CENTER CYCLICITY FOR SOME NILPOTENT SINGULARITIES INCLUDING THE Z 2 -EQUIVARIANT CLASS

. This work concerns with polynomial families of real planar vector ﬁelds having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum (1 ,n )-quasihomogeneous weighted degree, being n the Andreev number of the singularity. These families strictly include the case n = 2 and also the Z 2 -equivariant families. In some cases for such families we solve, under additional assumptions, the local Hilbert 16th problem giving global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family. Several examples are given.


Introduction
We shall consider analytic families of planar vector fields possessing a nilpotent singularity, that is the linearization of the field around it does not vanish but the associated eigenvalues do. It is well known that by an affine change of coordinates and a time rescaling the singularity can be placed at the origin and the linearization adopts a canonical form such that the vector field is written in the form (1)ẋ = y + P (x, y; λ),ẏ = Q(x, y; λ), where P and Q are analytic functions at (x, y) = (0, 0) without constant or linear terms. The parameters of the family are denoted by λ ∈ E ⊂ R p . We assume the parameter space E is chosen in such a way that the origin of (1) is monodromic for the full family so that the local orbits of (1) rotate about the origin and consequently there is a local Poincaré return. Since (1) is analytic, it follows that the local qualitative phase portrait of (1) corresponds to either a center or a focus, see for example two independent proofs of this fact in [18] and [33]. Finding out which of the two behaviors actually occurs is known as the center-focus problem.
The characterization of the nilpotent monodromic singularities was made by Andreev in [8] and essentially depends on two positive integer numbers n and β with n ≥ 2 and β ≥ n−1 (including also the possibility β = ∞), see the forthcoming Theorem 32. We call n the Andreev number associated to an analytic monodromic nilpotent singularity and in [22] it is proved that n is invariant under analytic orbital equivalency. A by product of the Andreev's characterization [8] is that any system (1) with a monodromic singularity at the origin of class (n, β) is analytically conjugate to the Andreev's canonical form (2)ẋ = yA(x, y; λ),ẏ = B(x, y; λ), where A(0, 0; λ) = −1, B(x, 0; λ) = x 2n−1 + O(x 2n ) with n the Andreev number and ϕ(x; λ) = ∂ y B(x, 0; λ) = O(x β ) with β ≥ n − 1 or ϕ(x; λ) ≡ 0 (corresponding this last case with β = ∞). In the special case that the order at x = 0 of ϕ reach the minimum value β = n − 1 we will use throughout the work the parameter notation λ = (ω, ν) ∈ E to emphasize that ϕ(x; λ) = ωx n−1 + O(x n ) with ω ∈ R * = R\{0} it follows that the parameter space E of family (2) is where Ω n = {ω ∈ R * : ω 2 < 4n}. Observe that we treat ω as a special parameter when β = n − 1 not only because it cannot take any real value but also because it is present in the (1, n)-quasihomogeneous truncation of minimum degree of the vector field (2) and this fact causes serious problems in some points of this work. The restriction of family (2) to E implies that the origin remains always monodromic for the full family as Theorem 32 characterizes.
We emphasize here that nilpotent centers are not characterized by the existence of a formal first integral. Some exceptions to this rule are the trivial Hamiltonian centersẋ = ∂ y H(x, y),ẏ = −∂ x H(x, y), the monodromic time-reversible with respect to (x, y, t) → (x, −y, t) (see [10]) or the monodromic Z 2 -equivariant (those invariant under the representations of Z 2 acting on R 2 by (x, y) → (−x, −y)), see [4] and particular cases in [23]. Notice that these exceptions can occur in some specific system (1) but, for example, it never happens in the monodromy class β = n − 1 as we will see.
It has been proved in [22] that if an analytic vector field with a monodromic nilpotent singularity has a formal first integral then it also has an analytic inverse integrating factor. Anyway the existence of a formal inverse integrating factors also does not characterize the nilpotent centers although some exceptions appear like the minimum Andreev number n = 2, see the saga of works [35], [36] and [37] or see [22] for a simpler and unified proof.
Some relations arise like in the monodromy class n odd and β ≥ n for which in the work [5] it is proved that if there is a formal inverse integrating factor then the singularity is a center if and only if it is formally integrable. But besides the trivial case of having a formal inverse integrating factor non-vanishing at the origin so that one can construct an associated formal first integral, hence a local analytic first integral by the results in [40], no other relationships are known.
In the literature there are several techniques specifically designed to solve the nilpotent center-focus problem. As it is explained in [7], one of them consists in applying the normal form theory to orbitally conjugate the vector field, up to some order, into certain Liénard normal form. Another method consists in to embed the nilpotent singularity as the limit of a 1-parameter family of nondegenerate centers, see the initial source [30] until the final corrected version [25]. Instead we will use an approach following ideas of Lyapunov [39] which are explained in, for example, [6] and [27]. It consists on blowing-up the singularity by means of generalized polar coordinates, see section §4.2 for details, in such a way that a local analytic Poincaré return map Π : Σ × E → R is defined on a transversal section Σ ⊆ R to the flow with endpoint at the singularity. Thus one can construct the analytic expansion Π(r 0 ; λ) = j≥1 v j (λ)r j 0 where r 0 ≥ 0 is the parameterization of Σ. The small amplitude (surrounding the singularity) periodic orbits of (2) correspond to positive fixed points of Π(.; λ). We are interested in the analysis of the so-called local Hilbert 16th problem at the singularity or, in other words, to determine the maximum cyclicity of the origin inside the vector field family X associated to (2). We recall that the cyclicity of the specific nilpotent monodromic singularity at the origin of system (2) with parameters λ = λ † ∈ E is, roughly speaking, the maximum number of limit cycles that can bifurcate from it for all the parameters λ ∈ E with λ − λ † 1. We will de note by Cyc(X , 0) the maximum cyclicity of any center at the origin and this number is the most difficult to obtain. The cyclicity notion for the nondegenerate (hence not nilpotent) center problem was introduced in [19].
From now we will focus on polynomial families (2) parameterized by its coefficients, or more generally analytic families (2) depending linearly with their parameters. Also we restrict our selves to the monodromic points out of class β = n − 1 and n odd because there the singularity is always a focus with v 1 = 0 and no small amplitude limit cycle can bifurcate. The aim of this work is to obtain algebraic procedures capable of producing an upper bound on Cyc(X , 0) for some families (2) characterized by the fact that any center has associated a formal inverse integrating factor vanishing at the singularity with a leading term of minimum (1, n)quasihomogeneous weighted degree. These families are characterized by I = I even and sometimes by the stronger condition I odd = {0}, see Definition 7 to understand this notation.
A first result of this work is the development of a new and efficient tool to analyze the nilpotent center-focus problem of that families that shares a computational simplicity analogous to the computation of the focus quantities arising in the construction of a formal Lyapunov function in the simpler classical non-degenerate Poincaré center-focus problem, see the book [42] for example. The algorithm only involves algebraic calculations and is presented in Theorem 3 whereas the allowed families are implicit in Corollary 10(iii). In the algorithm there appears what we call the integrability focus quantities {η j (λ)} j∈N which represents the obstructions to the existence of the above mentioned formal inverse integrating factors. It can be checked that the η j only depend on the parameters of the family and they lie in some Noetherian ring: either the polynomial ring R[λ] when β = n − 1 or the ring R(ω)[ν] (the set of polynomials in ν with rational coefficients in ω) otherwise. Here we interpret the elements of R(ω) as just formal expressions so R(ω) is a field, hence R(ω)[ν] is a Noetherian ring. Taking advantage of these algebraic structures we can use some tools of algebraic geometry to construct bounds on Cyc(X , 0) where the analyzed families strictly contain the Z 2 -equivariant family (as follows by Theorem 12 and Proposition 13) and also the specific case n = 2, see [24]. A consequence of the work is the existence of a center variety V C ⊂ R p associated to the origin of the studied families (16) which is an algebraic variety in the parameter space having the property that the center corresponds with the parameters that lie in V C ∩ E. We note that V C is the variety of the polynomial Bautin ideal B = v j : j ∈ N only when β = n − 1 and is the variety of certain polynomial ideal (called G is this work) associated to B otherwise.
We recall that the finite-codimension Hopf bifurcations that lead to the analysis of the cyclicity of nilpotent foci is much easier than the infinite-codimension center bifurcations that we perform in this work, which are needed to obtain the cyclicity of a nilpotent centers. The main reason is that the study of center bifurcations needs information on the Bautin ideal B in the corresponding ring rather than only on its associated real variety V R (B) ⊂ R p . This is why only few general results are known on the nilpotent center cyclicity problem. The techniques introduced in this work generalize to those given in [23], [24] and [29].
The paper is organized as follows. In §2 we present the main theoretical results of the work, being subsection 2.1 focusing on its cyclicity bounds applications when I = I even while subsection 2.2 explains the difficulties arising otherwise when I = I even . Next §3 is devoted to present some examples illustrating the theory. In §4 we introduce the background and preliminaries needed to follow later the proofs that are relegated to the §5. In the last §6 we analyze the behaviour and the properties of the examples presented before.

