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CYCLICITY OF NILPOTENT CENTERS WITH MINIMUM ANDREEV NUMBER

We consider polynomial families of real planar vector fields for which the origin is a monodromic nilpotent singularity having minimum Andree’s number. There the centers are characterized by the existence of a formal inverse integrating factor. For such families we give, under some assumptions, global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family.


Introduction
A singular point of an analytic planar vector field is nilpotent when the linearization around it does not vanish but the associated eigenvalues do. By an affine change of coordinates and a time rescaling an analytic planar vector field with a nilpotent singularity can be placed in the form (1)ẋ = y + P (x, y),ẏ = Q(x, y), where P and Q are analytic functions near the origin without constant or linear terms. In this work we focus on the monodromic case for which the local flow of (1) near the origin rotate about it. It is well known that, in this case, the local qualitative phase portrait corresponds to either a center or a focus. You can see two independent proofs of this fact in [14] and [24]. Andreev in [6] characterizes analytic systems (1) for which the origin is monodromic. This characterization depends on two positive integer numbers (n,β) ∈ N 2 with n ≥ 2 andβ ≥ n − 1, see the forthcoming Theorem 22. We call n the Andreev number associated to an analytic monodromic nilpotent singularity which is invariant under analytic orbital equivalency, see [16].
Following Andreev's ideas [6], see §3.2, any system (1) with a monodromic singularity at the origin with (n,β) = (2, 1) is analytically conjugate to the Andreev's canonical form (3)ẋ = −y + yP (x, y; ν),ẏ = x 3 + ωxy +Q(x, y; ν), whereP andQ denote terms with (1, 2)-quasihomogeneous degrees greater than 0 and 3, respectively, and with parameter space E = {(ω, ν) ∈ Ω × R p } and nilpotent center at the origin if and only if each element in both sequences vanishes at (ω * , ν * ). We want to stress here that the integrability focus quantities are much easier to work with than the Poincaré-Lyapunov quantities. The reason for this to happen is because theη j are computed via an algebraic algorithmic easy to use with a computer algebra system while the calculation of the v j needs quadratures of Lyapunov generalized trigonometric functions, see for example the computations in [4] and some of the results in [21].
Nonetheless we will relate both sequences {η j (ω, ν)} j∈N and {v j (ω, ν)} j∈N by means of computational algebra techniques in such a way that we can in some cases bound the cyclicity of centers at the origin of polynomial families (3). Recall that the cyclicity of the nilpotent monodromic singularity at the origin of system (3) with (ω, ν) = (ω † , ν † ) ∈ E is the maximum number of limit cycles that can bifurcate from it for parameters (ω, ν) ∈ E such that (ω, ν) − (ω † , ν † ) 1. Although there are in the literature studies on the cyclicity of nilpotent focus, the cyclicity of centers is harder to analyze. The main reason is that Hopf bifurcation from foci is a finite-codimension bifurcation in contrast with the center bifurcation. In some papers like in [33] the Hopf bifurcation at nilpotent center is analyzed under the restriction that the family is close to a Hamiltonian center by introducing a small perturbation parameter. Notice that this restriction is never satisfied by our systems (2) or (3). As far as I know this is the first work where a general method for computing global upper bound on the nilpotent center cyclicity is obtained for the monodromic class (n,β) = (2, 1). By the way we will observe the troublesome role that the parameter ω plays, particularly when we need to complexify the parameter space in order to analyze the cyclicity problem. This kind of problems are very similar to those encountered in the paper [20] related to the cyclicity of nondegenerate centers on center manifolds for 3-dimensional systems.

Main results
Our first result is in fact the definition of integrability focus quantitiesη j (ω, ν) associated to Andreev's canonical family (3). Theorem 1. LetX be the vector field associated to the Andreev's canonical form (3). Then, there is a unique formal series W (x, y) = k≥4 W k (x, y) where W k are (1, 2)-quasihomogeneous polynomials of weighted degree k with leading term W 4 (x, y) = x 4 + 2y 2 + ωx 2 y and such that Moreover, for any polynomial family (3) parameterized by its coefficients, we havê η j (ω, ν) ∈ R(ω)[ν] and they are well defined for ω ∈ Ω.
The next result shows the relationship between the v j (ω, ν) and theη j (ω, ν) quantities of system (3). Statement (i) was already proved in a more general context in [19] and we write it only for the sake of completeness.
We will use from now the following standard algebraic notation. 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). In this work we will only use either K = R or K = C.
The next result is a corollary of Theorem 2.
2.1. The Bautin ideal and the cyclicity of the nilpotent singularity. We define the displacement map δ associated to family (3) as (6) δ(r 0 ; ω, ν) = Π(r 0 ; ω, ν) − r 0 , where Π is the Poincaré return map associated to the origin of (3) which is computed using the parameterization of a transversal section to the flow as it is detailed in §3.3. So the fixed points of Π become zeroes of δ and correspond with the small amplitude periodic orbits of (3). Now we are going to perform a Bautin-type analysis [7] (see [31] for a modern exposition) to get upper bounds on the number of positive roots of δ(.; ω, ν) close to the origin. Assume first that system (3) with (ω, ν) = (ω † , ν † ) ∈ Ω × R p possesses a focus at the origin. Then there is a positive integer k called the order of the focus such that v j (ω † , ν † ) = 0 for j 2k − 1 but v 2k (ω † , ν † ) = 0. In this situation, for (ω, ν) sufficiently close to (ω † , ν † ) we have v 2k (ω, ν) = 0 by continuity and consequently we can write Now using standard arguments (see for example Proposition 6.1.2 of [31]) we known that an upper bound on the cyclicity of the focus at the origin with respect to perturbation within the family (3) is given by 2(k − 1). Actually, this bound can be improved sometimes using a special basis of the idealÎ k which we simply call "minimal basis" but is, in fact, a minimal basis with respect to the ordered integrability focus quantities.
