ON THE CENTER-FOCUS PROBLEM FOR THE EQUATION

— We study irreducible components of the set of polynomial plane differential systems with a center, which can be seen as a modern formulation of the classical center-focus problem. The emphasis is given on the interrelation between the geometry of the center set and the Picard–lefschetz theory of the bifurcation (or Poincaré–Pontryagin–Melnikov) functions. Our main illustrative example is the center-focus problem for the Abel equation on a segment, which is compared to the related polynomial Liénard equation. Résumé. — Nous étudions les composantes irréductibles de l’ensemble des champs polynomiaux plans avec centre, ce qui peut être vu comme une formulation moderne du problème classique du centre-foyer de Poincaré. L’accent est mis sur l’interrelation entre la géométrie de l’ensemble des centres et la théorie de Picard–Lefschetz des fonctions de bifurcation (ou de Poincaré–Pontryagin–Melnikov). Notre exemple principal est le problème du centre-foyer pour l’équation d’Abel sur un segment, comparée à l’équation de Liénard associée.


The center-focus problem
The plane differential systeṁ Note on the terminology. We do not specify here the category to which belong P, Q, f . They will be either analytic or polynomial, depending on the context. The base field will be either R or C depending on the context too. Most results will be valid for both. Thus, the definition of a center for (1.2) is the same in the real and in the complex case. In the case of an analytic complex plane vector field (1.1) the "complex" definition of a center is less straightforward. We say that the origin is a non-degenerate center, if the vector field has an analytic first integral with a Morse critical point at the origin. If this is the case, we shall also say that (1.1) has a Morse singular point, e.g. [CLN96,Dul08]. We recall therefore Definition 1.1. -The analytic complex vector field (1.1) is said to have a Morse singular point, if it allows an analytic first integral in a neighbourhood of this point, which has a Morse type singularity.
If (1.1) has a Morse singular point, then the linear part of (1.1) is diagonalisable with non-zero eigenvalues, that is to say the singular point of the vector field is non-degenerate.
An example is the saddle x = x, y = −y which has an analytic first integral xy of Morse type, and hence a Morse critical point. Of course, it is linearly equivalent (over C) to x = y, y = −x with first integral x 2 + y 2 which is the usual linear real center. The advantage to study Morse critical points over C is that we can use complex analysis and complex algebraic geometry. This is the point of view adopted in these notes.
The two equations (1.1) and (1.2) are closely related. First, a polar change of variables transforms a plane system (1.1) with a center, to equivalent equation of the form (1.2) with a center along the interval [0, 2π]. Second, if the family of functions f ( · , y), x ∈ [0, 1] is replaced by its Fourier series f ( · , y) (so f (x + 1, y) = f (x, y)) and the equation (1.2) has a center at y = 0, then the new system dy dx + f (x, y) = 0, (x, y) ∈ R/Z × R (1.3) will have all its orbits starting near the periodic solution y = 0 on the cylinder R/Z × R, periodic too. Of course, if the smooth function f is non-periodic, then the function f is only piece-wise continuous in x. The transport map of (1.2) along [0, 1] becomes a return map for (1.3) and the definition of a limit cycle for (1.2) is straightforward too. Actually, the scalar equation (1.2) in which f is a regular function, should be considered as a simplified model of the eventually singular equation dy dx = P (x, y) Q(x, y) .
We resume the above considerations in the following definitions, which make sense both on R or C: (i) The solution ϕ = ϕ( · ; x 0 , y 0 ) is said to be periodic iff ϕ(x 1 ; x 0 , y 0 ) = y 0 (ii) The solution ϕ = ϕ( · ; x 0 , y 0 ) is said to be a limit cycle, provided that it is periodic and isolated, that is to say there is a neighbourhood of its orbit on S 1 × R free of periodic solutions. (iii) the map y → ϕ(x 1 ; x 0 , y) is the first return map of (1.2) in a neighbourhood of y = y 0 . (iv) The equation (1.2) defines a center in a neighbourhood of the periodic solution ϕ provided that the first return map is the identity map in a neighbourhood of y 0 . If the return map is not the identity map, then we say that (1.2) defines a focus at the periodic solution ϕ.