Main theoretical results
We recall here that a formal series V ∈ R[[x, y]] with V ≡ 0 that formally satisfies X (V ) − V div(X ) = 0 (being div(.) the divergence operator) is called a formal inverse integrating factor of the planar vector field X because div(X /V ) ≡ 0 holds at the formal level. First we want to emphasize the following statement whose proof is given in §5.1.

Proposition 1.
There are analytic nilpotent centers without formal first integral neither formal inverse integrating factor.
We start by defining quasihomogeneous functions and vector fields.
Definition 2. We denote by P (1,n) k ⊂ R[x, y] the set of (1, n)-quasihomogeneous polynomials of weighted degree k. That is, p k (x, y) ∈ P (1,n) k if p k (ξx, ξ n y) = ξ k p k (x, y) for all ξ ∈ R. Moreover, a vector field X i = p i+1 ∂ x + q i+n ∂ y is a (1, n)quasihomogeneous polynomial vector field of weighted degree i if p i+1 ∈ P (1,n) i+1 and q i+n ∈ P (1,n) i+n . The following theorem shows the existence of certain families of formal power series V ∈ R[[x, y]] with specific leading terms that have associated some sequence {η j (λ)} j∈N of functions in the parameters λ of the Andreev canonical form (2) in such a way that V becomes a formal inverse integrating factor of (2) under the conditions η j = 0 for all j ∈ N. Theorem 3 will be proved in §5.2.
Theorem 3. Let X be the vector field associated to the Andreev canonical form (2) with Andreev number n. Then there are formal series V (x, y) = j≥2n V j (x, y) where V j ∈ P (1,n) j , with leading term V 2n (x, y) = x 2n + ny 2 + ωx n y with ω ∈ Ω n when β = n − 1 and V 2n (x, y) = x 2n + ny 2 otherwise, such that V formally satisfies Moreover V is unique if we set the monomials V j (0, y) = τ j (λ)y j for j > 2n. Additionally, for any polynomial family (2) parameterized by its coefficients one has that either η j ∈ R(ω)[ν] are well defined for ω ∈ Ω n when β = n − 1 or We emphasize that when η j = 0 for all j ≥ 3n, equation (4) implies that V is a formal inverse integrating factor of (2) which, since V (0, 0) = 0 does not implies the existence of formal first integral.
Definition 4. We call the {η j (λ)} j≥3n appearing in (4) the integrability focus quantities associated to the origin of the Andreev canonical form (2).