Definition 5. Consider the idealÎ k in the Noetherian ring R(ω)[ν]. For some s ≤ k we say that the basis B = {η j1 , . . . ,η js } ofÎ k is minimal if it satisfies the following properties: Of course similar definitions apply to obtain minimal basis of B 2k , B orÎ.
Let {η j1 , . . . ,η js } be a minimal basis of the idealÎ k having s elements and where j s = k. Then s ≤ k and near (ω † , ν † ) we can rearrange δ as producing the upper bound s − 1 on the cyclicity of the focus. On the contrary, now we assume that system (3) with (ω, ν) = (ω * , ν * ) ∈ Ω × R p has a center at the origin, that is v j (ω * , ν * ) = 0 for all j ∈ N. Let m ∈ N denote the cardinality of the minimal basis {v j1 , . . . , v jm } of the Bautin ideal B. This m is called the the Bautin depth of B. Then, for (ω, ν) sufficiently close to (ω * , ν * ), the displacement map can be expressed as where ψ k (r 0 ; ω, ν) are analytic functions at r 0 = 0. The reader can see a proof of this rearrangement in Lemma 6.1.6 of [31]. From (7) and by a repeated use Rolle's Theorem the following result follows.
Theorem 6. Assume the origin is a center of system . Then the cyclicity of the center with respect to perturbations within the family (3) is at most m − 1.
Although valuable from a theoretical point of view, in general it is very difficult to compute the Bautin depth of B in practical cases, so we need to develop other tools besides Theorem 6 to bound the center cyclicity of (3). This is the goal of the next section.
2.2. The ideas of [20] adapted to the nilpotent singularity. Following exactly the same ideas than in [20] we can obtain a center cyclicity bound computed with polynomial ideals. The proofs are essentially the same (therefore we do not reproduce them here) if we replace the concepts Hopf singularity in R 3 , formal first integral, focal values and admissible parameter λ ∈ R * by nilpotent singularity with Andreev number (n,β) = (2, 1), formal inverse integrating factor, integrating focal values and admissible parameter ω ∈ Ω, respectively.
The precise statement requires first some definitions. Since for a polynomial family (3) both v j andη j are in R(ω)[ν], they can be expressed like and where the roots of D j are not in Ω. Similarly, Definition 7. We define the polynomial ideals G = g j : j ∈ N , G k = g 1 , g 2 , . . . , g k , It was proved in [20] that if {V j1 , . . . , V jm } is the minimal basis of the polynomial ideal V with respect to the ordered basis {V j : j 2} then {v j1 , . . . , v jm } is a basis (not necessarily minimal) of the Bautin ideal B that satisfies the retention condition with respect to the ordered basis {v j : j 2}. Therefore the Bautin depth of the Bautin ideal B is at most m. Combining this fact together with Theorem 6 produces the following bound for the center cyclicity in terms of polynomial ideals.
Theorem 8. Let m be the cardinality of the minimal basis of the polynomial ideal V = V j : j ∈ N (equivalently, G = g j : j ∈ N ) associated to a polynomial family (3). Then for any system (3) corresponding to parameter values (ω, ν) ∈ V R (V) with ω ∈ Ω, the cyclicity of the center at the origin, with respect to perturbation within the family (3), is at most m − 1.
Theorem 8 is theoretically interesting but hard to use in concrete applications.
The key point is how to compute the cardinality of the minimal basis of V or G. As we will see later, to do that it is necessary to move over the complexes in order to use the completeness of C and Hilbert Nullstellensatz. Hence, we allow (ω, ν) ∈ C × C p and we regard G = g j : j ∈ N as an ideal in the complex polynomial ring C[ω, ν], see the forthcoming Theorem 14. However, after this complexification some complications arise such as we lose the geometrical ideas (for example symmetries) for identifying existence of a formal inverse integrating factor. For our purposes the most important step consists on checking that the equality of real varieties V R (G) = V R (G k ) for some k ∈ N implies the equality of complex varieties V C (G) = V C (G k ). The analysis of the former implication gets complicated mainly because when ω ∈ C\Ω then additional parameter restrictions may appear in order to obtain a formal inverse integrating factor. The outcome is that the variety V C (G) need not exactly pick out systems with a formal inverse integrating factor. Actually, when ω ∈ C\Ω it may happen that complex families with a formal inverse integrating factor are proper subsets of V C (G). In summary, we do not have a characterization in terms of formal inverse integrating factors of the variety V C (G) because there are points (ω, ν) on it with ω ∈ C\Ω for which the associated complex system (3) has no formal inverse integrating factor. The next example just illustrates what we have said.  y]] is an inverse integrating factor of (10) in C 2 .
