_{p}}, j = 1, 2, …, d and suppose that the ordering by the index j is the same as the ordering of corresponding saddle points in the line l_{p} = 0. We define

δ^{l_{p}} = ∑_{j=1}^{d}(−1)^{j}δ_{j}^{l_{p}} ∈ G, p = 0, 1, …, d

Now reindex all relatively compact polygons by U_{i}, 1 ≤ i ≤ a_{1} + a_{3}. In our example the polygons are triangles or cubics. For any polygon U_{i} we denote by δ^{i} (∈ G) the sum of vanishing cycles in the vertices of U_{i}. Also we denote by δ_{i} the vanishing cycle associated to U_{i}

Lemma 2.1. The cycles δ^{l_{p}}, 1 ≤ p ≤ d and δ^{i}, 1 ≤ i ≤ a_{1} + a_{3} generate all saddle vanishing cycles in G ⊗ ℚ and so they are linearly independent in G ⊗ ℚ.

Note that the number of the cycles considered in the lemma is equal to the number of saddle vanishing cycles.

Proof. This is a nice high school problem. It would be more difficult if we assume only that the lines l_{p} are in a general position in ℝ^{2} i.e., they are distinct and no three of them have a common intersection point. For our example we give the following hint: 1. First draw the lines for a small value of d 2. Let δ_{p,p+1} denote the vanishing cycle associated to the intersection of l_{p} and l_{p+1}, p = 0, 1, …, d − 1. Try to write d.δ_{p,p+1} as an integral sum of δ^{l_{p}} and δ^{l_{p+1}} and δ^{i}, where i runs through the index of all polygons between (the angle less than 90^{∘}) the lines l_{p} and l_{p+1}. 3. Now it is easy to conclude that every vanishing cycle associated to the intersection points multiplied by d can be written as an integral sum of δ^{l_{p}}, 0 ≤ p ≤ d and δ^{i}, 1 ≤ i ≤ a_{1} + a_{3}. 4. After choosing a proper sign for δ^{l_{p}}, 0 ≤ p ≤ d prove that ∑_{i=0}^{d} δ^{l_{p}} = 0.

Now let us state the geometric meaning of δ^{i} and δ^{l_{p}}.

Lemma 2.2. We have

1. δ^{i} = δ_{i} − h_{0}(δ_{i}), where h_{0} is the monodromy around 0;

2. I ⊗ ℚ is generated by the cycles δ^{l_{p}}, 1 ≤ p ≤ d.

Proof. The first part is a direct consequence of Picard-Lefschetz formula and Theorem 2.1. By Theorems 2.1 and 2.2 we have < δ^{l_{p}}, δ′ >= 0, ∀δ′ ∈ H_{1}(L_{b}, ℤ) and so δ^{l_{p}} is in I. By Lemma 2.1 δ^{l_{p}}, k = 1, 2, …, d are linearly independent in G and we know that I is freely generated of rank d. Therefore δ^{l_{p}}, k = 1, 2, …, d generate I ⊗ ℚ freely.

Because of the symmetry in Example 1.3, one may conjecture that δ^{l_{p}} is the cycle around l_{p} ∩ 𝕃 in \overline{L_{b}} (multiplied by a rational number). Since we don’t need this statement we don’t try to prove it.

Theorem 2.3. In the Example 1.3 the action of the monodromy on a Lefschetz vanishing cycle generates the homology H_{1}(\overline{L_{b}}, ℚ).

Proof. First consider the case where δ is a center vanishing cycle. By Theorem 2.1 and Picard-Lefschetz formula the action of the monodromy generates δ_{i} and then δ_{i}’s. By Lemma 2.1 and Theorem 2.2 these cycles generate H_{1}(\overline{L_{b}}, ℚ). Now suppose that δ is a saddle vanishing cycle. There is a center vanishing cycle δ′ such that < δ′, δ >≠ 0. Performing a monodromy of δ around the critical value associated to δ′ and subtracting the obtained cycle by δ we obtain δ′ and so we fall in the first case.

In the beginning of this section we defined the degree of a polynomial 1-form ω = P dy − Qdx to be the maximum of deg(F) and deg(Q). This definition is no more useful when we look at ω as a meromorphic 1-form in ℙ^{2} or when we consider the foliation induced by ω in ℙ^{2}. Let us introduce a new definition of degree. For a polynomial 1-form ω we define deg_{1}(ω) to be the order of ω along the line at infinity mines two. We can see easily that if deg_{1}(ω) ≤ d then ω = P dy − Qdx + G(xdy − ydx), where P, Q are two polynomials of degree at most d and G is zero or a homogeneous polynomial of degree d. Therefore deg_{1}(ω) − deg(ω) = 0, 1. Naturally for a ω ∈ H we define deg_{1}(ω) to be the minimum of the deg_{1}’s of the elements of ω. Let q be an indeterminacy point of f at the line at infinity 𝕃. Recall that the fibers of f intersect 𝕃 transversally. Now we can choose a continuous family of cycles {δ_{t}}_{t∈ℂ} such that δ_{t} is a cycle in L_{t} around q. Latter we will need the following lemma.