Remark 5.
Under the assumption η j = 0 for any j ≥ 3n we have the following straightforward consequences derived from [22]: (i) If either β = n − 1 and n is even or β ≥ n or ϕ(x) ≡ 0 then the origin is a center. The converse does not hold in general. (ii) If β = n − 1 then V is the unique formal inverse integrating factor of X up to multiplicative constants.
We state in the next proposition what we call the trivial case.
Proposition 6. We consider any family (1) with associated Andreev number n and β = n − 1. If n is odd then the origin is a focus and no limit cycle of (1) can bifurcate from the origin varying the parameters of the family.
To continue we need some definitions and notations. We will use from now the following standard algebraic notation, see for example [15]. The ideal generated by elements r 1 , . . . , r s of a ring R will be denoted by r 1 , . . . , r s . For a field K the variety in K n of an ideal I ⊂ K[x 1 , . . . , x n ] will be denoted V K (I). We remember here that, for two ideals I and J the sum ideal I + J is defined as I + J = {p + q : p ∈ I, q ∈ J}. Two properties are that a basis of I + J is the union of the basis of I and J and the equality of varieties V K (I + J) = V K (I) ∩ V K (J) holds. In this work we will only use either K = R or K = C and all the ideals lie in a polynomial ring over a field so that by the Hilbert Basis Theorem they are finitely generated. Definition 7. We define several ideals associated with the polynomial Andreev canonical family (2) with Andreev number n in the Noetherian rings R(ω)[ν] or R[λ] according to whether β = n − 1 and n even or not, respectively.
(a) The Bautin ideal B = v j : j ∈ N and the ideal B k = v 2 , . . . , v k . (b) I = η j : j ∈ N and I k = η 3n , η 3n+1 , . . . , η k with k ≥ 3n. (c) I even = η 2j : 2j ≥ 3n , I odd = η 2j+1 : 2j + 1 ≥ 3n so that I = I even + I odd , I even The next result shows the relationship between the v j and the η j quantities of system (2) under some assumptions. Statement (i) was already proved in [27] and we write it only for the sake of completeness.
Remark 9. A consequence of Theorem 8 is the knowledge of the initial string of identically zero Poincaré-Lyapunov quantities v j with j ≥ 2. To know this gap, let s ∈ N be such that η 3n+s ≡ 0 is the first non-identically zero integrability focus quantity. Then by Theorem 8 it follows that B 2+s = {0} if n + s is odd whereas B 2+s = v 2+s = η 3n+s when n + s is even.
In fact the proof of Theorem 8 shows that we have more information than the stated in Theorem 8 as the next simple corollary shows.
The strong requirement I odd = {0}, which implies in particular the weaker condition I = I even in Corollary 10(iii), is always reached (choosing adequate freedoms τ j (λ)) in case of having the minimum Andreev number n = 2 and β = 1, see [24]. In the next theorem we will see how to detect other cases with I odd = {0} under some symmetry conditions on the Andreev canonical form (2). Before the statement of Theorem 12 we recall some concepts on equivariant dynamics.
Let GL(2) be the group of all invertible linear transformations acting on R 2 and Γ ⊂ GL(2) a (linear) Lie group. Let Γ be a Lie group acting on the vector space R 2 . A vector field X on R 2 is Γ-equivariant if it commutes with Γ, that is, X • γ = γX for all γ ∈ Γ. Given a diffeomorphism σ on R 2 , the vector field X is said to be σ-equivariant if σ * (X) = X, where σ * is the push-forward associated to σ, that is, σ * (X) = (Dσ X) • σ −1 . We will say that σ is a symmetry of X. Notice that for any γ ∈ Γ one has γ * (X) = γX • γ −1 = X • γ • γ −1 = X, hence γ is a symmetry of X.
In the following we focus on the specific element This γ is present in the representation of several important Lie groups. For example γ ∈ O(2), the orthogonal group O(2) acting on the plane and also γ ∈ SO(2) ⊂ O(2) the special orthogonal group since γ corresponds with a rotation of angle π. But it is well known, see for example [32], that when SO(2) act on R 2 ≡ C in the standard way φ θ (z) = exp(iθ)z then every smooth SO(2)-equivariant planar vector field g : R 2 → R 2 has de form g(z) = p(zz)z + q(zz)iz where p and q are realvalued SO(2)-invariant functions. Some computations show that this implies that the linear part of a singularity at the origin of g cannot be a nilpotent matrix, hence the Andreev canonical form (2) is never SO(2)-equivariant. But (2) can be Z 2 -equivariant where Z 2 is the cyclic group acting on R 2 and represented by Z 2 = {I 2 , γ} with I 2 the identity 2 × 2 matrix since γ is an involution. In [4] it has been proved that the origin of any Z 2 -equivariant nilpotent vector field is a center if, and only if, it is locally analytically integrable.
Remark 11. We note that if the Andreev canonical form (2) admits the symmetry γ given in (5), that is, it is Z 2 -equivariant, then the function ϕ must be even. So if ϕ ≡ 0 then β is even and therefore, by Proposition 6, the case β = n − 1 corresponds to a trivial case since n would be odd. In consequence, either the Z 2 -equivariant Andreev polynomial canonical form (2) has a focus at the origin with zero cyclicity when β = n − 1 or the associated integrability focus quantities The next theorem states a key property of the Z 2 -equivariant vector fields (2).
Theorem 12. Let X be the polynomial vector field associated to a Z 2 -equivariant Andreev canonical form (2), hence with β ≥ n or ϕ(x) ≡ 0. Then there is a formal series V † (x, y) that satisfies (4) in such a way that From the computational point of view, Theorem 12 tell us that for the Z 2equivariant Andreev canonical vector fields we can select the freedom coefficients τ j in the formal series V (x, y) that satisfies (4) in such a way that I odd = {0}. However a natural question arises: are Z 2 -equivariant the only families (2) for which I odd = {0} holds? The answer is no as the following example shows.
Proposition 13. The 1-parameter family is not Z 2 -equivariant but its associated ideal I odd in the ring R[λ 4 ] is I odd = {0}.
We will see how to derive, sometimes, upper bounds on the cyclicity of nilpotent centers inside polynomial Andreev canonical families (2) under the main assumption that I = I even . Recall that families satisfying this condition strictly contain the Z 2 -equivariant family and they give rise to the title of this work. Recall that the families with I = I even and β = n − 1 are not Z 2 -equivariant as we emphasize in Remark 11.

Remark 14.
For the sake of brevity we do not give the explicit proofs of the forthcoming Theorem 17, Proposition 19, Theorem 21 and Theorem 22. The ideas involved in these proof are based on the adaptation of known results to our framework, which is rather straightforward under the assumption I = I even which implies B = I taking into account our Theorem 8. Besides using general facts established in books such as [42], [43] or [34], these proofs mainly apply constructions from [29], [23] and [24], this last reference strongly based on the work [28]. Theorem 22 ultimately relies on results in [13]. Essentially we need to replace the concepts of (i) nondegenerate planar monodromic singularity or Hopf singularity in R 3 ; (ii) formal first integral; (iii) focal values; and (iv) admissible parameter in the linear part of the vector field in R 3 lying in R * by (i) monodromic nilpotent singularity with I = I even ; (ii) formal inverse integrating factor; (iii) integrating focal values; (iv) when β = n−1 the admissible parameter in the (1, n)-quasihomogeneous terms of the vector field lying in Ω n . This approach has already been used successfully in [24] when n = 2.
Each monodromy class has its own characteristics, which implies a personalized treatment of each of them, although there are many similarities too. To tackle the cyclicity problem we will need to complexify the parameter space (to use the completeness of C and Hilbert Nullstellensatz) and along this process the parameter ω (when it exists because ϕ(x) ≡ 0) causes several new troubles with respect to the other monodromy classes, see [24] for the specific case n = 2. Before starting we need the concept of minimal basis.