Proof. When (ω, ν) ∈ E we find the first integrating focus quantity η 1 = −2Aω/15. Therefore A = 0 is a necessary center condition which turns out to be sufficient because (10) is time reversible with respect to the symmetry (x, y, t) → (−x, y, −t). This proves (i) since centers are characterized by existence of a formal inverse integrating factor in family (10). Statement (ii) is just straightforward. Now we present a technical proposition useful in applications of the forthcoming Theorem 14 when the allowed perturbations in family (3) keep the value of ω constant. Before, we need the following definition.
The next proposition is the adaptation of Proposition 29 in [20] to our setting.
in C p when ω is assigned to any fixed value ω * ∈ Ω.
Remark 12. Although implicit in the proofs of [20] we want here to point out that Proposition 11 remains true if we substitute the condition where E is the parameter space of (3). This refinement, although unimportant in the examples presented in [20] is crucial in some examples of this work.
Remark 13. In applications often the set Σ in condition (c) of Proposition 11 satisfies Σ ⊂ ∪ j = (C j ∩ C ) where the sets C s are the irreducible components of V C (G k ). Since by definition the Zariski closure of a set is the smallest variety that contains that set, in such a case the condition The following theorem is the version of Theorem 26 in [20] adapted to the nilpotent monodromic singularity class (n,β) = (2, 1) which is actually based on some ideas of the papers [25] and [15]. Theorem 14. Let (3) be a complex polynomial family on C 2 with parameters (ω, ν) ∈ C × C p . Assume the equality V C (G) = V C (G k ) holds and that the minimal basis of G k has cardinality m.
(i) If G k is radical ( √ G k = G k ) then G = G k and for any (ω * , ν * ) ∈ V R (G) with ω * ∈ Ω the cyclicity of the center at the origin perturbing within the real family (3) on R 2 is at most m − 1.
(ii) Let G k = R ∩ N be a primary decomposition of G k where R is the intersection of the ideals in the decomposition that are prime (hence is radical) and N is the intersection of the remaining ideals. Then for any system of the real family (3) on R 2 corresponding to parameters (ω * , ν * ) ∈ V R (G) \ V R (N ) with ω * ∈ Ω the cyclicity of the center at the origin is at most m − 1.
Theorem 15. Let C ⊂ V R (G) be an irreducible component of the center variety associated to the origin of the polynomial family (3). Let P = (ω * , ν * ) ∈ C ∩ E with parameter space E = {(ω, ν) ∈ Ω × R p } be a point such that rank(d P G κ ) = κ and κ p + 1. Then the following holds: (i) There exists a neighborhood U of P in R p+1 such that C ∩ U is a submanifold of R p+1 of codimension at least κ and there exist bifurcations of (3) producing κ − 1 small amplitude limit cycles from the origin for parameter values with (ω, ν) 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.

2.3.
Examples. In this section we apply the above developed theory to some concrete families (3).
In these examples an intensive use of several routines in the primdec.lib library [13] of Singular [12] is necessary. Thus, given a polynomial ideal I, we can use the routine minAssChar to obtain the prime decomposition of √ I. We also can check whether a polynomial ideal I is radical or not thanks to the primdecGTZ or primdecSY routines.
Once we know the cyclicity bound of a focus due to its order or the center cyclicity thanks to the Bautin depth, the next natural step is trying to prove that the bound is indeed sharp. Then we need to make a concrete perturbation within the family that creates the maximum number of limit cycles. If it exists, the details on the choice of that perturbations are explained below.
Computing the functions F , f and ϕ of Theorem 22 for this system gives F (x) ≡ 0, f (x) = −x 3 and ϕ(x) = kx. Therefore the origin is a monodromic nilpotent singularity if and only if 8 − k 2 > 0. Moreover, is of type (n,β) = (2, 1) if and only if, additionally, k = 0. The center problem at the origin of the full family (12) was solved in [1]. Restricting their result to k = 0, it follows that the only center corresponds to the time-reversible case a = c = 0. In the following theorem we analyze the cyclicity problem at the origin.
Theorem 17. The center variety V C associated to the origin of family (12) with The origin is a center of (12) if and only if (ω, ν) ∈ V C ∩E = V R (G 2 )∩E. Moreover the following holds: (i) The cyclicity of any center at the origin under perturbations within family (12) is 1; (ii) The cyclicity of any focus of (12) at the origin is bounded by 1 and there are foci with cyclicity 1.

Example 2.
Consider the polynomial Andreev's canonical family (3) withP andQ general (1, 2)-quasihomogeneous polynomials of minimal degrees, that is, degrees 1 and 4 respectively. This family is given by Theorem 18. The center variety V C associated to the origin of family (14) has 7 irreducible components: The origin is a center of (14) with parameter space Moreover the following holds: (i) The cyclicity of any center at the origin of (14) having parameter values (ω, ν) ∈ C j ∩ E with j = 3 under perturbations within family (14) is exactly 2 except when ν = (0, 0, 0, 0); (ii) The cyclicity of any center at the origin of (14) having parameter values (ω, ν) ∈ C 3 ∩E perturbed within family (14) is exactly 1 under the restriction (A + 3D)(2A + 3D) = 0; (iii) The cyclicity of the center at the origin of (14) when ν = (0, 0, 0, 0) perturbed within family (14) is zero provided α(ω, ν) ≡ 0 where the expression of α is given in (57). (iv) The cyclicity of any focus of (14) at the origin is bounded by 2 and there are foci with cyclicity 2.