Lemma 2.3. For ω ∈ H the integral ∫_{δ_{t}} ω is a polynomial of degree at most [\frac{n}{d+1}], where n − 2 = deg_{1}(ω) and d + 1 = deg(f). ∇^{i}ω, i > [\frac{n}{d+1}] restricted to each fiber \overline{f^{−1}(t)} has not residues in R and hence is a 1-form of the second type.

Proof. We have p(t) := ∫_{δ_{t}} ω = t^{\frac{n}{d+1}} ∫_{δ_{t}} _{\frac{ω}{f ^{\frac{n}{d+1}}}} . Since the 1-form _{\frac{ω}{f ^{\frac{n}{d+1}}}} has not pole along

the line at infinity, \frac{p(t)}{t^{\frac{n}{m}}} has finite growth at t = ∞. Since p(t) is is holomorphic in ℂ, we conclude that p(t) is a polynomial of degree at most [\frac{n}{d+1}]. The second part is a direct consequence of the first one and the formula (4).

Let f be a polynomial and ω be a 1-form in ℂ^{2}. ω is called relatively exact modulo f, or simply relatively exact if the underlying f is known, if the restriction of ω to each fiber L_{b} is exact. 1-forms of the type dP + Qdf, where P,Q are polynomials in ℂ^{2}, are relatively exact. Latter we will need the following lemma:

Lemma 2.4. If L_{t} is connected for all t ∈ ℂ then every relatively exact 1-form ω with deg_{1}(ω) = d is of the form dP +Qdf, where P, Q are two polynomials in ℂ^{2} with deg(P) = deg_{1}(ω) + 2 and deg(Q) = deg_{1}(ω) + 2 − deg(f).

The proof can be found in [BD], [Bo]. (For the assertion about the degrees see [Mo]). The main objective of this section is the following theorem.

Theorem 2.4. In the Example 1.3 let δ_{t} be a continuous family of vanishing cycles and ω be a degree d 1-form in ℂ^{2} such that ∫_{δ_{t}} ω = 0, t ∈ (ℂ, b). Then ω is of the form

ω = l_{0} ⋯ l_{d}α + d(P), α = ∑_{i=0}^{d} λ_{i} \frac{dl_{i}}{l_{i}} , λ_{i} ∈ ℂ

where P is a polynomial of degree not greater than d + 1.

Proof. By the hypothesis and (4) we have ∫_{δ_{t}} ∇^{2}ω = 0, t ∈ (ℂ, b). Lemma 2.3 implies that ∇^{2}ω is a 1-form without residue in each fiber and Theorem 2.3 implies that ∇^{2}ω is a relatively exact 1-form and hence by Lemma 2.4 it is zero in H̃. Since L_{0} is the only reducible fiber of f, by Corollary 1.1 ω must be of the form f ∑_{i=0}^{d} λ_{i} \frac{dl_{i}}{l_{i}} + dP + Qdf, where P and Q are two polynomials in ℂ^{2} and λ_{i} is a complex number. Recall that ω has degree less than d. By Lemma 2.4 we can suppose that P (resp. Q) has degree less than d + 2 (resp. 1). We have dP_{d+2} + Q_{1}df_{d+1} = 0, where P_{d+2} denote the homogeneous part of P of degree d + 2 and so on. It is easy to check that if Q_{1} ≢ 0 then we have f_{d+1} = (ax + by)^{d+1} which is not our case. Therefore Q must be constant and P of degree less than d + 1. We substitute P + Qf by P and so we can assume that Q is zero. We obtain the desired equality.

3 Deformation and Melnikov functions

As a first attempt to prove Theorem 0.1 one may fix a generic F ∈ L(d_{1}, …, d_{s}) and perform a deformation such that one of the center singularities of F persists. Then one may try to find some tools for finding a logarithmic first integral for the deformed foliation. These tools in the Ilyashenko’s case L(d + 1) were Picard-Lefschetz theory of a polynomial and the classification of relatively exact 1-forms modulo the polynomial. Developing these tools for a generic point of other irreducible components of M(d), for instance L(d_{1}, …, d_{s}), seems to be difficult and wasting the time.

In this section we want to explain this idea that it is not necessary to take a generic point of L(d_{1}, …, d_{s}). For instance suppose that the variety X_{1} = L(d_{1}, …, d_{s}) has a point F(df) which is Hamiltonian. This point may lie in other irreducible components of M(d). Let X_{1}, X_{2}, ⋯, X_{q} be all irreducible components of M(d) containing F. Now the idea is to deform F in such a way that one of its centers persists and since we can develop our tools for the Hamiltonian system F(df), we try to prove that the deformed foliation lies in one of X_{i}’s. Note that in the same time we prove that X_{1}, X_{2}, …, X_{q} are irreducible components of M(d). Now we are going to realize this idea for the Hamiltonian foliation with the polynomial in Example 1.3. This will lead to the proof of our main theorem. In this section L denotes the set of meromorphic 1-forms of the type ∑_{i=0}^{d} λ_{i} \frac{dl_{i}}{l_{i}} and P_{n} (resp. P_{∗}) denotes the set of polynomials of degree not greater than n (resp. arbitrary degree) in ℂ^{2}. I = {0, 1, …, d} and we denote by J an equivalence relation in I. If J_{1}, J_{2}, …, J_{s_{J}} are all equivalence classes of J then we define f_{i}^{J} = Π_{j∈J_{i}}l_{j}, i = 1, 2, …, s_{J}.