Definition 15.
When v 1 = 1 (that is out of the trivial case β = n − 1 and n odd), the minimal basis of the Bautin ideal B with respect to an ordered basis Notice that the Noetherian property assures that B admits a unique minimal basis. Clearly analogous definitions apply to obtain minimal basis of the other ideals defined in Definition 7. Performing computations we shall use the following notation.
Definition 16. We defineη j to be the remainder of η j upon division by a Gröbner basis of the ideal I j−1 in the corresponding ring and we will use the notation η j ≡η j mod I j−1 .
We recall that complexification of Bautin ideal and divisions based on Gröbner basis related to certain Bautin ideal in the spirit of Definitions 15 and 16 where used in [20].
2.1.1. The monodromy class β ≥ n or ϕ(x) ≡ 0. In this case the Bautin ideal lies in the polynomial ring R[λ]. Notice that since we will assume I = I even then B = I and, by the Noetherian property of the involved rings, there is an even number k ∈ N such that A first result is the following theorem.
Theorem 17. Let X be the vector field associated to a polynomial Andreev canonical family (2) with parameter space E, Andreev number n and β ≥ n or ϕ ≡ 0. Assume that I = I even and the center problem at the origin has been solved being k the minimum even number satisfying (7). Assume the equality (7) holds in the complex setting, i.e., suppose that V C (B) = V C (I k ) and let m be the cardinality of a minimal basis of I even k . Then, the following holds: (i) If I k is a radical ideal then B = I k and Cyc(X , 0) ≤ m − 1.
(ii) Suppose a primary decomposition of I k can be written as I k = R ∩N where R is the intersection of the ideals in the decomposition that are prime and N is the intersection of the remaining ideals in the decomposition. Then for any vector field X * of family (2) corresponding to Notice that we obtain cyclicity bounds for any point of the center variety only in case we are lucky and either I k = √ I k or V R (N ) = ∅. Moreover, the application of Theorem 17 to practical cases has several pitfalls to avoid. First we need to overcome the difficulty (only appearing when the family is not Z 2 -equivariant) consisting in to prove that I = I even is true, see for example the proof of Proposition 13. Secondly we have to prove the complex equality V C (B) = V C (I k ) under the real condition (7). The equality of real varieties (7) does not imply, in general, the equality of complex varieties in the assumptions of Theorem 17. In the following results we show some cases where we succeed to establish this correspondence between real an complex varieties.
Proposition 18. Assume that (7) holds for a polynomial Andreev canonical family (2) having β ≥ n or ϕ(x) ≡ 0 and with I = I even . Then V C (B) = V C (I k ) provided for any λ * ∈ V C (I k ) one of the following situations occur for the complexification of (2) on C 2 with λ = λ * : (i) There is a complex formal inverse integrating factor of the form The following proposition is an adaptation of Proposition 29 in [28] to the nilpotent monodromy class β = n − 1, that is when the Bautin ideal is polynomial.
be a set of polynomials satisfying the following conditions: (a) r p; The monodromy class β = n−1 and n even. In order to analyze the cyclicity problem in this case we proceed like in [24]. Since the η j ∈ R(ω)[ν] for any j ≥ 3n are well defined for any ω ∈ Ω n , they can be expressed as Then we can define the polynomial ideals G = g j : j ∈ N and G k = g 1 , g 2 , . . . , g k in the ring R[ω, ν]. The final summary of this approach is contained in the following result. To interpret the result correctly we need the following definition.
Definition 20. Given a field k and an ideal that is generated by the polynomials f * j (ν) = f j (ω * , ν) that arise when the indeterminate ω is replaced by the fixed element ω * of k.
Theorem 21. For the polynomial Andreev canonical family (2) with β = n − 1 and n even, Theorem 17, Proposition 18 still work after the replacement of the ideals B and I k by G and G k , respectively. With the same modification, Proposition 19 also applies but only gives the conclusion V C (G * k ) = V C (G * ) in C p−1 when ω is assigned to any fixed value ω * ∈ Ω n .
It is worth to emphasize that there is no analog to Proposition 18 in case β = n−1 because the variety V C (G) is not characterized by the existence of formal inverse integrating factors of the associated complex system (2). Indeed there are points (ω, ν) ∈ V C (G) with ω ∈ C\Ω n having associated a system (2) on C 2 without formal inverse integrating factor, see for example Proposition 2.9 in [24].
2.1.3. Irreducible components of the center variety in any monodromy class. There is a different approach (based on the ideas in [13]) for the center cyclicity problem which focus on the irreducible components of the center variety. For any κ ∈ N less or equal to the Bautin depth of B, we denote by d P G κ the Jacobian matrix of the mapping evaluated at a point P ∈ E, where {η j1 , . . . , η jκ } is the minimal basis of I jκ . Notice that G κ is a real polynomial map when β ≥ n or ϕ(x) ≡ 0 but it is just real analytic otherwise.
Theorem 22. Let C be an irreducible component of the center variety associated to the origin of the polynomial family (2) Then the following holds: (i) There exists a neighborhood U of P such that C ∩ U is a submanifold of codimension at least κ and there exist bifurcations of (2) producing κ−1 small amplitude limit cycles from the origin for parameter values with λ sufficiently close to P . (ii) If moreover codim(C) = κ then P is a smooth point of C and the cyclicity of P and also of any point in a relatively dense open subset of C is exactly κ − 1.
2.2. Some remarks when I = I even . We include this section so that the reader intuits the additional complications that appear in the cyclicity problem when I = I even . We only briefly comment the case of having a polynomial Bautin ideal by the sake of simplicity. By Theorem 8 and the Noetherian property of the rings involved there is always an even number k ∈ N such . Therefore, even when I = I even holds in family (2) we can still perform some computations around the center problem. In this direction, we can analyze the restricted family X | V which is defined as the polynomial family (2) whose coefficients are confined to lie in E ∩ V where V = V R (I odd k−1 ) with k the minimum integer defined by (10). Indeed, the equality (10) gives a solution of the center problem for such a restricted family Focusing on the cyclicity problem we can prove the following analogous result to Theorem 17. We state it although we will see below the difficulties that its application in practical problems has as consequence of having I = I even . Anyway it is instructive for the reader to emphasize these problems since in this way the inherent difficulties of this type of calculations can be glimpsed.
Theorem 23. Let X be the vector field associated to a polynomial Andreev canonical family (2) with parameter space E, Andreev number n and β ≥ n or β = n − 1 and n even or ϕ ≡ 0. Assume the center problem at the origin has been solved for the restricted family X | V with parameters lying in E ∩ V where V = V R (I odd k−1 ) and k is the minimum even number satisfying (10). Assume the equality (10) holds in the complex setting, i.e., suppose that V C (B + I odd k−1 ) = V C (I k ) and let m be the cardinality of a minimal basis of I even k . Denote by Cyc(X | V , 0) the cyclicity of the origin under perturbations within family X | V . Then, if I k is a radical ideal, Unfortunately the former Theorem 23 has no practical value at the moment and we explain why below. The fact that the equality of real varieties (10) holds does not imply, in general, the equality of complex varieties V C (B +I odd k−1 ) = V C (I k ) which is one of the assumptions in the above result. Actually, k−1 so we only need to prove the reverse inclusion of complex varieties. Thus, we must check (quite difficult) whether V C (I k ) ⊂ V C (B). Unfortunately now B = I since I = I even and we have not been able to characterize those points belonging to V C (B) in any satisfactory way.