The center focus-problem for the equation (1.2) or (1.1) is, roughly speaking, to distinguish between a center and a focus. The algebro-geometric content of the problem is as follows. Suppose, that (1.2) is polynomial, more precisely dy dx + As we shall see in Theorem 2.3, under the condition a 1 = 0, the coefficients c n = c n (a), n 1, are polynomials in the coefficients of a j = a j (x), j n. The condition that ϕ(1; 0, · ) is the identity map determines an infinite number of polynomial relations {c n (a) = 0} on the coefficients of the polynomials a j . By the Hilbert basis theorem, only a finite number of them are relevant, and they define an algebraic variety (the so called center variety C m,n ) in the vector space of all coefficients of the polynomials a j . The problem is therefore (as formulated by Lins Neto [LN14] in the context of a polynomial foliation induced by (1.1)): Describe the irreducible components of C m,n . TOME 3 (2020) The content of the paper is as follows. In Section 2 we give first an explicit formula for the general solution of the equation in terms of iterated path integrals, see Theorem 2.1. As a by-product we obtain a formula for the first return map, and explicit center conditions found first by Brudnyi, see Theorem 2.3. Section 3 is devoted to the perturbation theory of the integrable Abel equation dy dx = a(x)y 2 with first integral It is assumed that A(0) = A(1), so the equation has a center along [0,1]. We are interested in the number of limit cycles (isolated solutions, such that y(0) = y(1)) which the perturbed equation dy dx = a(x)y 2 + . . . can have. The center-focus problem for this perturbed equation leads to a well known polynomial moment problem. Under general assumptions this problem has an elegant solution, due to Colin Christopher, Theorem 3.2, which is presented here in the setting of Abelian integrals of dimension zero.
In Section 4 we study irreducible components of the center variety of the polynomial Abel equation (on the interval [0, 1]) dy dx = p(x)y 2 + q(x)y 3 (1.5) as well center variety of the related Liénard equation (by "center" we mean the usual Morse center in a neighbourhood of the origin in C 2 ) In Section 4.1 we prove that the set of Abel equations coming from "pull back" provide irreducible components of the center set, Theorem 4.1. These results are inspired by previous contributions of Movasati. In Sections 4.2 we revisit the classical center-focus for quadratic vector fields, with special attention to the Q 4 component of the center set.
In Section 4.3 we give a full description of the center set of Liénard type equations (1.6). These results belong mainly to Cherkas and Christopher, but we present them in the broader context of the present notes. In particular, the base field will be C, see Theorem 4.8 and 4.10. The centers found in this way are always of "pull back" type. This suggests that the only centers of the related Abel equation (1.5) are of "pull back" type too, which is the content of the so called Composition Conjecture for the Abel equation (1.5) [BRY10,p. 444] to be discussed in Section 4.4. In this last section we show, however, that there are scalar Abel equations with a center along [0, 1], which can not be obtained by a "pull back". These equations have a Darboux type first integral, and their construction is inspired by the study of the Q 4 component in Section 4.2. Among them we find the recent counter-example to the Composition Conjecture mentioned above, found first by Giné, Grau and Santallusia [GGS19]. (1)

Acknowledgement
The author is obliged to Jean-Pierre Françoise for the illuminating discussions on the center-focus problem. I thank also the anonymous referees for several suggestions, which helped to improve the text.

The first return map and the Brudnyi formula
In this section we shall describe the return map of (1.4) as a power series involving iterated path integrals. We prove an explicit formula, due to Brudnyi [Bru06], which amounts to solve the differential equation. The classical approach to do this is by the Picard iteration method. If y 0 is the initial condition at x 0 of the differential equation dy = f (x, y)dx then the Picard iteration is where y n tends to the solution of the equation as n → ∞. We illustrate this on the example dy = ydx. If y 0 is the initial condition at x = 0 then

As
. . . 0 tn ··· t 1 x y 0 dt 1 . . . dt n = y 0 x n n! we get y(x) = y 0 e x as expected. The multiple (or iterated) integrals above appear in a similar way in the non-autonomous linear dy = a(x)ydx, or even non-linear case dy = f (x, y)dx. The non-linear case is more involved, it is reduced to the linear one, but after introducing infinitely many new variables y, y 2 , y 3 , . . . . To get around this reduction we shall use a simple Ansatz, for which we need a formal definition of iterated integral.
Let Ass ω be the graded free associative algebra generated by the infinite dimensional vector space of differential one-forms ω = a(x, y)dx, a ∈ C{x, y}. Its elements are non-commutative polynomials in such one-forms. The differential operator (1) The present paper is an extended version of two lectures given during the Zagreb Dynamical Systems Workshop, October 22-26, 2018.
induces a differential operator on Ass ω which acts by the Leibnitz rule. The readers familiar with the Picard-Lefschetz theory will recognize in D an avatar of the covariant derivative of an Abelian differential on the level sets {y = c} c . To save brackets, it is convenient to introduce the following notation so that (using brackets) and If we use the notation . . .
The iterated integral allows also a recursive definition (hence the name): where in the case n = 1 we have the Riemann integral x x 0 ω 1 . We note, that the usual notation for the multiple integral (2.2) is x x 0 ω n ω n−1 . . . ω 1 on the place of x x 0 ω 1 ω 2 . . . ω n , see Chen [Che77] or Hain [Hai87]. The reason to prefer the definition (2.3) is that it is better adapted to applications in differential equation, e.g. [Gav05]. Recall in this context, that For a short summary of properties of iterated integrals which we use, see [Gav05,Appendix], [GMN09, Section 2].