Let f be the polynomial considered in Example 1.3 and

F_{ϵ} : ω_{ϵ} = df + ϵ^{k}ω_{k} + ϵ^{k+1}ω_{k+1} + ⋯ + ϵ^{2k}ω_{2k} + h.o.t. = 0, ω_{k} ≠ 0 (9)

where ω_{i}, i = k, k + 1, … are polynomial 1-forms in ℂ^{2} and k is a natural number, be a one parameter degree d deformation of F(df).

Definition 3.1. Let p be a center singularity of F = F(df). p is called a persistent center if there exists a sequence ϵ_{i}, i = 1, 2, 3, …, ϵ_{i} → 0 such that the singularity of F_{ϵ_{i}} near p_{0} = p, namely p_{ϵ_{i}}, is center for all i.

Latter we will see that if p is persistent then p_{ϵ} is center for all ϵ. Our main theorem in this section is the following:

Theorem 3.1. If p is a persistent center in the degree d deformation (9) then ω_{k} is of the form

ω_{k} = f ∑_{J} ∑_{i=1}^{s_{J}}(λ_{i}^{J} \frac{df_{i}^{J}}{f_{i}^{J}} + \frac{A_{i}^{J}}{f_{i}^{J}})

where in the first sum J runs through all equivalence relations in I, for each J the complex numbers λ_{i}^{J}, i = 1, 2, …, s_{J} are distinct and A_{i}^{J} ∈ P_{deg(f_{i}^{J})}.

Let δ_{t}, t ∈ (ℂ, b) be a continuous family of vanishing cycles around p and Σ be a transverse section to F at some point of δ_{b}. We write the Taylor expansion of the deformed holonomy h_{ϵ}(t)

h_{ϵ}(t) − t = M_{1}(t)ϵ + M_{2}(t)ϵ^{2} + ⋯ + M_{i}(t)ϵ^{i} + h.o.t.

M_{i}(t) is called the i-th Melnikov function of the deformation. If the center p is persistent under the deformation then M_{i} = 0 for all i. But we don’t need to use all these equalities. For instance in Ilyashenko’s case L(d + 1), k = 1 we need only M_{1} = 0. To prove Theorem 3.1 and our main theorem we will need to use M_{k} = M_{k+1} = ⋯ = M_{2k} ≡ 0.

Lemma 3.1. In the above situation 1) M_{1} = M_{2} = ⋯ = M_{k−1} ≡ 0 and 2) if Σ is parameterized by the image of f then

M_{k}(t) = − ∫_{δ_{t}} ω_{k}

3) If M_{k} = M_{k+1} = ⋯ = M_{2k−1} ≡ 0 then

ω_{i} = fα_{i} + P_{i}, P_{i} ∈ P_{d+1}, α_{i} ∈ L, k ≤ i ≤ 2k − 1

4) Moreover if the transverse section is parameterized by the image of a branch of lnf then

M_{2k}(t) = − ∫_{δ_{t}} \frac{ω_{2k}}{f} − \frac{P_{k}}{f} .α_{k}

Proof. Let δ_{t,h_{ϵ}(t)} be the path connecting t and h_{ϵ}(t) along δ_{b} in the leaf of F_{ϵ} through t. We take the integral ∫_{δ_{t,hϵ(t)}} of (9). Now the equalities associated to the coefficients of

ϵ^{i}, 1 ≤ i ≤ k − 1 prove the first part. The equality associated to the coefficient of ϵ^{k} proves the second part (for more detail see [Ro][Mo]). We prove the third and fourth part by induction on i. First i = k. M_{k} ≡ 0 implies that ∫_{δ_{t}} ω_{k} = 0, t ∈ (ℂ, b) and so by Theorem 2.4 ω_{k} is of the form

ω_{k} = fα_{k} + P_{k}, α_{k} ∈ L, P_{k} ∈ P_{d+1} (10)

Now let us suppose that

ω_{j} = fα_{j} + P_{j}, α_{j} = ∑_{p=0}^{d} λ_{j,p} \frac{dl_{p}}{l_{p}} , k ≤ j ≤ i

Let ω̃_{ϵ} = ω_{ϵ}/f, ω̃ = ω/f and so on. With this new notation we have

ω̃_{j} = α_{j} + d̃P_{j} = d(P̃_{j} + lnl_{1}^{λ_{j,1}} ⋯ l_{d+1}^{λ_{j,d+1}}) + P̃_{j}d̃f = dg_{j} + P̃_{j}d̃f