Examples
In this section we analyze some families (2) under the light of the developed theory.
Remark 24. It is convenient to use an computer algebra system such as Maple or Mathematica to obtain the expressions of η j . Also, the computations need intensive use of several routines incorporated in the primdec.lib library [17] of Singular [16]. As a taste, we use the routine minAssChar to obtain the prime decomposition of √ J for some polynomial ideals J. We also use the routines primdecGTZ or primdecSY to check whether J is radical or not for example.
Of course when these cyclicity lower bounds reach the upper bounds of Theorem 17 for β = n − 1 or Theorem 21 otherwise then we obtain exactly the cyclicity.
3.1. Example 1. We present one example with n = 2 and β = 1 studied in the work [31]. Using his notation, the 1-parameter family is given bẏ and the method used in [31] only can establish that (13) −98 + 47a 2 + 20a 4 = 0 is a necessary center condition for the origin of (12). We will go further using our method.
Proposition 26. The origin is monodromic for (12) when 4a 2 − 16 < 0, in which case it is always a focus.
This family has Andreev number n = 3 and ϕ(x) ≡ 0 and in [4] it is proved that the only centers at the origin lie in the time-reversible subfamily invariant under (x, y, t) → (−x, y, −t) and that all of them are analytically integrable. The following result confirm this and gives moreover either the cyclicity or an upper bound of it except for the Hamiltonian centerẋ = y,ẏ = −x 5 corresponding to (18) with λ = 0. Proposition 30. The center variety V C associated to the origin of family (18) has one unique codimension 4 irreducible component The origin is a center of (18) if and only if λ ∈ V C = V R (I even 38 ). Moreover the cyclicity of the origin under perturbations within family (18), denoted by Cyc(X , 0), satisfies the following restrictions: (i) Any center at the origin with 195λ 1 + 11λ 6 = 0 has Cyc(X , 0) = 4; (ii) Any center at the origin with 195λ 1 + 11λ 6 = 0 has Cyc(X , 0) ≥ 3. If moreover λ = 0 then Cyc(X , 0) ≤ 6.
(iii) The cyclicity of any focus of (18) at the origin is bounded by 6 and there are foci with cyclicity 6.
3.6. Example 6. Another Z 2 -equivariant nilpotent family considered in [4] iṡ x = y + y(λ 1 x 2 + λ 2 xy + λ 3 y 2 + λ 4 x 3 y + λ 5 x 2 y 2 + λ 6 xy 3 + λ 7 y 4 ), The family has Andreev number n = 2. In fact in [4] the center problem is solved in the larger family containing a term ωx 2 y in the second component of this vector field. In that enlarged family β = 2 if the coefficient ω = 0, but therefore the origin is a focus by Proposition 27(i) so we kill that monomial and only consider (20) corresponding to the ϕ(x) ≡ 0 case. Proposition 31. The center variety V C associated to the origin of family (20) decomposes into the irreducible components The origin is a center of (20) if and only if λ ∈ V C = V R (I even 20 ). Moreover the cyclicity of the origin has the following properties: (i) Any center at the origin with parameters on C 3 has Cyc(X , 0) = 6 provided λ 8 λ 9 = 0. Moreover Cyc(X , 0) ≥ 4 when λ 8 = 0 but λ 9 = 0 and Cyc(X , 0) ≥ 3 when λ 9 = 0; (ii) Any center at the origin with parameters on C 2 has Cyc(X , 0) ≥ 4 provided λ 9 = 0 and Cyc(X , 0) ≥ 3 otherwise. (iii) Any center at the origin with parameters on C 1 has Cyc(X , 0) = 4 when λ 9 = 0 and Cyc(X , 0) ≥ 3 otherwise. (iv) The cyclicity of any focus of (18) at the origin is bounded by 6 and there are foci with cyclicity 6.
The following theorem of Andreev [8] characterizes analytic systems (1) for which the origin is monodromic.
Any analytic vector field with a monodromic nilpotent singularity is locally analytically conjugated to the Andreev's canonical form (2). The conjugation is done performing first the analytic change of variables (x, y) → (x, y − F (x; ω, ν)) where the function F is defined in Theorem 32 and next making the rescaling (x, y) → (ξ x, −ξ y) with ξ = (−1/a) 1/(2−2n) which simply introduce a minus sign in the first equation of the differential system just for convenience in the next section.
Remark 33. It was proved in [27] that when (2) with λ = λ † ∈ E possesses a focus at the origin with v 1 (ω † ) = 1 then the first subindex k with v k (λ * ) = 0 has the same parity than the Andreev number n (both are even or odd). This is just the information given in Theorem 8(i).
The computation of the v j is cumbersome and depend on the first terms of the Taylor expansion at r = 0 of the function F(r, θ; λ) = F 1 (θ; ω)r + j≥2 F j (θ; λ)r j defined in (23). As a taste, v 1 (ω) = exp if β = n − 1, 0 otherwise.
Next, assuming that v 1 = 1, there are positive constant γ j (n) such that the next Poincaré-Lyapunov quantities are v 2 = γ 2 (n) ω = 0 if β = n and n is even, 0 otherwise (provided β ≥ n), if β = n and n is odd, γ 3 (n) ω = 0 if β = n + 1 and n is odd, 0 otherwise (provided β ≥ n + 1), Notice that the origin is always a focus of (2) when either β = n − 1 and n is odd (because v 1 = 1) or β = n and n is even (because v 2 = 0) or β = n + 1 and n is odd (because v 3 = 0). This rule will be generalized in Proposition 27(i). The most complicated case arises when β = n−1 and n even because, as pointed out in [6], we do not know how to compute the primitive to get the value of In [29] it is proved that when (2) is a polynomial family parameterized by the set of admissible coefficients then the Poincaré-Lyapunov quantities v k are polynomials in the parameters if and only if ω is a fixed constant, not a parameter.