Theorem 2.1. -With the notation (2.1), a first integral of the differential equation dy + f (x, y)dx = 0 is given by the following recursively defined convergent series where ω = f (x, y)dx. The general solution of (1.2) with initial condition (x 0 , y 0 ) is given by and the general solution is In the quadratic case dy + 2xy 2 dx = 0, ω = 2xy 2 dx we compute recursively Therefore we get the first integral ϕ(x 0 ; x, y) = y + y 2 (x 2 − x 2 0 ) + y 3 (x 2 − x 2 0 ) 2 + . . . and the corresponding general solution is .
Proof of Theorem 2.1. We first verify, that for every fixed x 0 , the function ϕ(x 0 ; x, y) is a first integral : DωDωDω + . . .
As ϕ(x 0 ; x 0 , y 0 ) = y 0 then the level set {(x, y) : ϕ(x 0 ; x, y) = y 0 } contains both (x 0 , y 0 ) and (x, y). By symmetry is the solution of (1.2) with initial condition y(x 0 ) = y 0 . The convergency proof is by standard a priori estimates (omitted) TOME 3 (2020) Note that for fixed x 0 , x 1 the two return maps are mutually inverse. Therefore ϕ(x 1 ; x 0 , · ) = id if and only if ϕ(x 0 ; x 1 , · ) = id. Using Theorem 2.1 we can give explicit center conditions. Assume that and develop the return map ϕ(x 0 ; x 1 , y) as a power series in y If we denote, by abuse of notations, a i = a i (x)dx then we get for the first few coefficients c n (a) (compare to [BRY10,p. 450]) x 1 x 0 a 4 + 2a 3 a 1 + 3a 2 2 + 4a 1 a 3 + 6a 2 a 2 1 + 8a 1 a 2 a 1 + 12a 2 1 a 2 + 24a 4 1 and so on. The general form of the coefficients c n (a) is found immediately from Theorem 2.1. We resume this in the following Theorem 2.3 (Brudnyi's formula [Bru06]). -The coefficients c n (a) of the first return map (2.5) for the differential equation are given by the formulae . . .  Example 2.5. -Suppose that the equation dy dx + a 1 (x)y 2 + a 2 (x)y 3 + · · · = 0 has a center on the interval [x 0 , x 1 ]. Then, using as above the notation a i = a i (x)dx we have implies then, that x 1 x 0 a 2 = 0. If we consider more specifically the Abel equation Therefore a necessary condition for the Abel equation (2.6) to have a center on [ If we suppose that a 1 , a 2 are polynomials of degree at most two, these conditions are also sufficient. The case deg a 1 , a 2 = 3 can be studied similarly, see [BFY98, BFY99, BFY00].
In general, an obvious sufficient condition to have a center is therefore Centers with the property (2.8) were called universal in [Bru06].
Consider, more specifically, the following equation with polynomial coefficients a i via a suitable polynomial map ξ = ξ(x) having the property ξ(x 0 ) = ξ(x 1 ).
Not all centers of (2.9) are universal, as discovered recently in [GGS19].

Bifurcation functions related to Abel equation and a Theorem of Christopher
In this section we study the following perturbed Abel differential equation on the Question. -How many limit cycles has the perturbed system (3.1) on the interval [0, 1]?
The number of the limit cycles in a compact set are bounded by the number of the zeros of the so called bifurcation function, which we define bellow. A limit cycle which remains bounded when ε → 0, tends to a periodic solution of the non perturbed system. If the non-perturbed system (ε = 0) has a periodic solution, then necessarily A(0) = A(1), which already implies that it has a center. For this reason we assume from now on that has a center along 0 x 1. The perturbed equation can be written For a solution y(x), let P ε be the first return map which sends the initial condition y 0 = y(0) to y 1 = y(1). We parameterise P ε by h = 1 y = H(0, y) = H(1, y) and note that P ε is analytic both in h and ε (close to zero). We have therefore for the first return map The function M k is the bifurcation function, associated to the equation (3.1). It is also known as "first non-zero Melnikov function". The reader may compare this to (2.4) which is another representation of the first return map, defined for small y. As we shall see, the bifurcation function is globally defined. Therefore for every compact set K, [0, 1] ⊂ K ⊂ R 2 and all sufficiently small |ε|, the number of the limit cycles of (3.1) in K is bounded by the number of the zeros of the bifurcation function M k (counted with multiplicity). M k allows an integral representation where the integration is along the level set The differential form Ω k is computed by the classical Françoise's recursion formula [Fra96,Ili98,Rou98] as follows: If k = 1 then Ω 1 = ω 1 , otherwise The first order Melnikov function M 1 was computed first by Lins Neto [LN80, Section 3], see also [BFY00]. We have M 1 vanishes identically if and only if 1 0 p 1 (x)dx = 0 and which is the content of the famous polynomial moment problem for q 1 and A, solved in full generality by Pakovich and Muzychuk [PM09], see also [BFY98, BFY99, BFY00, Chr00, Yom03]. If M 1 = 0 by the above formula we get where r 1 is computed from the identity ω 1 = dR 1 + r 1 dH. As dω 1 = dr 1 ∧ dH then dr 1 = ω 1 = dω 1 dH is the Gelfand-Leray form of ω 1 . From the identity H(x, y(x, h)) ≡ h we have ∂y ∂h = −y 2 and hence r(x, y) = We conclude -Under the hypothesis M 1 = 0 the second Melnikov function is given by the following iterated integral of length two The hypothesis M 1 = 0 is of interest for us, as it will allow to compute the tangent space to the center set at the point (a, 0), see the next Section 4. For our purposes the polynomial a(x) can be taken in a general position, in which case the polynomial moment problem for q(x), A(x) has the following elegant solution vanishes identically, if and only if the polynomials Q = q and A satisfy the following "Polynomial Composition Condition" (PCC): There exist polynomials Q, A, W , such that Before recalling the proof of Christopher, we put I in the broader context of the Picard-Lefschetz theory.
