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 Andreev 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 forṁ 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 rotates about it. It is well known that, in this case, the local qualitative phase portrait corresponds to either a center or a focus. One can find 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 3.1. We call n the Andreev number associated to an analytic monodromic nilpotent singularity which is invariant under analytic orbital equivalency, see [16]. From now on we restrict ourselves to families in the class of monodromic nilpotent singularities with minimum Andreev number (n,β) = (2, 1), that is, those that fall in case (iii) of Theorem 3.1. This monodromic family is characterized by the following form:ẋ = y + ax 2 + P * (x, y; ζ ),ẏ = bx 3 + cx y + Q * (x, y; ζ ), (2) where the parameter restrictions b − ac < 0, 2a + c = 0 and 8b + (c − 2a) 2 < 0 hold by monodromy, and P * and Q * denote (1, 2)-quasihomogeneous higher order terms in x and y with vector parameter ζ . Following Andreev's ideas [6], see Sect. 3.2, any system (1) with a monodromic singularity at the origin with (n,β) = (2, 1) is analytically conjugate to the Andreev canonical forṁ x = − y + y P(x, y; ν),ẏ = x 3 + ωx y + Q(x, y; ν), where P and Q denote terms with (1, 2)-quasihomogeneous degrees greater than 0 and 3, respectively, with parameter space E = {(ω, ν) ∈ × R p } and = {ω ∈ R * : ω 2 − 8 < 0}.
Therefore we see that ω is a special parameter because it appears in the (1, 2)-quasihomogeneous truncation of minimum degree of the vector field (3).
In what follows we will denote by X and X the vector fields associated to (1) (or (2) depending on the context) and (3), respectively. Hence X is the linear differential operator X = y + P(x, y)∂ x + Q(x, y)∂ y and div(X) its divergence.
It is well known that nilpotent centers, unlike the non-degenerate centers with associated non-vanishing pure imaginary eigenvalues, are not characterized by the existence of a formal first integral. Some exceptions are the trivial Hamiltonian centerṡ x = ∂ y H (x, y),ẏ = − ∂ x H (x, y), the monodromic time-reversible with respect to (x, y, t) → (x, − y, t), see [8], or the monodromic Z 2 -symmetric (those invariant under (x, y) → (− x, − y)), see [2]. Notice that these exceptions may happen in a general system (1) but never occur in systems (2) or (3) for which (n,β) = (2, 1).
However, nilpotent centers with a minimum Andreev number n = 2 are characterized by the existence of a formal inverse integrating factor. This fact stems from the series of three works [26][27][28], while a different, simpler and unified proof can be found in [16]. We remark that the nilpotent monodromic class (n,β) = (2, 1) is not analyzed in [3], and is out of the range of applicability for the families studied in [21] and [17].
It is known that a formal Lyapunov function exists for any non-degenerate monodromic singularity, giving rise to the so-called focal quantities, which are nothing more than the obstructions to the formal integrability. In an analogous way, for the class of monodromic nilpotent singularities with (n,β) = (2, 1) (and in particular for family (3)), we can find the obstructionsη j (ω, ν) to the existence of an inverse formal integrating factor by means of an algorithm that only involves algebraic calculations.
Here, R(ω)[ν] denotes the set of polynomials in ν with rational coefficients in ω. Since a polynomial ring over a field is Noetherian and the elements of R(ω) are just formal expressions so it is a field, hence R(ω)[ν] is a Noetherian ring. System (3) with (ω, ν) = (ω * , ν * ) ∈ E has a center at the origin if and only ifη j (ω * , ν * ) = 0 for any j ∈ N, in which case there is a formal inverse integrating factor of (3).
On the other hand, using adequate coordinates, we can find the Poincaré return map : ⊆ R → R defined by an analytic diffeomorphism on a transversal section to the flow with endpoint at the nilpotent singularity. More precisely, using ideas of Lyapunov [29], we can parameterize so that (r ) = r + j 2 v j (ω, ν)r j where the elements of the sequence {v j (ω, ν)} j∈N ⊂ R(ω) [ν] are called Poincaré-Lyapunov quantities. Both sequences of integrability focus quantities and Poincaré-Lyapunov quantities share the property that system (3) with (ω, ν) = (ω * , ν * ) ∈ E has a 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 algorithm 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 in the literature there are 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 we are aware 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 non-degenerate 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 canonical family (3). Theorem 2.1 Let X be the vector field associated to the Andreev 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 + ω x 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 ∈ .
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 on 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 by V K (I ). In this work we will only use either K = R or K = C. Definition 2. 3 We define the Bautin ideal B associated with a polynomial Andreev canonical family (3) as the ideal B = v j (ω, ν) : j ∈ N in the Noetherian ring R(ω) [ν]. Also, we denote by B 2k the ideal B 2k = v 2 , v 4 , . . . , v 2k . We define analogously the ideals I = η j (ω, ν) : j ∈ N in the ring R(ω)[ν] and I k = η 1 ,η 2 , . . . ,η k . The next result is a corollary of Theorem 2.2.