The Poincaré map and inverse integrating factors.
An inverse integrating factor of (23) is a function v : C → R of class C 1 (C), hence a T n -periodic function in the variable θ, which is non-locally null and satisfies the partial differential equation: If V (x, y) is an inverse integrating factor of the Andreev canonical form (2) defined in a neighborhood of the nilpotent monodromic singularity at the origin then it is easy to prove that an inverse integrating factor v(r, θ) of the corresponding equation (23) is (30) v(r, θ) = V (r Cs θ, r n Sn θ) r n Θ(r, θ) .
In [26] it is also proved the following fundamental result: where the prime denotes derivative with respect to r 0 .

Proof of Proposition 1.
Proof. We give just one example of this kind of systems. In [5] it is proved that system with Andreev number n = 4 does not have a formal inverse integrating factor but the origin is a center since it is monodromic and time-reversible (invariant with respect to the involution (x, y, t) → (−x, y, −t)). We transform first the system to the Liénard form with a(x) = x 7 + 4x 11 andb(x) = 10x 5 by the change (x, y) → (x, y − x 6 ).

5.2.
Proof of Theorem 3. Before proving Theorem 3, we need a general result for nilpotent vector fields (1) independent of the monodromy.
Proposition 34. Let X be the analytic vector field associated to (1) where the origin can be monodromic or not. Then, there are formal series W (x, y) = y 2 + j≥3 W j (x, y) where the W j are homogeneous polynomials of degree j satisfying Moreover, if we set the monomials W j (0, y) then W is unique.
Proof. Let X = j≥1 X j with X 1 = y∂ x and X j = P j (x, y)∂ x + Q j (x, y)∂ y homogeneous polynomial fields of degree j. Consider a formal series W (x, y) = j≥2 W j (x, y) with W 2 (x, y) = y 2 and W j homogeneous polynomial of degree j. Associate to it the formal series F(x, y) = It follows that F 3 (x, y) = X 1 (W 3 (x, y)) + X 2 (W 2 (x, y)) = y(∂ x (W 3 (x, y)) + 2Q 2 (x, y)). Therefore, F 3 (x, y) ≡ 0 if and only if W 3 (x, y) = xK 2 (x, y) + τ 3 y 3 with τ 3 ∈ R an arbitrary coefficient and K 2 the homogeneous polynomial of degree 2 given by xK 2 (x, y) = −2 x 0 Q 2 (s, y)ds. Let us consider now F k (x, y) with k ≥ 4. We have by definition where G k (x, y) is a given homogeneous polynomial of degree k that only depends on the previous W (x, y) with = 2, . . . , k − 1. Then F k (x, y) = y(∂ x (W k (x, y)) + 2Q k−1 (x, y)) + G k (x, y). Since there is a homogeneous polynomial H k−1 (x, y) of degree k − 1 and a coefficient ξ 1 (s, y))ds. This shows that we can construct the series W of the theorem step by step. Observe that if we set the values of the freedoms τ k corresponding to the coefficients of the monomials W k (0, y) then W is unique. Theorem 3 shows some extra structure on the formal series W and the quantities ξ j of Proposition 34 in case that the nilpotent singularity be monodromic and the vector field X be in the Andreev canonical form (2).
Proof of Theorem 3. Indeed, we will see that V (x, y) = nW (x, y) where W is the series of Proposition 34. We introduce the quasihomogeneous structure by writing the analytic Andreev canonical form (2) as In other words, if we let X to denote the associated vector field to the Andreev canonical form, then X = i≥n−1 X i where X i = p i+1 (x, y)∂ x + q i+n (x, y)∂ y denotes a (1, n)-quasihomogeneous polynomial vector field of weighted degree i. Of course X n−1 = p n (x, y)∂ x + q 2n−1 (x, y)∂ y and p i+1 (x, 0) ≡ 0.
Consider the formal series V (x, y) = ny 2 + · · · = nW (x, y) and rewrite it as . In particular η j = 0 for any j < 3n − 1 and F 3n−1 = X n−1 (V 2n ) − V 2n div(X n−1 ) = η 3n−1 x 3n−1 . Inserting in this last equation the expression V 2n (x, y) = ax 2n +bx n y+ ny 2 with arbitrary coefficients a and b and equating the same coefficients of each monomial we get that η 3n−1 = 0, a = 1 and either b = ω or b = 0 depending on whether β = n − 1 or β ≥ n, respectively. Hence the V 2n (x, y) is as stated in the theorem and is an inverse integrating factor of X n−1 .
The fact that setting the monomials V j (0, y) yield uniqueness of the series V is a consequence of Proposition 34.
From the structure of (4) it is straightforward to see that the η j lie in either R(ω) [ν] or R[λ] depending on the relation between n and β.

Proof of Proposition 6.
Proof. Family (1) is locally analytically conjugated to the Andreev's canonical form (2). We emphasize that n is an invariant under conjugation but β is not in general. Fortunately, the explicit conjugation between (1) and (2) showed in §4.1 also keeps the value of β. The cyclicity is also invariant by conjugation so we restrict ourselves to analyze the cyclicity of the origin in the associated Andreev canonical form (2) with β = n − 1 and n odd which we write as (36) with ω ∈ Ω n and the dots denote higher order (1, n)-quasihomogeneous terms.
First we claim that the origin of (36) is a focus when n is odd. The proof is based on proving first that the origin of the leading (1, n)-quasihomogeneous vector field X n+1 = −y∂ x + (x 2n−1 + ωx n−1 y)∂ y is a focus and next we can use Theorem 5 of [1] to conclude that the origin of (36) is a focus too. To do it we assume that the origin is a center of X n+1 and we let γ be a T -periodic orbit inside the period annulus. It is clear that γ is not hyperbolic and therefore it must satisfy J γ = T 0 div(X n+1 ) • γ(t) dt = 0, see for example the book [41]. But this is a contradiction when n is odd since then divX n+1 = ωx n−1 and consequently J γ = 0. Therefore the claim is proved.
To check that the cyclicity of the focus at the origin of (36) with n odd is zero we check that the Poincaré map has the form Π(r 0 ; ω, ν) = v 1 (ω)r 0 + O(r 2 0 ) with v 1 (ω) = 1, see (28). From here the proposition follows.

5.4.
A break before the forthcoming proof of Theorem 8. We make a break in the discussion before the proof of Theorem 8 in the next subsection to see how a generalization of the Bendixson-Dulac criteria to multiple connected domains can be used in our context to give a much weaker version of Theorem 8.