From now on suppose that Σ is parameterized by the image of lnf. We have

(1 − P̃_{k}ϵ^{k}) ⋯ (1 − P̃_{i}ϵ^{i})ω̃_{ϵ}
(1 − P̃_{k}ϵ^{k}) ⋯ (1 − P̃_{i}ϵ^{i})(d̃f + ϵ^{k}ω̃_{k} + ϵ^{k+1}ω̃_{k+1} + ⋯ + ϵ^{2k}ω̃_{2k} + h.o.t.)
= d̃f + ϵ^{k}dg_{k} + ϵ^{k+1}dg_{k+1} + ⋯ + ϵ^{i}dg_{i}

+ϵ^{i+1}ω̃_{i+1} + ⋯ + ϵ^{2k−1}ω̃_{2k−1} + ϵ^{2k}(ω̃_{2k} − P̃_{k}ω̃_{k}) + h.o.t. (11)

We take the integral ∫_{δ_{t,hϵ(t)}} of (11). Now the equality associated to the coefficients of ϵ^{i+1} is M_{i+1}(t) + ∫_{δ_{t}} ω_{i+1} = 0. M_{i+1} ≡ 0 and Theorem 2.4 imply that ω_{i+1} is of the desired form. In the last step i = 2k − 1 the fourth part of the lemma is proved. Note that ∫_{δ_{t}} ω̃_{2k} − P̃_{k}ω̃_{k} = ∫_{δ_{t}} ω̃_{2k} − P̃_{k}α_{k}.

Now M_{2k} = 0 implies that ∫_{δ_{t}} fω_{2k} − P_{k}fα_{k} = 0. ω_{2k} − P_{k}fα_{k} has a pole of order at most 2d + 3 at the line at infinity. Therefore by Lemma 2.3 ∇^{3}(ω_{2k} − P_{k}fα_{k}) restricted to fibers \overline{f^{−1}(b)} has not residues. By Theorem 2.3, Corollary 1.1 we conclude that

fω_{2k} − P_{k}.fα_{k} = α_{1}.f + α_{2}f^{2} + dg + pdf, g, p ∈ P_{∗}, α_{i} ∈ L, i = 1, 2

The restriction of the above equality to the L_{0} = f^{−1}(0) implies that g is constant on L_{0}. Since L_{0} is connected in ℂ^{2}, we conclude that dg = d(fg′), g′ ∈ P_{∗}. From now on we write λ_{k,i} = λ_{i}. The above equality modulo multiplications by l_{i} gives us

l_{i} | λ_{i}P_{k} + g′, i = 0, 1, …, d (12)

Let I = {0, 1, …, d}. Define i ≅ j if λ_{i} = λ_{j}. ≅ is an equivalence relation. Let J_{1}, J_{2}, …, J_{s} be the equivalence classes of ≅. We define f_{i} = Π_{j∈J_{i}}l_{j}. Note that we have f = f_{1}f_{2} ⋯ f_{s}. The following lemma finishes the proof of Theorem 3.1.

Lemma 3.2. In the above situation, P_{k} must be of the form

P_{k} = f ∑_{i=1}^{s} \frac{A_{i}}{f_{i}} , A_{i} ∈ P_{#J_{i}} (13)

Proof. Let d_{i} = deg(f_{i}). (12) implies that P_{k} is zero in {f_{i} = 0} ∩ {f_{j} = 0}. The space of degree d + 1 polynomials in ℂ^{2} vanishing in {f_{i} = 0} ∩ {f_{j} = 0} for all 1 ≤ i < j ≤ s, namely G, is of dimension \frac{(d+2)(d+3)}{2} − ∑_{1≤i<j≤s} d_{i}d_{j}. But the space of polynomials in

(13) is a subset of G and has dimension −1 + ∑_{i=1}^{s} \frac{(a_{i}+1)(a_{i}+2)}{2}. Since d + 1 = ∑_{i=1}^{s} d_{i}, these two numbers are equal and so P_{k} must be of the form (13).

4 Proof of Theorem 0.1

Let (X, 0) be a germ of an analytic variety in (ℂ^{n}, 0). The tangent cone T C_{0}X of X at 0 is defined as follows: Let γ : (ℂ, 0) → (ℂ^{n}, 0) be an analytic map such that its image lies in X and has the Taylor series γ = ωϵ^{l} + ω′ϵ^{l+1} + ⋯, ω, ω′, … ∈ ℂ^{n}. T_{l} is the set of all ω and T C_{0}X = ∪_{i=1}^{∞}T_{l}.