Corollary 2.4
The relations v 2k+ j ∈ I k for j ∈ {0, 1} and any k 1 hold. In particular I k = B 2k and I = B.

The Bautin ideal and the cyclicity of the nilpotent singularity
We define the displacement map δ associated to family (3) as where 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 Sect. 3.3. So the fixed points of 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 † ) ∈ × R p possesses a focus at the origin. Then there is a positive integer k called the order of the focus such 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 [31, Proposition 6.1.2]) we know that an upper bound on the cyclicity of the focus at the origin with respect to perturbation within family (3) is given by 2(k −1). Actually, this bound can be improved sometimes using a special basis of the ideal I k which we simply call "minimal basis" but is, in fact, a minimal basis with respect to the ordered integrability focus quantities.

Definition 2.5
Consider the ideal I k in the Noetherian ring R(ω) [ν]. For some s k we say that the basis B = {η j 1 , . . . ,η j s } of I k is minimal if it satisfies the following properties: Of course similar definitions apply to obtain minimal basis of B 2k , B or I. Let k be the order of the focus at the origin of (3) with (ω, ν) = (ω † , ν † ) such that, according to Theorem 2.2,η j (ω † , ν † ) = 0 for j k − 1 andη k (ω † , ν † ) = 0. Let {η j 1 , . . . ,η j s } be a minimal basis of the ideal I 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 * ) ∈ × 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 j 1 , . . . , v j m } 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 find a proof of this rearrangement in [31,Lemma 6.1.6]. From (5) and by a repeated use Rolle's Theorem the following result is implied.

Theorem 2.6 Assume the origin is a center of system
. Then the cyclicity of the center with respect to perturbations within 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 2.6 to bound the center cyclicity of (3). This is the goal of the next section.

The ideas of [20] adapted to the nilpotent singularity
Following exactly the same ideas as 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 ∈ , respectively.
The precise statement requires first some definitions. Since for a polynomial family and where the roots of D j are not in . Similarly,

Definition 2.7
We define the polynomial ideals G = g j : j ∈ N , G k = g 1 , g 2 , . . . , It was proved in [20] that if {V j 1 , . . . , V j m } is the minimal basis of the polynomial ideal V with respect to the ordered basis {V j : j 2} then {v j 1 , . . . , v j m } 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 2.6 produces the following bound for the center cyclicity in terms of polynomial ideals.

Theorem 2.8
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 ∈ , the cyclicity of the center at the origin, with respect to perturbation within family (3), is at most m − 1.
Theorem 2.8 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 2.14. 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 of 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\ 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\ 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\ for which the associated complex system (3) has no formal inverse integrating factor. The next example just illustrates what we have said.

Proposition 2.9
The polynomial familẏ Proof When (ω, ν) ∈ E we find the first integrating focus quantity η 1 = − 2 Aω/15. Therefore A = 0 is a necessary center condition which turns out to be sufficient because (6) 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 (6). Statement (ii) is just straightforward. Now we present a technical proposition useful in applications of the forthcoming Theorem 2.14 when the allowed perturbations in family (3) keep the value of ω constant. First, we need the following definition.

Definition 2.10 Given a field K and an ideal
The next proposition is the adaptation of [20, Proposition 29] to our setting.
when ω is assigned to any fixed value ω * ∈ .

Remark 2.12
Although implicit in the proofs of [20] we want here to point out that Proposition 2.11 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 2.13
In applications often the set in condition (c) of Proposition 2.11 satisfies ⊂ 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 [20,Theorem 26] 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 2.14 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.
then G = G k and for any (ω * , ν * ) ∈ V R (G) with 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 * ∈ the cyclicity of the center at the origin is at most m − 1.
Another approach for the center cyclicity focusing on the irreducible components of the center variety is the following one. For any κ ∈ N less or equal to the Bautin depth of B, we denote by d P G κ the κ × ( p +1) Jacobian matrix of the real analytic mapping Then, adapting [20, Theorem 32] to our setting we obtain the following result which is in fact based on results in [10].
be an irreducible component of the center variety associated to the origin of the polynomial family (3).
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.

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.