2.3.3. Example 3. In Theorem 4.1 of [5] the center problem of the following polynomial Andreev's canonical family (3) was analyzed: (15)ẋ = −y,ẏ = x 3 + ωxy + Ax 2 y + By 2 + Cxy 2 + Dy 3 , with parameters ω ∈ Ω and ν = (A, B, C, D) ∈ R 4 . There it is proved that the origin is a center if and only if A = B = D = 0, which corresponds to the timereversible stratum with symmetry (x, y, t) → (−x, y, −t) and, moreover, that there exist systems inside family (15) with 3 small amplitude limit cycles bifurcating from a nilpotent focus at the origin, see Remark 33. We expand this study analyzing the cyclicity of the origin in (15).
Theorem 19. The center variety V C associated to the origin of family (15) with The origin is a center of (15) if and only if (ω, ν) Moreover the following holds: (i) If C = 0 then the cyclicity of any center at the origin under perturbations within family (15) is exactly 2; (ii) If C = 0 then there are perturbations within family (15) producing 1 limit cycle bifurcating from the center at the origin. (iii) The cyclicity of any focus of (15) at the origin is bounded by 3 and there are foci with cyclicity 3.

Example 4.
We will analyze Liénard polynomial families of arbitrary degree d ≥ 4 with linear damping inside the Andreev's canonical form (3). This family has the form with polynomials ν j x j and parameters ω ∈ Ω and ν = (ν 4 , . . . , Theorem 20. The center variety V C associated to the origin of family (17) with Moreover the following holds: (i) The cyclicity of any center at the origin under perturbations within family (17) is exactly m − 1; (ii) The cyclicity of any focus of (17) at the origin is bounded by m − 1 and there are foci with cyclicity m − 1.
2.3.5. Example 5. We study family (3) withQ(x, y; ν) ≡ 0 andP (x, y; ν) an arbitrary polynomial of degree 2, that is, Theorem 21. The center variety V C associated to the origin of family (20) with Moreover, the cyclicity of any symmetric center at the origin with ω 2 = 7 under perturbations within family (20) is exactly 1.
If this conjecture is true then the center problem for the origin of family (20) is completely solved being the center variety

Andreev's characterization of monodromic nilpotent singularities.
The following theorem of Andreev [6] characterizes analytic systems (1) for which the origin is monodromic.
Theorem 22. For an analytic system of the form (1) with an isolated singularity at the origin let y = F (x) be the unique solution of y + P (x, y) = 0 such that F (0) = F (0) = 0 and let and ϕ(x) = (∂P/∂x + ∂Q/∂y)(x, F (x)).
Let a = 0 and α ≥ 2 be such that f (x) = ax α + · · · . When ϕ is not identically zero let b = 0 andβ ≥ 1 be such that ϕ(x) = b xβ + · · · . Then the origin of (1) is monodromic if and only if α = 2n − 1 is an odd integer, a < 0, and one of the following conditions holds: In this work we will focus on the former monodromic case (iii) with Andreev's number n = 2. In the study of the monodromic nilpotent singularities it is useful and natural to introduce (1, n)-quasihomogeneous polynomials with n the associated Andreev's number.
In consequence we get p k (x, y) = i+nj=k a ij x i y j for certain coefficients a ij ∈ R. On the other hand, a vector field X j = p j+1 ∂ x + q j+n ∂ y is a (1, n)-quasihomogeneous polynomial vector field of weighted degree j if its components p j+1 and q j+n are (1, n)quasihomogeneous polynomials of weighted degrees j + 1 and j + n, respectively.
3.2. The Andreev's canonical form. We consider system (1) and we assume that the origin is a nilpotent monodromic singular point with (n,β) = (2, 1), that is, we consider case (iii) in Theorem 22. Then, considering the function F and the value a ∈ R of Theorem 22, we perform first the analytic change of variables (x, y) → (x, y − F (x)) and next the rescaling (x, y) → (α x, −α y) with α = (−1/a) −1/2 bringing system (1) to the Andreev canonical form (3).
Let Ψ(θ; r 0 , ω, ν) be the solution of (23) with initial condition Ψ(0; r 0 , ω, ν) = r 0 . Then we can define the Poincaré map Π : Σ ⊆ R → R as Π(r 0 ; ω, ν) = Ψ(T ; r 0 , ω, ν). Thus Π becomes an analytic diffeomorphism and, from the results of [18] and [19], it follows that Π admits the Taylor expansion where v j are the Poincaré-Lyapunov quantities. In order to exemplify the difficulties we encounter when we try to compute the Poincaré-Lyapunov quantities, we see that [4] for details. In [21] it is proved that when (3) 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. More precisely, we have that v k ∈ R(ω)[ν].
In [18] it is also proved the following fundamental result: where the prime denotes derivative with respect to r 0 .

4.1.
Proof of Theorem 1. Before giving the proof of Theorem 1 we need Lemma 25. This lemma appears in [1] but we reprove it again with our notations for the sake of completeness. 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.