We have ℂ.T C_{0}X = T C_{0}X therefore we can projectivize T C_{0}X and obtain a subset, namely Y, of ℙ^{n−1}. Suppose that X is irreducible. Let π : M → (ℂ^{n}, 0) be a blowup at 0 with the divisor π^{−1}(0) ≅ ℙ^{n−1}. The closure X̃ of π^{−1}(X − {0}) in M is an irreducible analytic set and we can see easily that Y is isomorphic to the intersection of X̃ and ℙ^{n−1} ⊂ M, and so it is algebraic compact subset of ℙ^{n−1}. Moreover since dim(Y) ≥ dim(X̃) + dimℙ^{n−1} − n (see [Ke] Theorem 3.6.1) and Y cannot be the whole ℙ^{n−1}, Y is of pure dimension dimX−1, i.e. each irreducible component of Y is of dimension dimX−1. We conclude that T C_{0}X is an algebraic subset of ℂ^{n} of pure dimension dim(X). If X is smooth then T C_{0}X is the usual tangent space of X at 0 and hence it is a vector space. For more information about the tangent cone of a singularity and its definition by the leading terms of the polynomial defining the singularity see [Mu] and [Ke] Section 6.2.

The variety L(d_{1}, ⋯, d_{s}) is parameterized by

τ : ℂ^{s} × P_{d_{1}} × ⋯ × P_{d_{s}} → F(d)

τ(λ_{1}, …, λ_{s}, f_{1}, …, f_{s}) = f_{1} ⋯ f_{s} ∑_{i=1}^{s} \frac{df_{i}}{f_{i}} (14)

and so it is irreducible. Let J be an equivalence relation in I = {0, 1, …, d} with s equivalence classes, namely J_{1}, …, J_{s}. Let also f be the polynomial in Example 1.3 and F_{0} = F(df). In a neighborhood of F_{0} in F(d), L(d_{1}, ⋯, d_{s}) has many irreducible components corresponding to the J’s as follows: The above parameterization near (1, …, 1, Π_{i∈J_{1}}l_{i}, …, Π_{i∈J_{s}}l_{i}) determines an irreducible component, namely L(d_{1}, ⋯, d_{s})_{J}, of (L(d_{1}, ⋯, d_{s}) , F_{0}) corresponding to J. Fix one of these branches and name it X. Now to prove our main theorem it is enough to prove that X is an irreducible component of (M(d), F_{0}).

What we have proved in Theorem 3.1 is:

T C_{F_{0}}M(d) = ∪T C_{F_{0}}L(d_{1}, ⋯, d_{s})_{J}

where the union is taken over all 1 ≤ d_{i} ≤ d + 1, 1 ≤ i ≤ s ≤ d + 1, ∑_{i=1}^{s} d_{i} = d + 1 and all equivalence relations J. Now let X ⊂ X′, where X′ is an irreducible component of (M(d), F_{0}). Since the above union is the decomposition of T C_{F_{0}}M(d) to irreducible components, the irreducible component of T C_{F_{0}}X′ containing T C_{F_{0}}X must be a subset of T C_{F_{0}}X and so is equal T C_{F_{0}}X. An irreducible component of T C_{F_{0}}X′ is of dimension dim(X) and so X′ is of dimension dim(X). Since X ⊂ X′ and X, X′ are irreducible, we conclude that X = X′.

5 Limit cycles and Bautin Ideals

Let F_{0} ∈ M(d), p_{0} be a center singularity of F_{0}, δ_{t}, t ∈ (ℂ, 0) be a continuous family of cycles invariant by F_{0} around p_{0} and Σ be a transverse section to F_{0} at some point of δ_{0}. Let also (ℂ^{μ}, ψ) be an affine chart of F(d) with ψ(F_{0}) = 0. We use also ψ for the points in ℂ^{μ}. For instance we denote by F_{ψ} the foliation associated to ψ ∈ ℂ^{μ} by this affine chart.

The holonomy of F_{0} along δ_{0} in Σ is identity. Now considering a ψ near 0, we have the holonomy h_{ψ} of F_{ψ} along δ_{0} in Σ which is called the deformed holonomy. We write the Taylor expansion

h_{ψ}(t) − t = ∑_{i=0}^{∞} a_{i}(ψ)t^{i}

The ideal generated by a_{i}(ψ), 0 ≤ i is called the Bautin ideal of δ_{0} in the deformation space F(d). If ψ ∈ Zero(I) then the holonomy of F_{ψ} along δ_{0} is identity. Using the below lemma we conclude that zero(I) ⊂ M(d).

Lemma 5.1. In the above situation, the singularity p_{ψ} near p_{0} is center and so F_{ψ} ∈ M(d).