The function I(h) is well defined for sufficiently big h, and has an analytic continuation in a complex domain to certain multivalued function. It is in fact an Abelian integral depending on a parameter. More precisely, consider the genus zero affine curve It is a Riemann sphere with n + 2 removed points, provided that h = 0. The removed points correspond to ( h we define a singularized algebraic curve Γ sing h , see Figure 3.1. As a topological space it is just the curve Γ h with the two points P 0 and P 1 identified to a point m. The structural sheaf of Γ sing h is the same as the structural We note that given an arbitrary effective divisor m = P 0 + P 1 + . . . P k on Γ h , one constructs in a similar way a singularized curve Γ sing h , which is the natural framework of the generalized center problem for the Abel equation, see [BFY99, Conjecture 1.7] and [BRY10]. As the integral I(h) is constant, it follows that where is a primitive of q, and x i (h) are the roots of the polynomial and call J an Abelian integral of dimension zero along the zero-cycle Definition 1]). If the Abelian integral I(h) vanishes identically, then the same holds true for J (h), hence J(h) = const. and it is easy to check that the constant is zero, is a subfield of the field of all rational functions C(x). By the Lüroth theorem this subfield is of the form C(W ) for suitable rational function W . It follows that are polynomials. For this reason we may suppose that Q, A, W are polynomials. If , then clearly W (0) = W (1) and the Theorem 3.2 is proved.
We may reason then by induction on A, Q which have a smaller degree than A, Q respectively. Thus this process must stop and we get W with W (x i (h)) ≡ W (x j (h)), and hence W (0) = W (1).

Irreducible components of the Center set
An affine algebraic variety V in C n is the common zero locus of a finite collection of polynomials f i ∈ C[z 1 , . . . , z n ]. The variety V is said to be irreducible, if for any pair of closed varieties In this case we may ask whether V 1 and V 2 are further reducible and so on. It is a basic fact of commutative algebra that in this way only a finitely many irreducible subvarieties V i ⊂ V can be found, and more precisely: Any variety V can be uniquely expressed as a finite union of irreducible varieties V i with V i V j for i = j, e.g. [Har95]. The varieties V i which appear in the finite decomposition Let W ⊂ V be another algebraic variety. Is W an irreducible component of V ? It is usually easy to verify, whether W is irreducible. It is much harder to check that W is an irreducible component of V . Indeed, it might happen that W V i where V i is an irreducible component of V . To verify this, one may compare the dimensions of the tangent spaces T x W and T x V at some smooth point x ∈ V ∩ W (one point x is enough!). Then W V i if and only if T x W T x V . Of course, there might be no way to know that x is a smooth point, in which case we use the tangent cones T C x W and The choice of x ∈ W is irrelevant, which allows a great flexibility.
The above observation will be applied in the case when V is the center set of the equation (1.2), and W is a subset of equations with a center. In the planar case (1.1) this approach was developed by Movasati [Mov04]. He observed that the vanishing of the first Melnikov function, related to one-parameter deformations (arcs) of systems (1.1) with a center, provides equations for the tangent space T x W , while the vanishing of the second Melnikov function provides equations for the tangent cone T C x W . This remarkable connection between algebraic geometry and dynamics will allow us to go farther in the description of irreducible components of the center set. We adapt the approach of Movasati [Mov04] and Zare [Zar19] to (1.2) in the context of the set A n of Abel differential equations dy dx = a(x)y 2 + b(x)y 3 (4.1) parameterised by the polynomials a(x), b(x) of degree at most n. They form therefore a vector space of dimension 2n + 2, and consider the subset C n ⊂ A n of Abel differential equations having a center on the interval 0 x 1. As we saw in the preceding section, C n is defined by finitely many polynomial relations c n (a, b) = 0 and therefore is an algebraic set.