Proposition 35.
Assume that there is a k ∈ N such that η j (λ † ) = 0 for all 3n ≤ j ≤ 2k − 1 and η 2k (λ † ) = 0. Then the Andreev canonical form (2) with λ = λ † has a focus at the origin and the number of small amplitude limit cycles near the origin is at most 1.
Proof. Again as in the proof of Theorem 8 the series defining V in Theorem 3 is truncated at a sufficiently large order but we omit it for simplicity. So V (x, y) = j≥2n V j (x, y) where the level sets of V 2n are ovals around the origin which implies that the zero set V −1 (0) has an isolated point at (x, y) = (0, 0) and therefore the function 1/V is well defined in a punctured neighborhood U * of the origin. Let X be the vector field associated to (2) with λ = λ † . Using (4) and the hypothesis of the proposition we obtain div with η 2k = 0. Therefore, taking U * sufficiently small, the function 1/V is of class C 1 (U * ) and div (X /V ) | U * is sign defined. Indeed, div (X /V ) | U * ≤ 0 or div (X /V ) | U * ≥ 0 according to whether η 2k > 0 or η 2k < 0, respectively. In short, 1/V is a Dulac function for X in the 1-connected region U * . Notice moreover that the set {(x, y) ∈ U * : div (X /V ) = 0} is a zero Lebesgue measure set that does not contain ovals. Then by the extension of the Bendixson-Dulac criteria to more general domains (allowing -connected domains and not only the simply connected ones), see for example [38] or [11], the maximum number of periodic orbits of X contained in U * is = 1 finishing the proof.
Indeed in Theorem 8 we will generalize Proposition 35 obtaining as byproduct that the focus of Proposition 35 cannot have any small amplitude limit cycles surrounding it. 5.5. Proof of Theorem 8. We omit in this section the parameters λ for the sake of simplicity except when we want to emphasize its dependence. Before proving Theorem 8 we need the following preliminary result. Using the change of variables Φ on R 2 \{(0, 0)} defined by (22) that transforms (x, y) → (r, θ) with Jacobian r n and next the time-rescaling ξ = 1/Θ(r, θ) it follows that the vector field X associated to the Andreev canonical form (2) and the vector fieldX = ∂ θ +F(r, θ)∂ r with F defined at (23) are orbitally equivalent on a sufficiently small punctured neighborhood of the origin. We state before proving Theorem 8 the following corollary of Lemma 2.3 in [2], see also Lemma 4.1 in [24].
Given a diffeomorphism Φ on U ⊂ R 2 with det(DΦ)| U = 0, we can apply the change of variables x → y defined by x = Φ(y) to a C 1 vector field X and a scalar function f , both defined on U . The outcome is the vector field Φ * (X ) and the function Φ * (f ) defined via the pull-back Φ * (X ) := (DΦ) −1 X • Φ and the function Φ * (f ) := f • Φ.
We consider the vector fieldX = ∂ θ + F(r, θ)∂ r of Corollary 36 and any (at the moment) real function v(r, θ) belonging to the set of T n -periodic functions of θ such that the scaled vector fieldX /v is of class C 1 (D ∪ ∂D). Taking the positive anti-clock-wise orientation in ∂D, we shall use Green's theorem in the domain D to the vector fieldX /v with associated differential 1-form Ω = (Fdθ − dr)/v: dr v(r, T n ) .
From (24) we know that where Θ n−1 is defined in (25) and satisfies Θ n−1 (θ) > 0 for any θ ∈ [0, T n ). Consider the series V (x, y) = j≥2n V j (x, y) of Theorem 3 with V 2n (x, y) = x 2n +ny 2 +ωx n y with ω ∈ Ω n if β = n − 1 and V 2n (x, y) = x 2n + ny 2 otherwise when β ≥ n. Since the series defining V in Theorem 3 is only formal we truncate it at a sufficiently large order but for simplicity of expression we will suppress any reference to this truncation in the following notation. Taking into account the fundamental relation Cs 2n θ + n Sn 2 θ = 1 we see that in any case V 2n (r Cs θ, r n Sn θ) = Θ n−1 (θ)r 2n . Now we define the function v(r, θ) via the relation (30) and we obtain Remark 37. Notice that if η j = 0 for any j ∈ N then v(r, θ) is an inverse integrating factor ofX , that is div X /v ≡ 0, hence the fundamental relation (31) can be applied and both limits in (40) give the same result. Thus (39) is a trivial equality.
From the Taylor expansion (26) we have Π(r 0 ) = r 0 + k≥2 v k r k 0 . Here it is very important to emphasize that in this step is crucial the hypothesis that either β ≥ n or ϕ ≡ 0 or β = n − 1 and n even which are equivalent to guarantee that v 1 = 1, see for example the formula (28). Using the series of Π together with (42) yields with g 0 = −v 2 and, for any k ≥ 1, Now we shall analyze the integrand div X /v in (39). We have where we have used Corollary 36 in the last equality. Inserting (42) and (41) into (45) leads to where f 0 (θ) = −η 3n Cs 3n θ/Θ 2 n−1 (θ) and (47) The left-hand side of (39) becomes where the expansion Ψ(θ; r 0 ) = j≥1 Ψ j (θ)r j 0 was applied in the last equality. The first coefficients h k are From these expressions and using (47) we get that h 0 = β 0 η 3n and where we have defined dθ for any j ∈ N and with Ψ 1 (θ; ω) = exp θ 0 F 1 (σ; ω)dσ .
Remark 38. We emphasize the following properties of these numbers β j : (a) β j < 0 when 3n + j is even; (b) β j = 0 when β ≥ n and 3n + j is odd.
Dividing both sides in (39) by ∆r 0 and taking the limit when ∆r 0 → 0 (using (40) and (43) for the right-hand side of (39) and (48) for the left-hand side of (39)) gives Now we are going to equate the coefficients in the above series and use (44) and (49). The result is that −v 2 = β 0 η 3n and, for any j ∈ N, Combining (52) and (I)-(II) together with Remark 33 we obtain the conclusions (ii) and (iii) of the theorem after defining the positive constants α j = −β j /(j + 1) when β j < 0. Notice that when β ≥ n the β j are constants independent of ω.