Proof. Let (D_{a} × D_{a}, (x, y)) be a coordinate system around p_{0} such that p_{0} = (0, 0) and the foliations is given by f_{0} = xy = const., where D_{a} = {z ∈ ℂ || z |≤ a}. Let K = \overline{(D_{a/2} − D_{a/3}) × D_{a/2} ∪ D_{a/2} × (D_{a/2} − D_{a/3})} and Σ be a transverse section to F_{0} at a point in (D_{a/2} − D_{a/3}) × {0}. Since K is compact and the holonomy of F_{ψ} along a closed path in (D_{a/2} − D_{a/3}) × {0} is identity, there exists 0 < ϵ such that for all | ψ |< ϵ we can associate to each point (x, y) of K the intersection point, namely A(x, y), of the leaf passing through p and Σ. Now f_{ψ} = f_{0}(A(x, y)) is a holomorphic map in K and so by Hartogs’s theorem it extends to a neighborhood of p_{0}. Since f_{ψ} |_{ψ=0}= xy and f_{ψ} is constant in the leaves of F_{ψ}, we conclude that f_{ψ} has a unique non-degenerate critical point near p_{0} which is p_{ψ}.

The center p_{0} of F_{0} is called stable if for any deformation F_{τ} , τ ∈ (ℂ^{k}, 0) of F_{0} inside M(d), the deformed singularity p_{τ} is again a center.

Let F_{0} ∈ M(d) and F_{τ} be a deformation of F_{0} inside M(d). Since each F_{τ} has at least on center, there is a sequence p_{τ_{i}} of centers converging to a singularity of F_{0}. We conclude that the deformed holonomy along the vanishing cycles around p_{0} is identity and so by the above lemma we conclude that p_{τ} is center for all τ. From this argument we conclude that every F_{0} with (M(d), F_{0}) irreducible has at least one stable center. In particular generic points of irreducible components have stable centers. It is an interesting problem to show that a generic point of L(d_{1}, …, d_{s}) has d^{2} − ∑_{i<j} d_{i}d_{j} stable centers. For the stable center p_{0} we have zero(I) = (M(d), F_{0}), where I is the Bautin ideal associated to a vanishing cycle around p_{0} and the deformation space F(d).

Now let X be an irreducible component of M(d), F ∈ X − sing(M(d)) be a real foliation, i.e. its equation has real coefficients, p be a real center singularity and δ_{t}, t ∈ (ℝ, 0) be a family of real vanishing cycles around p.

Theorem 5.1. The cyclicity of δ_{0} in a deformation of F inside F(d) is greater than codim_{F(d)}(X) − 1

Roughly speaking, the cyclicity of δ_{0} is the maximum number of limit cycles appearing near δ_{0} after a deformation of F in F(d). The proof of this Theorem and the exact definition of cyclicity can be found in [Ro]. codim_{F(d)}(L(d + 1)) − 1 = \frac{(d+2)(d−1)}{2} and this is the number obtained by Yu. Ilyashenko in [Il].

Now Let X = L(d_{1}, …, d_{s}) and F = F(f ∑_{i=1}^{s} λ_{i} \frac{df_{i}}{f_{i}}) ∈ L(d_{1}, …, d_{s}) − sing(M(d)). Suppose that λ_{i}’s and the coefficients of f_{i}’s are real numbers and F has a (real) center singularity at 0 ∈ ℝ^{2}. As a direct consequence of Theorem 0.1 and Theorem 5.1.

Corollary 5.1. Suppose that s > 1. The cyclycity of δ_{0} in a deformation of F in F(d) is not less than

(d + 1)(d + 2) − ∑_{i=1}^{s}(\frac{(d_{i} + 1)(d_{i} + 2)}{2}) − 1

The above lower bound reaches to its maximum when d_{1} = d_{2} = … = d_{s} = 1, s = d + 1. In this case the cyclicity of δ_{0} is not less than d^{2} − 2.

Until the time of writing this paper, the best upper bound for the cyclicity of a vanishing cycle of a Hamiltonian equation is the P. Mardesic’s result \frac{d^{4}+d^{2}−2}{2} in [Ma]. Nothing is known about the exact calculation of these numbers.

6 Challenge

In this section I would like to give some highlights into the difficulty and bigness of the center conditions problem.

All the varieties L(d_{1}, …, d_{s}) pass through F = F(df), where f is the polynomial in Example 1.3.

Problem 6.1. Are L(d_{1}, …, d_{s})’s all irreducible components of M(d) through F?

Note that the argument used in § 4 does not give an answer to this problem. There may be an irreducible component of M(d) through F such that its tangent cone at F is a subset of ∪T C_{F_{0}}L(d_{1}, ⋯, d_{s})_{J}. In this case the definition of other notions of tangent cone based on higher order 1-forms in the deformation (9) seems to be necessary.

Problem 6.2. For i = 0, …, d let l_{i} be a line in the real plane and m_{i} be an integer number. Put f = l_{0}^{m_{0}} ⋯ l_{d}^{m_{d}}. Find all irreducible components of M(d) through F(df).