Universal centers of the Abel equation define irreducible components of the center set
If the integer k > 1 divides n + 1, then we denote by U n/k ⊂ C n ⊂ A n the algebraic closure of the set of pairs of polynomials (a, b) (or Abel equations (4.1)), such that the following Polynomial Composition Condition (PCC) is satisfied There exist polynomials A, B, W of degrees (n + 1)/k, (n + 1)/k, k, such that The differential form associated to (4.1) is a pull back of the differential form Indeed, the iterated integrals x 1 x 0 a i 1 · · · a i k vanish, because they are pull backs under W of iterated integrals along an interval, contractible to the point W (x 0 ) = W (x 1 ). Following Brudnuyi [Bru06], we say that (4.1) determines an universal center if and only if It is shown then that a center is universal, if and only if the corresponding equation (4.1) is a pull back under an appropriate polynomial as above, see Brudnyi [Bru06, Corollary 1.20]. Thus, the universal centers are exactly those, obtained by a polynomial pull back in the sense (4.2), see the Polynomial Composition Condition (PCC).
Note that the universal center set U n/k is an irreducible algebraic variety, as a Zariski open subset of it is parametrized by the polynomials A, B, W of degrees respectively (n + 1)/k, (n + 1)/k, k. The main result of the section is Proposition 4.2. -The tangent space T (a,0) U n/n+1 is a vector space of dimension n + 1, which consists of pairs of polynomials (p, q) of degree at most n, such that q and a are co-linear polynomials, and 1 0 p(x)dx = 0. The proof is left to the reader. Next, we compute the tangent cone T C (a,0) C n at (a, 0) to the center set C n . To avoid complications, we choose a to be a non-composite polynomial. The Melnikov function M 1 , according to Section 3, is computed to be Assuming that for all sufficiently small ε the deformed Abel equation belongs to the center set C n , implies M 1 = 0, which on its turn imposes rather severe conditions on the polynomials p, q. First, 1 0 p(x)dx = 0 as follows already from (2.7). The second condition is well studied in a number of articles, and is known as the polynomial moment problem, e.g. Note that in full generality, a vector (p, q) which belongs to the tangent cone is a vector, such that there is a one-parameter deformation ε → (a + ε k p + . . . , ε k q + . . . ) at the point (a, 0) which belongs to the center set C n . The same arguments give the same constraints to the vector (p, q).
Proof of Theorem 4.1 in the general case. -Assume that the integer k > 1 divides n + 1 and consider the algebraic set U n/k of Abel differential equations, at y = 0 along [0, 1]. The proof follows the same lines as the case k = n, with the notable difference that the second Melnikov function M 2 will be needed.
We compute first the tangent space to U n/k at a general point (a, 0). Consider for this purpose the one-parameter deformation (4.4) F ε : dy are polynomial one-forms, deg p i n, deg q i n. As before we denote The point (a, 0) belongs to U n/k if and only if A = A • W for some degree k polynomial W .
Proposition 4.4. -The tangent space T (a,0) U n/k is the vector space of polynomials (p 1 , q 1 ) such that where P 1 , Q 1 are arbitrary polynomials of degree at most (n + 1)/k and R = R(x) is any degree k polynomial, such that R(0) = R(1).
The proof is straightforward, it suffices to consider the first order approximation in ε of the general deformation Next, we study the tangent cone T C (a,0) C n . We need to compare the affine varieties T (a,0) U n/k ⊂ T C (a,0) C n .
Proposition 4.5. -In a sufficiently small neighbourhood of every general point (p, q) ∈ T (a,0) U n/k the tangent cones T C (a,0) C n and T (a,0) U n/k coincide.
The above Proposition 4.5 shows that there is no irreducible component of T C (a,0) C n which contains an irreducible component of T (a,0) U n/k of strictly smaller dimension. This would imply Theorem 4.1.
The first Melnikov function, as in the case k = n, is M 1 = 1 0 p 1 dx + yq 1 dy. By Christopher's theorem M 1 = 0 implies that q 1 satisfies the composition condition Additional obstructions on the form of p 1 will be found by inspecting the second Melnikov function M 2 . Under the condition that M 1 = 0 we find [Gav05, Formula (2.8)] where the derivative is with respect to the parameter h. The identity h = A(x) + 1 y shows that y = −y 2 and ω 1 = −y 2 dx, it is clearly a covariant derivative in a cohomology bundle (although we do not need this interpretation here). Therefore, for the iterated integral of length two we find where P 1 is a primitive of p 1 . Indeed, M 1 = 0 implies the composition condition for Q 1 = q 1 and A, that is to say the integral {H=h} yq 1 dx vanishes as a pull back. The same then holds true for its derivative {H=h} y 2 q 1 dx as well for the iterated integral {H=h} (yq 1 dx))(y 2 q 1 dx). Further, by the shuffle relation for iterated integrals Further, for 1 0 ω 2 we find so that under the condition M 1 = 0 implies We apply Christopher's theorem to M 2 and conclude that the primitive of the polynomial q 1 (x)P 1 (x) − Q 2 (x)a(x) is a composite polynomial, it can be expressed as a polynomial function in W (x), and therefore for certain polynomial R 1 . Assuming that Q 1 and A are mutually prime, there exist polynomials R 2 , R 3 such that This implies finally that A (W (x)) divides P 1 (x) − R 2 (W (x)) and Proposition 4.4, and hence Theorem 4.1 is proved.