Proof of Theorem 12.
Proof. Let V be one of the formal series of Theorem 3, hence satisfying the relation (4) for some sequence {η j } j≥3n of integrability focus quantities. We define the We follow [2] where it is proved that since V satisfies X (V ) − V div(X ) = f for some power series f thenV satisfies the linear partial differential equation Y(V ) −V div(Y) = g where g is the power series g = (f • γ)/ det(γ) and Y = γ * X is the conjugated field. Taking into account that X is Z 2 -equivariant it follows that in fact Y = X and, consequently, eliminating the non-vanishing factor det(γ) we see that V γ satisfies where f γ = f • γ and f (x, y) = j≥3n η j x j , being η j the integrability focus quantities associated to V . By hypothesis on γ we obtain f γ (x, y) = f (−x, −y), hence is V 2n (x, y) = x 2n + ny 2 when β ≥ n or ϕ(x) ≡ 0. Therefore, the leading terms of V and V γ coincide.
Finally, using the linearity of the differential operator in the left-hand side of (4) we deduce that where V † = 1 2 (V + V γ ) has the same leading term V 2n than V . In consequence the associated ideal I odd = {0}, hence the Bautin ideal is B = I even from Corollary 10(iii).
To prove the former we will use the matrix representation A k of the linear operator L in the canonical basis of P (1,3) k and P (1,3) k+2 . The size of A k when k ≡ 0 mod 3 is ( + 2) × ( + 1) and has the following tridiagonal structure according to whether k ≡ 1 mod 3 or k ≡ 2 mod 3, respectively. Using the canonical basis of P Proof. Since B = I when I = I even one has V C (B) ⊂ V C (I k ) because I k ⊂ B so we only need to prove the reverse inclusion of complex varieties. Thus, we must check whether V C (I k ) ⊂ V C (B). To prove the last inclusion of complex varieties it is necessary to check that if λ * ∈ V C (I k ) then η j (λ * ) = 0 for any j ≥ 3n because the complexification of (2) on C 2 with parameters λ = λ * ∈ C p+1 has a complex formal inverse integrating factor V (x, y) = j≥2n V j (x, y) ∈ C[[x, y]] with V 2n ≡ 0. This proves statement (i).
The main result Theorem 1.3 of [3] adapted to the Andreev canonical family (2) with β = n − 1 implies that the associated complex system (2) is analytically integrable if, and only if, there exists a formal inverse integrating factor V like in part (i). Therefore part (ii) is proved.

Proof of Theorem 23.
Proof. We need to move to the complex setting because there the assumption √ I k by the Strong Hilbert Nullstellensatz. Then we obtain the following chain where the last equality follows by hypothesis. The reverse inclusion I k ⊂ B + I odd Using Theorem 8 and following the ideas of the Bautin's seminal work [9] (see also the books [34], [42] and [43] and for a modern point of view) the displacement map d(r 0 ; λ) = Π(r 0 ; λ) − r 0 = j≥2 v j (λ)r j 0 of X | V can be rearranged in some neighborhoods of r 0 = 0 and λ = λ * ∈ V R (B) ∩ E ∩ V into the form where the ψ j are analytic functions such that ψ j (0; λ * ) > 0 (these constants are related to the positive constants of Theorem 8) and with exponents { j } m j=1 defining an strictly increasing sequence of positives integer. From the expression (55) we can derive the cyclicity bound Cyc(X | V , 0) ≤ m − 1, see for example the arguments exposed in Lemma 6.1.6 and Theorem 6.1.7 of [42]. Proof. The system (12) is not in the Andreev canonical form (2). The function F in Theorem 32 is F (x) = (−1 + √ 1 − 20ax 3 )/(10x) = −ax 2 + O(x 3 ). As indicated in §4.1, we perform the change of variables (x, y) → (x, y − F (x)) and next the rescaling (x, y) → √ 2(x, y) to obtain the system in the form (2) with n = 2, β = 1, and ω = −a √ 2. Observe that the restriction ω ∈ Ω 2 = {ω ∈ R * : ω 2 < 8} coincides with the above monodromic condition 4a 2 − 16 < 0 as it must be.

Proof of Proposition 27.
Proof. Under the conditions β = n + j (so ϕ(x; ω, ν) ≡ 0) and n ≥ 2 + j of part (i) the computations show that the first non-identically zero integrability focus quantity is η 3n+j = −ω = 0 up to positive constant factors. Then the origin is a focus if β is even using Proposition 35 because the subindex 3n + j is even too. Moreover, from Remark 9, the Poincaré map is Π(r 0 ; ω, ν) = r 0 +v 2+j (ω)r 2+j 0 +O(r 3+j 0 ) where v 2+j = 0, hence no limit cycle can bifurcate from the origin in the whole family statement (i) is proved.

Proof of Proposition 28.
Proof. As in the proof of Proposition 27(ii), we have η 3n = η 3n+1 = · · · = η 4n−2 = 0 and from the freedom τ 3 we can take η 4n−1 = 0. The first center condition appears has the inverse integrating factor exp(−(ν 1 + 2ν 5 )x) which does not vanish at the origin. In any case we observe that all the centers are analytically integrable.
The previous work tell us that V R (I even ) = V R (I even 38 ) and that I even 38 = λ 2 , λ 4 , λ 5 , λ 7 . Therefore V C (I even 38 ) = V C (λ 2 , λ 4 , λ 5 , λ 7 ). Considering the map F : C 7 → C 4 : λ → (λ 2 , λ 4 , λ 5 , λ 7 ), we obtain rank(d P F ) = 4 at every point P ∈ V C (I even 38 ). Therefore, the set Σ of Proposition 19 is empty and the Zariski closure condition in Proposition 19 holds trivially so that V C (I even ) = V C (I even 38 ). A primary decomposition is I even the origin of R 7 . Taking into account that the cardinality of a minimal basis of the ideal I even 38 is 7 now we are in position to apply Theorem 17(ii) to conclude that the center at the origin of (18) with parameters λ ∈ V C \ {0} has cyclicity bounded by 6 and this gives the upper cyclicity bound of (ii).
Therefore we see that the irreducible components of I even 20 are C j = V R (J j ). We check that C 1 and C 2 correspond to the reversible (with respect to the symmetry (x, y, t) → (−x, y, −t)) and Hamiltonian component, respectively, of the center variety. Regarding family (20) with parameters on C 3 , in [4] the non-formal inverse integrating factor (1 + 2λ 8 x 2 ) 3/2 was obtained. In summary this proves V C = C 1 ∪ C 2 ∪ C 3 = V R (I even 20 ). From the above equality of varieties and the cardinality 7 of the basis B the upper bound 6 on the cyclicity of any focus at the origin is given. To reach this bound we will proceed as explained in Remark 25. Take a point λ † ∈ R 9 \V C corresponding to a focus of maximum order, that is, such thatη j (λ † ) = 0 for j = 8, 10, 12, 14, 16, 18, butη 20 (λ † ) = 0. One of these points is just λ † = (3, 0, 0, 0, 0, 0, 0, 1, 1). Now we take the parameterized curve ε → λ(ε) ∈ R 9 going through λ(0) = λ † define by so that the realization condition (11) holds with all the elements of the basis B, hence with s = 7. That means that the focus with parameters λ † has cyclicity 6 proving statement (iv).