Remark 2.16
Let k be maximum order of a focus at the origin of (3) with parameter space E. In other words, k is the minimum integer such that the equality . . ,η j s (γ (ε))} with j s = k be a minimal basis of the ideal I k of cardinality s k. Let us perturb along an analytic curve ε → γ (ε) = (ω(ε), ν(ε)) ⊂ E with γ (0) = (ω † , ν † ) in such a way that, for |ε| 1, the chain of inequalities Then by using standard arguments of bifurcation theory it follows that s − 1 small amplitude limit cycles can be made to bifurcate from the origin of (3) by perturbing in E from (ω † , ν † ). The limit cycle realization perturbing a center is basically equal. Assume that . . , m − 1, then, for |ε| 1, m − 1 limit cycles bifurcate from the center at the origin of (3) by perturbing in E from (ω * , ν * ).

Example 1
In [1] the authors consider the following polynomial family: Computing the functions F, f and ϕ of Theorem 3.1 for this system gives Therefore the origin is a monodromic nilpotent singularity if and only if 8 − k 2 > 0. Moreover, it is of type (n,β) = (2, 1) if and only if, additionally, k = 0. The center problem at the origin of the full family (7) was solved in [1]. Restricting that 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 2.17
The center variety V C associated to the origin of family (7) with param-

The origin is a center of (7) if and only if
Moreover the following holds: (i) The cyclicity of any center at the origin under perturbations within family (7) is 1.
(ii) The cyclicity of any focus of (7) at the origin is bounded by 1 and there are foci with cyclicity 1.

Example 2
Consider the polynomial Andreev canonical family (3) with P and Q general (1, 2)-quasihomogeneous polynomials of minimal degrees, that is, degrees 1 and 4 respectively. This family is given bẏ with parameters ω ∈ and ν = (A, D, E, F) ∈ R 4 .

Theorem 2.18
The center variety V C associated to the origin of family (8) has seven irreducible components: The origin is a center of (8) with parameter space Moreover the following holds: (i) The cyclicity of any center at the origin of (8) having parameter values (ω, ν) ∈ C j ∩ E with j = 3 under perturbations within family (8) is exactly 2 except when ν = (0, 0, 0, 0). (ii) The cyclicity of any center at the origin of (8) having parameter values (ω, ν) ∈ The cyclicity of the center at the origin of (8) when ν = (0, 0, 0, 0) perturbed within family (8) is zero provided α(ω, ν) ≡ 0 where the expression of α is given in (39). (iv) The cyclicity of any focus of (8) at the origin is bounded by 2 and there are foci with cyclicity 2.

Example 3
In [5, Theorem 4.1] the center problem of the following polynomial Andreev canonical family (3) was analyzed: with parameters ω ∈ and ν = (A, B, C, D) ∈ R 4 . There it was 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 (9) with three small amplitude limit cycles bifurcating from a nilpotent focus at the origin, see Remark 4.9. We expand this study analyzing the cyclicity of the origin in (9).

Theorem 2.19
The center variety V C associated to the origin of family (9) with param- The origin is a center of (9) Moreover the following holds: (i) If C = 0 then the cyclicity of any center at the origin under perturbations within family (9) is exactly 2. (ii) If C = 0 then there are perturbations within family (9) producing one limit cycle bifurcating from the center at the origin. (iii) The cyclicity of any focus of (9) 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 canonical form (3). This family has the forṁ with polynomials

Theorem 2.20
The center variety V C associated to the origin of family (10) with Moreover the following holds: (i) The cyclicity of any center at the origin under perturbations within family (10) is exactly m − 1. (ii) The cyclicity of any focus of (10) at the origin is bounded by m − 1 and there are foci with cyclicity m − 1.

Theorem 2.21
The center variety V C associated to the origin of family (12) with Moreover, the cyclicity of any symmetric center at the origin with ω 2 = 7 under perturbations within family (12) is exactly 1.
If this conjecture is true then the center problem for the origin of family (12) is completely solved being the center variety

Andreev characterization of monodromic nilpotent singularities
The following theorem of Andreev [6] characterizes analytic systems (1) for which the origin is monodromic.
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 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 number.
In consequence we get p k (x, y) = i+nj=k a i j x i y j for certain coefficients a i j ∈ 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.

The Andreev canonical form
We consider system (1) and assume that the origin is a nilpotent monodromic singular point with (n,β) = (2, 1), that is, we consider case (iii) in Theorem 3.1. Then, considering the function F and the value a ∈ R of Theorem 3.1, we perform first the analytic change of variables (x, y) → (x, y − F(x)) and next the rescaling

The Poincaré map and inverse integrating factors
An inverse integrating factor of (14) is a function v : C → R of class C 1 (C), hence a T -periodic function in the variable θ , which is non-locally null and satisfies the partial differential equation: If W (x, y; ω, ν) is an inverse integrating factor of the Andreev canonical form (3) 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 (14) is v(r , θ; ω, ν) = W (r Cs θ, r 2 Sn θ ; ω, ν) r 2 (r , θ; ω, ν) .
In [18] it is also proved the following fundamental result: where the prime denotes derivative with respect to r 0 .