In this problem the line l_{i} has multiplicity m_{i} and it would be interesting to see how the classification of irreducible components through F(df) depends on the different arrangements of the lines l_{i} in the real plane and the associated multiplicities. In particular we may allow several lines to pass through a point or to be parallel. When there are lines with negative multiplicities then we have a third kind of singularities {l_{i} = 0} ∩ {l_{j} = 0} called dicritical singularities, where l_{i} (resp. l_{j}) has positive (resp. negative) multiplicity. They are indeterminacy points of f and are characterized by this property that there are infinitely many leaves of the foliation passing through the singularity. Also in this case there are saddle critical points of f which are not due to the intersection points of the lines with positive (resp. negative) multiplicity. The reader may analyze the situation by the example f = \frac{l_{0}l_{1}}{l_{2}l_{3}} . As we saw in this paper, our problem breaks into two main parts 1. Study the action of Gauss-Manin connection on the Brieskorn lattice/Petrov module of f 2. Study the action of monodromy on the first homology group of a regular fiber of f. These two studies enable us to calculate Melnikov functions, as many as necessary, and obtain an explicit form for the 1-forms appearing in a deformation of F(df) with a persistent center. The reader may consult with [OrTe] about the arrangements of lines. May be one of the interesting parts of Problem 6.2 is:

Problem 6.3. Generalize A’Campo/Gusein zade theorem on calculating the Dynkin diagram of f in the Problem 6.2 with some m_{i}’s negative.

Another way to look for more irreducible components of M(d) is as follows. Take a polynomial of degree d in ℂ^{2} with the generic conditions considered by Ilyashenko (see Introduction). Now F(df) has degree d − 1. Let us suppose that we have compactified F(d) in a projective space by considering the low degree foliations.

Problem 6.4. Classify all irreducible components of M(d) through F(df).

All L(d_{1}, …, d_{s})’s pass through F(df). There are other candidates as follows

1. A_{i} = \{F(\frac{dp}{p} + d(\frac{q}{p^{i}})) | deg(p) = 1, deg(q) = d\} i = 0, 1, 2, …, d;

2. B_{1} = \{F(\frac{dq}{q} + d(p)) | deg(p) = 1, deg(q) = d\};

An element of A_{i} (resp. B_{1}) has a first integral of the type pe^{q/p^{i}} (resp. qe^{p}). These candidates are supported by Dulac’s classification (see [Du] and [CL] p.601) in the case d = 2.

In the beginning of this paper we defined the degree of a polynomial 1-form ω to be the maximum of degrees of underlying polynomials. But in § 2 we came to know deg_{1} which is more geometric and is due to to compactification of ℂ^{2} in ℙ^{2}. In fact it is the number of tangency points of the foliation induced by ω in ℙ^{2} with a generic line. Now we redefine F(d) to be the space of holomorphic foliations F in ℙ^{2} with deg_{1}(F) = d.

Problem 6.5. Prove a theorem similar to Theorem 0.1 in the above space of parameters.

For this problem a solution to Problem 6.3 is necessary. For more information about this point of view see [Mo].

A Center conditions are algebraic

Let k be an arbitrary algebraically closed field. We can develop our notations in Introduction for k except the center singularity. Now let us see how we can define the notion of a center singularity when we are working with k.

Suppose that the origin 0 = (0, 0) is a reduced singularity of F(ω) ∈ F(d). 0 is called the center singularity of ω if there is a formal power series

f = xy + f_{3} + f_{4} + ⋯ + f_{n} + ⋯ (15)

where f_{n} is a homogeneous polynomial of degree n and with coefficients in k, such that

df ∧ ω = 0

A singularity p of ω is called center if the origin is a center singularity of i^{∗}ω, where i is the linear transformation a → a + p in k^{2}. Our definition is complete.

Now let k = ℂ and the origin is a center singularity of a 1-form ω. By theorem A in [MaMo] the existence of a formal series (15) implies the existence of a convergent one, namely g. Using the complex Morse theorem we find a coordinates system (x̃, ỹ) around the origin such that g = x̃^{2} + ỹ^{2}. So our definition of a center singularity coincides with the definition in Introduction.

Theorem A.1. M(d) is an algebraic subset of F(d).

Proof. Let M_{0}(d) be the set of 1-forms ω in F(d) with a center at the origin. Since M(d) is obtained by the action of the group {a → a + p | p ∈ k^{2}} on M_{0}(d), it is enough to prove that M_{0}(d) is algebraic. Now we reserve the p for the characteristic number of the field k.

Let ω ∈ F(d) and ω = ω_{1} + ω_{2} + ω_{3} + … + ω_{d} be the homogeneous decomposition of ω, then

ω ∧ df = 0 ⇒ (ω_{1} + ω_{2} + ω_{3} + ⋯ + ω_{d+1}) ∧ (d(xy) + df_{3} + df_{4} + ⋯) = 0

Putting the homogeneous parts of the above equation equal to zero, we obtain

\{ \begin{matrix}ω_{1} ∧ d(xy) = 0 ⇒ ω_{1} = c.d(xy), c ∈ k \\ ω_{1} ∧ df_{3} = −ω_{2} ∧ d(xy) \\ … \\ ω_{1} ∧ df_{n} = −ω_{2} ∧ df_{n−1} − ⋯ − ω_{n−1} ∧ d(xy) \\ … \end{matrix} (16)