Lemma 25. Let X be a C 1 -vector field on an open set U ⊂ R 2 . Assume that a C 1 function V : U → R satisfies the linear partial differential equation for some scalar function f defined on U. Let Φ and ξ be a diffeomorphism and a scalar function on U such that the restrictions det(DΦ)| U = 0 and ξ| U = 0. We define the pull-back vector field Y = Φ * (ξX ), that is, Y and X are orbitally equivalent. Then, the function W = Φ * (ξ) Φ * (V )/ det(DΦ) satisfies the linear partial differential equation Proof. If we defineṼ = Φ * (V ) := V • Φ andỸ = Φ * (X ), first we claim that This equality follows by the the chain rule and the push-forward definition. In short, defining F (x) andF (y) as the components of X andỸ respectively, and using the dot to denote an ordinary scalar product, one has proving the claim. Now we defineJ(y) = det(DΦ(y)), so that W (y) =ξ(y) J −1 (y)Ṽ (y) wherẽ ξ = ξ • Φ and also J(x) =J(Φ −1 (x)). Then, calling G(x) the components of Y we have Now we analyze each one of these terms changing from variables y to x, using (29) and recalling the definition of Y. We obtaiñ In the last term we will apply the well-known relation (30) div(F ) = div(F ) +J −1 ∇ yJ .F see a proof in the book [30]. Thereforẽ Finally we obtain that proving the result.
Proof of Theorem 1. We make the proof with the help of the results proved in [26]. As noted in [26], after a nonlinear change of variables (x, y) → (αx, α(y+Kx 2 )) with convenient α, K ∈ R, any vector field X with a monodromic nilpotent singularity at the origin with minimum Andreev number (n,β) = (2, 1) is taken towards with parameter space {(λ, µ) ∈ R * × R p }, whereP andQ only contain (1, 2)quasihomogeneous terms of degrees greater than 2 and 3, respectively. Moreover, in Theorem 2.8 of [27] it is proved that if X is the vector field associated to (31) with λ = 0 then there is a unique formal series V (x, y) = x 4 + y 2 + j≥5 V j (x, y) where V j are (1, 2)-quasihomogeneous polynomials of weighted degree j such that Our proof is a consequence of Lemma 25 applied to the particular case in which X is the field associated to (31), the linear partial differential equation under consideration is (32), and the conjugate fieldX is in the Andreev's canonical form (3). In that case, the rescaling ξ ≡ 1. Moreover, if we compute the functions that appears in Theorem 22 in the particular case of a polynomial family (31) we obtain that the analytic function F depends polynomially on the parameters (λ, µ). More concretely the function F and f of Theorem 22 have the Taylor expansions In particular, according with §3.2, the conjugation (x, y) → Φ(x, y) with and α(λ) = 2(1 + λ 2 ) brings X toX , that is, transforms the polynomial family (31) into the analytic family of Andreev canonical forms (3) where ω(λ) = 4λ/α(λ). We emphasize that the parameters (ω, ν) of (3) depend on the parameters (λ, µ) of (31) polynomially in µ but not in λ.

Remark 27.
As a particular case of the results of [32] it is known that (3) is orbitally analytically equivalent to a Liénard system. Let us take a polynomial such family (34)ẋ = −y,ẏ = x 3 + yb(x; ω, ν), with b(x) = ωx + m j=2 ν j x j and ω ∈ Ω. Let X be the vector field associated to (34). Then, using the perspective of Theorem 1 and its proof we claim that there is a unique formal power series V (x, y) = k≥4 V k (x, y) where V k are certain (1, 2)quasihomogeneous polynomials of weighted degree k and V 4 (x, y) = x 4 +2y 2 +ωx 2 y such that Moreover, η 2k (ω * , ν * ) = 0 for all k ∈ N if and only b(x; ω * , ν * ) is an odd function, hence if and only if the origin is a center. A different proof of this claim consists on direct considerations using algebraic methods as we explain below. Let P (1,2) k ⊂ R[x, y] be the subspace of (1, 2)-quasihomogeneous polynomials of weighted degree k, that is p k (x, y) ∈ P and q i+2 ∈ P (1,2) i+2 . Let X = −y∂ x + (x 3 + yb(x; ω, ν)∂ y be the associated vector field to system (34). Then X = i≥1 X i where X i denotes a (1, 2)-quasihomogeneous polynomial vector field of weighted degree i. More specifically we have X 1 = −y∂ x + (x 3 + ωxy)∂ y and X i = ν i x i y∂ y for any integer i ≥ 2.
Let us consider the left hand side X (V ) − V div(X ) of (35). Taking into account that div(X ) = b(x; ω, ν) and the (1, 2)-quasihomogeneous expansions X = i≥1 X i Recalling that L(V 4 ) ≡ 0, the terms in P  y). Therefore (36) is written as Now we will prove the following claims.
(I) When k is even, an arbitrary element V k ∈ P (1,2) k has dim P (1,2) k arbitrary coefficients which can be selected (solving linear algebraic equation) in such a way that the same number dim P (1,2) k of coefficients of L(V k ) can be settled to have arbitrary values. In that case we will determine the unique coefficients of V k such that L(V k ) = F k+1 . (II) On the contrary, when k is odd, the dim P (1,2) k coefficients of an arbitrary polynomial V k ∈ P (1,2) k can be chosen (solving again linear equation) in order that the dim P (1,2) k + 1 coefficients of L(V k ) can be settled to belong to an arbitrary set of real numbers except one of them. This exceptional coefficient is just the coefficient of x k+1 . Thus, for k odd, we can take with uniqueness the coefficients of V k in such a way that L(V k ) − F k+1 = η k−3 x k+1 for certain integrating focal quantity η k−3 .