The center set of plane quadratic vector fields
Let A n be here the set of all polynomial vector fields of degree at most n. The only (non-trivial) case in which the center set C n ⊂ A n is completely known is the quadratic one, n = 2. For comprehensive description and historical comments concerning the center-focus problem in the quadratic case see Zoladek [Żoł94]. To the plane quadratic vector field (1.1) we associate a foliation F ω = {ω = 0} on C 2 , defined by the polynomial one-form The leaves of the foliation are the orbits of the plane vector field (1.1), and the restriction of the one-form ω on the leaves of F ω vanishes identically.
In this section we assume that the polynomials P, Q are of degree at most two, and the system has a center. As the foliation is over C we must be more careful in the definition. We shall say that a singular point is a center, if the point is nondegenerate, and has a local holomorphic first integral with a Morse critical point. Thus, in a neighbourhood of such a point, and up to a complex affine change of the variables, the system can be written in the form (4.5)ẋ = x + P 2 (x, y),ẏ = −y + Q 2 (x, y) for some homogeneous polynomials P 2 , Q 2 . The following classical result is implicit in Zoladek [Żoł94, Theorem 1] Theorem 4.6. -The center set C 2 of plane polynomial quadratic systems with a Morse center has four irreducible components.
The above claim is a modern interpretation of the Dulac's classification [Dul08] of such Morse centers in a complex domain, see Lins Neto [LN14, Theorem 1.1]. Indeed, it is easier to decide that a given variety is irreducible, than to decide that it is an irreducible component of some algebraic set. Sketch of the proof of Theorem 4.6 can be found in [FGX16,Appendix].
To describe explicitly the four components of the center variety C 2 , recall that the foliation F ω , respectively the vector field (1.1), is said to be logarithmic, if for suitable polynomials f i and exponents λ i . As then the logarithmic foliation F ω has a first integral of Darboux type Let L(d 1 , d 2 , . . . , d k ) denotes the set of such logarithmic foliations (or plane vector fields) with For generic polynomials f i of degree d i the degree of the associated vector field is d i − 1. Therefore L(d 1 , d 2 , . . . , d k ) is quadratic, provided that d 1 = 3 or d 1 = 1, d 2 = 2 or d 1 = d 2 = d 3 = 1. This defines three large irreducible components of the center set C 2 of quadratic systems with a Morse center, L(3), L(1, 2), L(1, 1) respectively. We have, however, one more irreducible component of C 2 which is Here L(2, 3) is the set of polynomial foliations as above, with a first integral f 3 2 /f 2 3 where deg f 2 = 2, deg f 3 = 3. Generically such a foliation is of degree four, but it happens that its intersection Q 4 with the space A 2 of quadratic foliations is non empty and it is an irreducible algebraic set. The notation Q 4 is introduced by Zoladek [Żoł94], the index 4 indicates the co-dimension of the set in the space of quadratic vector fields A 2 .
The exceptional set Q 4 might look not quite explicit, we investigate it in details below.
The space A n of polynomial vector fields of degree at most n are identified to a vector space of dimension (n + 1)(n + 2). On A n acts the affine group Aff 2 of affine transformations of K 2 (as usual K = R or K = C), as well the multiplicative group K * corresponding to "change of time", dim Aff 2 × K * = 7. Therefore the dimension of the orbit of a general polynomial vector field is 7. For this reason it is expected that the minimal dimension of a component of the center set C n is also 7. Such components, if exist, will be in a sense exceptional.
In the quadratic case n = 2 the dimension of the four components of C 2 are easily found. For instance, in the case L(1, 1, 1) ⊂ A 2 , and up to an affine changes of variables and time, one may suppose that the first integral is in the form xy λ (1 − x − y) µ . Therefore the dimension of L (1, 1, 1) is 2 + 7 = 9 and the codimension is 3 = 12 − 9. We find similarly that dim L(2, 1) = dim L(3) = 9.
The corresponding affine foliation on the chart C 2 defined by z = 1 (4.10) 3P 3 (x, y, 1)dP 2 (x, y, 1) − 2P 2 (x, y, 1)dP 3 (x, y, 1) = 0 is of degree 4. We may obtain a plane polynomial foliation of degree 2 by imposing the following additional conditions. Suppose first, that the infinite line {z = 0} is invariant, that is to say (up to affine change) This condition can be written as The foliation (4.9) takes the form z P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz = 0 where deg P, deg Q 3, so (4.10) is of degree 3. If we further suppose that z divides the homogeneous one form 3P 3 dP 2 − 2P 2 dP 3 then (4.9) takes the form z 2 P (x, y, z)dx + Q(x, y, z)dy + zR(x, y, z)dz = 0

ANNALES HENRI LEBESGUE
where deg P, deg Q 2, so (4.10) is a plane quadratic foliation. The condition that z 2 divides 2P 3 dP 2 − 3P 2 dP 3 can be written as or equivalently y) 2 a 1 (x, y) = 2b 3 (x, y)b 2 (x, y). (4.13) These polynomial relations can be further simplified by affine changes of the variables x, y. First, (4.12) implies that a 2 is a square of a linear function in x, y which we may suppose equal to x, that is to say The second condition (4.13) becomes 3xa 1 = 2b 2 where we may put a 1 = 2y, and hence a 1 (x, y) = 2y, b 2 (x, y) = 3xy.