Since 0 is a reduced singularity, we have c ≠ 0. Dividing the 1-form ω by c, we can assume that c = 1. Let P_{n} denote the set of homogeneous polynomials of degree n. Define the operator :

S_{n} : P_{n} → (P_{n}dx ∧ dy)
S_{n}(g) = ω_{1} ∧ d(g)

We have

S_{i+j}(x^{i}y^{j}) = d(xy) ∧ d(x^{i}y^{j}) = (xdy + ydx) ∧ (x^{i−1}y^{j−1}(jxdy + iydx))
= (j − i)x^{i}y^{j}dx ∧ dy

This implies that

Im(S_{n}) = A_{n}dx ∧ dy

where A_{n} is the subspace generated by the monomials x^{i}y^{j}, i ≠ j mod p. The existence of f_{n} implies that the coefficient of x^{i}y^{j} i = j mod p in

−ω_{2} ∧ df_{n−1} − ⋯ − ω_{n−1} ∧ d(xy)

which is a polynomial, say P_{ij}, with variables

coefficients of ω_{2} … ω_{n−1}, f_{2}, …, f_{n−1}

is zero. The coefficients of f_{i}, i ≤ n − 1 is recursively given as polynomials in coefficients of ω_{i}, i ≤ n − 1 and so the algebraic set

X : ∩_{i=j mod p}{P_{ij} = 0}

consists of all F(ω) in F(d) with a center singularity at the origin. Note that by Hilbert zeroes theorem, a finite number of P_{ij}’s defines M_{0}(d).

B Proof of Lemma 1.1

Let us compactify ℂ^{2} in ℙ^{2} = ℂ^{2} ∪ 𝕃, where 𝕃 is the line at infinity and look f as a meromorphic function in ℙ^{2} with indeterminacy points at 𝕃. Let F be the foliation in ℙ^{2} given by df = 0 and F̃ be the sheaf of holomorphic 1-forms in ℙ^{2} inducing the foliation F. The foliation F have two kind of singularities 1. the critical points of f 2. the indeterminacy points of f at the line at infinity. We denote by sing(F) the singular points of F.

It is easy to see that F̃ is a freely generated sheaf and hence it is coherent. Let Ω̃^{i}, i = 1, 2 denote the sheaf of i-forms in ℙ^{2}. Since the wedge product map ∧ : Ω̃^{1}×F̃ → Ω̃^{2} is an analytic map between two coherent sheaves, its image, namely F̃ ∧ Ω̃^{1}, is coherent (see [GR]). The sheaf

L = \frac{Ω̃^{2}}{F̃ ∧ Ω̃^{1}}

is a coherent discrete sheaf whose support is a subset of sing(F). If the foliation around a singular point p is given by P dx + Qdy = 0, g.c.d.(P, Q) = 1 then the stalk of L at p is \frac{O_{ℂ^{2},p}}{<P,Q>}. For an arbitrary analytic sheaf S on ℙ^{2} we define S(∗𝕃) to be the sheaf of meromorphic sections of S with poles of arbitrary order along 𝕃. The sheaf L(∗𝕃) is obtained from L by putting zero the nonzero stalks of L associated to the singularities of F at 𝕃. Therefore H^{0}(ℙ^{2}, L(∗𝕃)) is a vector space of dimension μ, where μ is the sum of local Milnor numbers of f.

Now F̃∧Ω̃^{1} is coherent and 𝕃 is a positive divisor in ℙ^{2}, therefore by Kodaira vanishing theorem H^{1}(ℙ^{2}, (F̃ ∧ Ω̃^{1}(∗𝕃)) = 0 (see [Nr]). Writing the long exact sequence associated to

0 → (F̃ ∧ Ω̃^{1})(∗𝕃) → Ω̃^{2}(∗𝕃) → L(∗𝕃) → 0

we conclude that \frac{H^{0}(ℙ^{2},Ω^{2}(∗𝕃))}{H^{0}(ℙ^{2},(F̃∧Ω^{1})(∗𝕃))} is a vector space of dimension μ. But H^{0}(ℙ^{2}, (F̃ ∧ Ω^{1})(∗𝕃)) = df ∧ Ω^{1} and H^{0}(ℙ^{2}, Ω^{2}(∗𝕃)) = Ω^{2}. So we have proved the first part.

For a critical value p_{i}, i = 1, …, s of f let V_{i} be the stalk of L in p_{i}. In fact in the above argument we have proved that the natural canonical map V → ⊕_{i=1}^{s}V_{i} is an isomorphism of vector spaces. V_{i}’s are invariant by the linear operator A : V → V associated to the multiplication by f. Also A − c_{i}I restricted to V_{i} is a nilpotent matrix (see [Br]), where c_{i} = f(p_{i}). This finishes the proof of the second part.

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Hossein Movasati Institute for studies in theoretical Physics and Mathematics, IPM School of Mathematics P.O.Box : 19395-5746 Tehran-Iran Instituto de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, RJ, Brazil E-mails: hossein@impa.br, movasati@ipm.ir