To prove the claims we will use induction over k. Denote by A k the matrix representation of the linear operator L : P (1,2) k → P (1,2) k+1 in the canonical basis. When k is even A k is a square matrix of order k/2 + 1 and when k is odd the size of A k is (k − 1)/2 + 2 rows and (k − 1)/2 + 1 columns. It is easy to check by mathematical induction that A k is a tridiagonal matrix, that is a band matrix that has nonzero elements only on the main diagonal (those entries located at equal row and column), the first diagonal below this, and the first diagonal above the main diagonal. More specifically we have, for k odd, , and, when k is even, First, we prove claim (II). To this end, we define the square matrix A * k as the submatrix obtained after deleting the first row of matrix A k with k odd. It follows that A * k is upper triangular and det A * k = k!! = k(k − 2)(k − 4) · · · 1 = 0, hence A * k is nonsingular. This property of A * k implies, in particular, claim (II). The reason is that the involved linear system to determine the coefficients c ij of V k (x, y) = i+2j=k c ij x i y j such that L(V k ) = F k+1 has associated matrix A k . Therefore that linear system is overdetermined and, in general, has no solution. But, the linear system that satisfy the coefficients of V k and the η k−3 such that L(V k ) = F k+1 + η k−3 x k+1 can be uniquely solved. This is because the subsystem obtained after deleting the first equation of that linear system has associated matrix A * k , hence it can be solved for all the coefficients of V k and, finally, we determine η k−3 from the first equation.
To prove claim (I) first we recall a result on determinants of tridiagonal matrices. Let B = (b i,j ) be a square tridiagonal matrix of order n and denote ∆ m (with m = 1, . . . , n) the principal minors, i.e., the determinants of the submatrices formed by the first m rows and columns of B. From linear algebra, it is known that the determinant det B = ∆ n can be computed from a three-term linear recurrence relation as follows: In our case, applying this recurrence to the square tridiagonal matrix A k with k even, we get for 3 ≤ m ≤ k/2 + 1 with initial conditions ∆ 1 = −ω and ∆ 2 = k. Clearly det A k = ∆ k/2+1 and claim (II) follows after proving that det A k = 0 for any k ≥ 6 even under the constraint ω ∈ Ω.

4.2.
Proof of Theorem 2. Before proving Theorem 2 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 2 and next the time-rescaling ξ = 1/Θ(r, θ) it follows that the vector field X associated to the Andreev canonical form (3) 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. Then, using Lemma 25 we can prove the following result.
Corollary 28. LetX = ∂ θ +F(r, θ)∂ r be the vector field defined by (23) and v(r, θ) given by (27) where W is defined in Theorem 1. Then Proof of Theorem 2. Let r 0 > 0 and ∆r 0 > 0 both sufficiently small and Ψ(θ; r 0 ) the solution of equation (23) with initial condition Ψ(0; r 0 ) = r 0 . We take a simply connected domain D in the coordinate plane (θ, r) whose boundary ∂D is the union of the curves Γ Regarding the right-hand side of (41) we note that .

Proof of Theorem 17.
Proof. Reversing the time t → −t and denoting ω = −k family (12) is written in the Andreev canonical form (3) with parameters ω and ν = (a, b, c) ∈ R 3 . We compute for this polynomial family the first integrability focus quantitiesη j ∈ R(ω)[ν] up to positive multiplicative constants. The outcome is the following one: Notice that the functionsη j are well defined for any ω ∈ Ω defined in (4) as it must be. The following definition will be used from now.
Definition 30. We defineη j =η j modÎ j−1 in the ring R(ω) [ν]. In other words, we letη j denote the remainder ofη j upon division by a Gröbner basis of the ideal I j−1 .
Since V C ∩ E = V R (G 2 ) ∩ E we know that 2 is an upper bound of the order of any foci, hence the cyclicity of any focus of (12) at the origin is bounded by 1.
Using the notation of Definition 7 we check that g 4 ∈ √ G 3 and √ G 4 = G 4 . To solve the center problem we use the routine minAssChar in the PRIMDEC.lib library of Singular ( [12]) to obtain the prime decomposition Notice that C j = V R (J j ). Therefore, we get Notice that any member of family (14) belongs to the trivial center strata defined by the Hamiltonian fields or the time-reversible with respect some coordinate axis, hence to know whether (ω, ν) ∈ C j ∩ E for j ∈ {1, . . . , 7} implies (ω, ν) ∈ V C ∩ E is, in general, a difficult problem. To solve it we will use the ideas explained in the following remark. (14) is a particular case of system (53)ẋ = yP 3 (x),ẏ = P 0 (x) + P 1 (x)y + P 2 (x)y 2 with P 3 (0) = 0. Performing the (analytic and tangent to the identity) Cherkas change of variables (x, y) → (x, y/ξ(x)) with

Remark 31. System
and rescaling the time (dividing the vector field by P 3 (x)ξ(x)), system (53) is transformed into the Liénard system .
When (53) comes from (14) we have P 3 (x) = −1 + Ax, P 0 (x) = x 3 + Dx 4 , P 1 (x) = ωx + Ex 2 and P 2 (x) = F . This choice gives ξ(x) = (1 − Ax) F/A if A = 0 and ξ(x) = exp(−F x) when A = 0 (which is just the limit when A → 0 of ξ(x)) and consequently for any value of A we get In Theorem 2.6 of [22] the authors make a generalization of a result of Cherkas [9] that characterizes the nondegenerate centers of Liénard equations to some class of degenerate singularities including our nilpotent class where f and g are analytic functions with starting terms like in (55). In our context their result is stated as follows: The origin of the analytic Liénard system with f and g as in (55) has a (nilpotent) center at the origin if and only if, for any x small enough, the system has a unique solution z(x) satisfying z(0) = 0 and z (0) < 0 where we have defined Now we continue the proof using the technique in Remark 31 to family (14) when the parameters lie on V R (J j ) ∩ E for j = 1, . . . , 7.