This is the exceptional co-dimension four component of Q 4 . The reader may check that the corresponding vector field x = xy − αx + 1, y = αx 2 + 2y 2 + αy − x has a Morse center at x = 1/α, y = 0 which is moreover a usual real center for α ∈ (1, 0). The above computation is suggested by [LN14] where, however, the modulus α is wrongly fixed equal to α = ∞). The foliation on P 2 corresponding to has two invariant lines {x = 0} and {z = 0}, in contrast to the general foliation defined by dH α (x, y) = 0 which has only one invariant line {z = 0}. We resume the above as follows Proposition 4.7. -Every polynomial vector field having a rational first integral of the form H(x, y) = a 0 (x, y) + a 1 (x, y) + a 2 (x, y) where the homogeneous polynomials a i , b j of degrees 0 i 2, 0 j 3 are subject to the relations is of degree two. The set of such quadratic vector fields form the irreducible component Q 4 of the center set C 2 . Up to an affine change of the variables x, y the polynomial H can be assumed in the form H(x, y) = (x 2 +2y+α) 3 (x 3 +3xy+1) 2 where α is a parameter. We conclude this section with the following remarkable property of Q 4 . One may check that general rational function of the form H(x, y) = P 3 2 /P 2 3 , where P 2 , P 3 are bi-variate polynomials of degree two and three, defines a pencil of genus four curves Γ t = {(x, y) : H(x, y) = t} on C 2 . However, the special rational function H α (4.14) defines an elliptic pencil, that is to say the level sets Γ t = (x, y) ∈ C 2 : H α (x, y) = t are genus one curves, see [GI09].

The center set of the polynomial Liénard equation
Consider the following polynomial Liénard equation The description of the non-degenerate centers of (4.16) is due to Cherkas [Che72] in the real analytic case, and to Christopher [Chr99] in the polynomial case.
Consider the following Polynomial Composition Condition (PCC) There exist polynomials P , Q, W such that . The Theorem of Cherkas and Christopher can be formulated as follows The proof of Theorem 4.8, see [Chr99,CL07], is based on the following simple observation due to Cherkas [Che72, Lemma 1] Lemma 4.9. -The real analytic equation has a center at the origin, if and only if a 2j = 0, ∀ j 1.
Indeed, the truncated equation is reversible in x, and hence it has a center at the origin. In a sufficiently small neighbourhood of the origin, (4.18) is rotated with respect to the vector field (4.17), unless a 2j = 0, ∀ j 1. The final argument of Christopher is to use the Lüroth theorem, to deduce the (PCC) condition. This topological argument of Cherkas does not apply in a complex domain. We shall prove, however, the following more general  2 y 2 + Q(x) has a Morse critical point at the origin and there exists a local bi-analytic change of the variable x → X such that 1 2 y 2 + Q(x(X)) = 1 2 (y 2 + X 2 ). Thus (4.21) ydy + (q(x) + yp(x))dx = 1 2 d(y 2 + X 2 ) + ydP (x(X)).
We expand and obtain the equivalent foliation By analogy to the Cherkas Lemma we shall prove Proof. -After rescaling (X, y) → ε(X, y) the foliation takes the form (4.23) and it suffices to prove that for sufficiently small ε it has a Morse critical point. Note first that the truncated foliation F t ε (4.24) F t ε : under the map π : (X, y) → (ξ, y), ξ = X 2 . The foliation (4.25) is regular at the origin and has a first integral where O(ε) is analytic in ε, ξ, Y , and vanishes as ε = 0. Thus F t ε has a first integral where O(ε) is analytic in ε, X 2 , Y , and vanishes as ε = 0. This also shows that the origin is a Morse critical point of the truncated foliation F t ε . As H ε is a first integral of F t ε then for every fixed ε we have Suppose now that for some j 1, a 2j = 0 and let j = k be the smallest integer with this property. We have (4.26) where by abuse of notations O(ε 2k+1 ) denotes an analytic function in X, y, ε which is divisible by ε 2k+1 . The origin is a Morse critical point if and only if the holonomy map of the two separatrices of F ε at the origin, are the identity maps. The holonomy map will be evaluated by the usual Poincaré-Pontryagin-Melnikov formula. The separatrices are tangent to the lines y ± iX = 0. We take a cross-section to one of the separatrices, parameterised by the restriction of H ε (x, y) on it. Let be a continuous family of closed loops vanishing at the origin as h → 0. The holonomy map of F ε , corresponding to this closed loop is By homogeneity of the polynomials As the homology of the algebraic curve {(y, X) ∈ C 2 : y 2 + X 2 = 2h} has one generator we can suppose that this generator is just the real circle γ 0 (1) = {(y, X) ∈ R 2 : y 2 + X 2 = 2} and in this case We conclude that if the holonomy map is the identity map, then a 2k = 0 which is the desirable contradiction. Lemma 4.11 is proved. Proof of Theorem 4.10. -Assuming that the Liénard equation has a Morse critical point, and hence Q(x) has a Morse critical point at the origin, denote x 1 (h), x 2 (h) the two roots of the polynomial Q(x) − h which vanish at 0 as h tends to 0. We have obviously that X(x 1 (h)) = −X(x 2 (h)). By Lemma 4.11 the analytic function P (x(X)) is even in X, and hence P (x 1 (h)) ≡ P (x 2 (h)). Following an idea of Christopher (already used at the end of Section 3), consider now the subfield C ⊂ C(x) formed by all rational functions R = R(x) ∈ C(x) satisfying the identity According to the Lüroth theorem, every subfield of C(x) is of the form C(W ) for some rational function W = W (x). Thus we have C = C(W ) where P, Q ∈ C. Therefore there exist rational functions P , Q such that Using the same argument as in the proof of Theorem 3.2 we may suppose that P , W, Q are polynomials, and hence P, Q satisfy (PCC) which completes the proof of the Theorem 4.10.