Let (ω, ν) ∈ C 1 , that is, D = −F/6, A = 3F/2 and E = −ωF/2. Then and the solution of (56) is If (ω, ν) ∈ C 2 we take D = 2F/3, A = −F and E = ωF . Then and the solution of (56) is Let (ω, ν) ∈ C 3 so that F = 2(A + D) and E = ωD. Then . We note that system (56) is written locally around (x, z) = (0, 0) as the analytic equations ∆(z, x) = x 2 − z 2 + · · · = 0, ∆ 2 (z, x) = 0, where the dots are higher order terms. Therefore we only need to analyze the local structure of the solutions of the first equation. There are several methods to do that and we opt by the way of singularities of smooth maps, see for example the book [23]. Using subscripts for partial derivatives, ∆(0, 0) = ∆ z (0, 0) = 0 hence the point (z, x) = (0, 0) is called a singularity of the map ∆ because the hypothesis of the Implicit Function Theorem fail at the singularity. We say that the maps ∆(z, x) and∆(z, x) are equivalent if there exist a local diffeomorphism of R 2 of the form (z, x) → (Z(z, x), X(x)) mapping the origin to (0, 0) and a positive function U (z, x) such that U (z, x) ∆(Z(z, x), X(x)) =∆(z, x) and where the diffeomorphism preserves the orientations of z and x, that is, the derivatives Z z (z, x) > 0 and X x (x) > 0. We call∆ a normal form of ∆ when it is the simplest representative from a whole equivalence class of mappings. Let D 2 ∆(0, 0) stands for the Hessian matrix of ∆ at (0, 0). Since in our case ∆ zz (0, 0) > 0, ∆ x (0, 0) = 0, and det(D 2 ∆(0, 0)) < 0 then the singularity is called simple and the normal form is just∆(z, x) = x 2 − z 2 . This proves that the equation ∆(z, x) = 0 only has two real solutions in a neighborhood of the origin having the form x + O(x 2 ) and Let (ω, ν) ∈ C 4 so that D = −2F/3, A = 3F/4 and E = −ωF/2. Then and the solution of (56) is Let (ω, ν) ∈ C 5 , hence D = −2F/3, A = F/3 and E = −ωF/3. Then and the solution of (56) is Let (ω, ν) ∈ C 6 , hence E = 0, D = −F/9 and A = F/3. Then and the solution of (56) is are even functions and therefore the solution of (56) is the trivial one z = −x and so C 7 ⊂ V C . This computation solves the center problem at the origin of family (14) which is the first part of the theorem.
Now we start the analysis of the cyclicity of the origin inside family (14). We will use Theorem 15. First we claim that codim(C j ) = 3 when j = 3 and codim(C 3 ) = 2.
The proof of the claim uses the fact that if the map F : R n → R m with n ≥ m is such that its differential d P F at P ∈ R n has maximal rank (in other words, F is a submersion at its regular point P ) then the origin of R m is a regular value of F restricted to some neighborhood U ⊂ R n of P and therefore U ∩ F −1 (0) is a smooth submanifold of R n of codimension m.
The above implies statements (i) and (ii) from Theorem 15. In summary, only the limit cycle bifurcations from the center at the origin when either ν = 0 or A = −3D, F = −4D and E = ωD or A = −3D/2, F = −D and E = ωD remains open. Statement (iii) analyze the firs case.
Computing the first polynomial ideals G k of Definition 7 we check that g 4 ∈ √ G 3 and √ G 4 = G 4 . Using again Singular ( [12]) we obtain the prime decomposition G 4 = J ∩ ∩ 6 j=1 J j where J = A, B, D and each J j has a generator given by a polynomial p j (ω) with the property p −1 j (0) ∈ Ω. Therefore, V C = V R (J ) and (58) ) ∩ E proving the first part of the theorem. Moreover, from this equality we obtain that the maximum order of a focus at the origin is 4, hence the cyclicity of the foci is at most 3 and there are examples in [5] with 3 limit cycles proving statement (iii). Now we are going to prove statements (i) and (ii) by using Theorem 15. Notice that the center variety only has one irreducible componentĈ = {A = B = D = 0} with codim(Ĉ) = 3. We define the map (ω, ν) → G j (ω, ν) = (η 1 (ω, ν), . . . ,η j (ω, ν)) whose Jacobian when j = 3 at an arbitrary point P = (ω, 0, 0, C, 0) ∈Ĉ is Taking into account that ω ∈ Ω we get rank(d P G 3 ) = 3 if C = 0 and rank(d P G 2 ) = 2 if C = 0. Consequently statements (i) and (ii) follow from Theorem 15.