Abel equations with Darboux type first integral
The polynomial Liénard equation with associated foliation ydy + (q(x) + yp(x))dx = 0, after the substitution y → 1/y, becomes the following Abel equation Equivalently, we consider the foliation Note that the Cherkas-Christopher theorem is for non-degenerate centers. The Composition Conjecture missed the possibility for the Abel or Liénard equations to have a Darboux type first integral, with resonant saddle point and characteristic ratio p : −q (instead of a non-degenerate center with 1 : −1 ratio). Incidentally, Liénard equations with a Darboux type first integral will produce counter-examples to the Composition Conjecture, which is the subject of the present section. We explain in this context the recent counter-example of Giné, Grau and Santallusia [GGS19].
The method of constructing such systems is based on the example of the co-dimension four center set Q 4 for quadratic system, as explained in Section 4.2. Let where a i , b j are polynomials, such that P p 2 = Q q 2 + O(y 3 ), where p, q are positive relatively prime integers. This implies that the corresponding one-form pQ 2 dP 2 − qP 2 dQ 2 is divisible by y 2 , and then the associated reduced foliation (after division by y 2 ) is of degree two in y, and moreover {y = 0} is a leaf. Therefore the foliation is defined as (4.30) (r 1 y + r 2 )dy = y(r 3 y + r 4 )dx = 0, where r 1 = 2(p − q)a 2 b 2 Note that if a 2 = const. = 0, b 2 = const. = 0 the foliation takes the Liénard form (4.31) (r 1 y + r 2 )dy = yr 4 dx, r 1 = const. .
Of course, it is not clear, whether such polynomials exist. To verify this we have to solve the equation a 0 (x) + a 1 (x)y + a 2 (x)y 2 p = b 0 (x) + b 1 (x)y + b 2 (x)y 2 q mod y 3 assuming that a i (x), b j (x) are polynomials, and a 2 , b 2 are constants. A first condition is given by Thus a i , b j are polynomials which satisfy the following redundant system of equations It follows that for some polynomial R, a 0 (x) = R(x) q , b 0 = R(x) p and moreover pa 2 R(x) −q − qb 2 R(x) −p is a square of a rational function, where we recall that a 2 = const., b 2 = const.. It is easy to check that this is only possible if, say p < q, and p = 2k − 1, q = 2k for an integer k 1. With this observation the analysis of the system is straightforward and is left to the reader. We formulate the final result in the following (1 − r(x) 2 ) 2k + 2kr(x)(1 − r(x) 2 ) k y + ky 2 2k−1 (1 − r(x) 2 ) 2k−1 + (2k − 1)r(x)(1 − r(x) 2 ) k−1 y + 2k−1 2 y 2 2k is the first integral of a Liénard type equation of the form It is clear that the above Liénard system is a polynomial pull back under x → r(x) of a simpler master Liénard system with first integral (4.33) H k (x, y) = (1 − x 2 ) 2k + 2kx(1 − x 2 ) k y + ky 2 2k−1 (1 − x 2 ) 2k−1 + (2k − 1)x(1 − x 2 ) k−1 y + 2k−1 2 y 2 2k which can not be further reduced. As the Liénard equation (4.32) is equivalent to the Abel equation where z = 1/Y and Y = y − r 2 (x) and p(x) = r 4 (x) + r 2 (x), q(x) = r 2 (x)r 4 (x) then we obtain for every k ∈ N * an Abel equation with a Darboux type first integral. Except in the case k = 1 these Abel equations will have a center along the interval [−1, 1].