Remark 34. In the sequel we prepare the use of Theorem 14 to analyze family (15). We recall that V R (G) ∩ E = V R (G 4 ) ∩ E from the proof of Theorem 19. Now we move to the complex setting and we want to see if the equality V C (G) = V C (G 4 ) holds. But we lower our pretensions and focus, in the light of Proposition 11 and Remark 12, into the verification of the weaker equality V C (G * ) = V C (G * 4 ) where G * 4 = g 1 (ω * , ν), . . . , g 4 (ω * , ν) in the ring R[ν] where ω takes any fixed value ω * ∈ Ω. To apply Proposition 11 we consider the map F : C 5 → C 4 defined by (ω, ν) → F (ω, ν) = (g 1 (ω, ν), . . . , g 4 (ω, ν)) such that, at any point P = (ω, ν) ∈ C 5 , we have the 4 × 5 Jacobian matrix d P F such that rank(d P F ) = 4 at any point P ∈ C 5 \ σ where σ = V C (δ 1 , . . . , δ 5 ) where δ j (ω, ν) ∈ R[ω, ν] denotes the determinant of the matrix d P F whose jth column has been removed. In particular, rank(d P F ) = 4 at P ∈ V C (G 4 ) \ Σ where Σ = V C (G 4 ) ∩ σ = V C (K) where K = g 1 , . . . , g 4 , δ 1 , . . . , δ 5 . Straightforward computations with Singular yield the prime decomposition √ K = J ∩ K 1 where J = A, B, D and K 1 = ω, A, D . In conclusion We get that Σ is a finite union of intersections of hyperplanes in C 5 which intersect transversally with V C (G 4 ), hence the Zariski closure V C (G 4 ) \ Σ = V C (G 4 ). Therefore, for any fixed value ω = ω * ∈ Ω we have by Proposition 11 that V C (G * 4 ) = V C (G * ) as claimed. We cannot apply Theorem 14(i) because √ G 4 = G 4 . Thus we try to apply Theorem 14(ii) with a fixed (but arbitrary) value ω = ω * ∈ Ω. Using the ring definition ring r = (0, w), (A, B, C, D) in Singular we find the primary decomposition of G * 4 using the routine primdecGTZ in the primdec.LIB library. The result is that G * 4 = N ∩ R where R is a prime ideal, N is primary but not prime, and In summary, since the minimal basis of G 4 has cardinality 4, Theorem 14(ii) tell us that the cyclicity of the center at the origin, only allowing perturbations inside family (15) keeping ω = ω * ∈ Ω constant, is at most 3 for the parameters belonging to (V C \ V R (N )) ∩ E = {(ω * , ν) ∈ V C ∩ E : C = 0}. Notice that this bound is not sharp due to statement (i) of Theorem 19 which is stronger than the former claim.
Proof. We claim that the origin is a nilpotent center if and only if g is an odd function, that is g(−x; ν) = g(x; ν), in which case they are time-reversible centers with respect to the symmetry (x, y, t) → (−x, y, −t). The reader can check the claim using for instance the criterium explained in Remark 31 taking into account that f is linear. Therefore the center variety V C is (19). The first integrability focus quantities reduced modulo the ideal generated by the previous one in the ring R(ω)[ν], up to positive multiplicative constants, are (59)η j (ω, ν) = ω ν 2(j+1) for all j = 1, . . . , m, hence V C ∩ E = V R (G m ) ∩ E and the m is as stated in the theorem. From here we see that the maximum order of a focus at the origin is m and from (59) that theη j are independent and can be adjusted with total freedom, hence the bound m − 1 of the focus cyclicity can be reached proving statement (ii).
We are going to study the cyclicity of the center at the origin of family (17) by means of Theorem 15. The center variety only has one irreducible component C = V C of codim(C) = m. We consider the map (ω, ν) → G m (ω, ν) = (η 1 (ω, ν), . . . ,η m (ω, ν)). Due to (59) it is easy to check that the m × (d − 2) Jacobian matrix d P G m has full rank (that is, rank(d P G m ) = m) at any point P ∈ C ∩E, and therefore statement (i) follows by Theorem 15(ii).
Remark 35. For family (17), {η 1 (ω, ν), . . . ,η m (ω, ν)} is a minimal bases of G m of cardinality m since in this example theη j lie in R[ω, ν] (look at (59)) rather than in the bigger ring R(ω)[ν] as usual. Also V R (G)∩E = V R (G m )∩E as we have seen in the proof of Theorem 20. At first glance it seems that family (17) is a good candidate to apply Theorem 14(i) because the monomial ideal G m is radical, i.e., G m = √ G m . But first we need to rely on Proposition 11 and, moving to the complex setting, we define the map F : C d−2 → C m via (ω, ν) → F (ω, ν) = (η 1 (ω, ν), . . . ,η m (ω, ν)). We see in the proof of Theorem 20 that F is the complexification of the map G m and then rank(d P F ) = m at any point P ∈ V C (G m ) \ Σ where Σ = {(ω, ν) ∈ C d−2 : ω = 0} V C (G m ) is a hyperplane in C d−2 . Unfortunately, in this example Σ is a complete irreducible component of the variety V C (G m ) because V C (G m ) = Σ ∪ V C . Hence the Zariski closure V C (G m ) \ Σ = V C = V C (G m ) and we cannot apply Proposition 11.

Proof of Theorem 21.
Proof. Taking into account that (20) is time-reversible with respect to the symmetry (x, y, t) → (−x, y, −t) when the parameters are in C sym it follows that (20) has a center at the origin when the parameters lie in C sym ∩ E.
Then rank(d P G 2 ) = 2 at any point P ∈ C sym ∩ E except when ω 2 = 7. Hence the theorem follows by Theorem 15(ii).