Proof of Theorem 2.1
Before giving the proof of Theorem 2.1 we need Lemma 4.1. This lemma appears in [1] but we reprove it again with our notations for the sake of completeness. Given a diffeomorphism on U ⊂ R 2 with det(D )| U = 0, we can apply the change of variables x → y defined by x = (y) to a C 1 -vector field X and a scalar function f , both defined on U . The outcome is the vector field * (X) and the function * ( f ) defined via the pull-back * (X) . . = (D ) −1 X • and the function * ( f ) . . = f • .

Lemma 4.1 Let X be a C 1 -vector field on an open set
Assume that a C 1function V : U → R satisfies the linear partial differential equation for some scalar function f defined on U. Let and ξ be a diffeomorphism and a scalar function on U such that the restrictions det(D )| U = 0 and ξ | U = 0. We define the pull-back vector field Y = * (ξ X), that is, Y and X are orbitally equivalent. Then, the function W = * (ξ ) * (V )/det(D )n satisfies the linear partial differential equation This equality follows by the the chain rule and the push-forward definition. In short, defining F(x) and F(y) as the components of X and Y respectively, and using the dot to denote an ordinary scalar product, one has proving the claim. Now we define J (y) = det(D (y)), so that W (y) = ξ(y)J −1 (y) V (y) where ξ = ξ • and also J (x) = J ( −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 (18) and recalling the definition of Y. We obtain In the last term we will apply the well-known relation div(F) = div( F) + J −1 ∇ y J . F, see a proof in the book [30]. Therefore Finally we obtain that Proof of Theorem 2. 1 We make use of the results proved in [26]. As noted in [26], after a nonlinear change of variables (x, y) → (αx, α(y + K x 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 towardṡ with parameter space {(λ, μ) ∈ R * × R p }, where P and Q only contain (1, 2)quasihomogeneous terms of degrees greater than 2 and 3, respectively. Moreover, in [27,Theorem 2.8] it is proved that if X is the vector field associated to (19) 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 4.1 applied to the particular case in which X is the field associated to (19), the linear partial differential equation under consideration is (20), and the conjugate field X is in the Andreev canonical form (3). In that case, the rescaling ξ ≡ 1. Moreover, if we compute the functions that appear in Theorem 3.1 in the particular case of a polynomial family (19) we obtain that the analytic function F depends polynomially on the parameters (λ, μ). More concretely the functions F and f of Theorem 3.1 have the Taylor expansions In particular, according with Sect. 3.2, the conjugation (x, y) → (x, y) with and α(λ) = 2(1 + λ 2 ) brings X to X, that is, transforms the polynomial family (19) 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 (19) polynomially in μ but not in λ.
Recall that the diffeomorphism used in Lemma 4.1 is just the inverse of the defined in (21), that is, ; λ, μ .

Remark 4.2
Of course, imitating the proof of Theorem 2.1, we are able to define integrability focus quantities η * j associated to family (2) and relate them to integrability focus quantities η j for family (19). More concretely, where Y is the vector field associated to (2), the formal series W (x, y) = k 4 W k (x, y) where W k are certain (1, 2)-quasihomogeneous polynomials of weighted degree k with W 4 (x, y) = bx 4 + (c − 2a)x 2 y − 2y 2 and where = (a, b, c, ζ ) is the vector parameter of (2). Moreover η * j = 0 if and only if η j = 0.

Remark 4.3
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 familẏ with b(x) = ωx + m j=2 ν j x j and ω ∈ . Let X be the vector field associated to (22). Then, using the perspective of Theorem 2.1 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 if 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 of 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 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 (22). 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 + ωx y) ∂ y and X i = ν i x i y ∂ y for any integer i 2.
(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 (these 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 i j of V k (x, y) = i+2 j=k c i j 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 satisfies 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 ) 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 ∈ .

Proof of Theorem 2.2
Before proving Theorem 2.2 we need the following preliminary result. Using the change of variables on R 2 \{(0, 0)} defined by (13) that transforms (x, y) → (r , θ) with Jacobian r 2 and next the time-rescaling ξ = 1/ (r , θ) it follows that the vector field X associated to the Andreev canonical form (3) and the vector field X = ∂ θ + F(r , θ)∂ r with F defined at (14) are orbitally equivalent on a sufficiently small punctured neighborhood of the origin. Then, using Lemma 4.1 we can prove the following result.

Corollary 4.4
Let X = ∂ θ + F(r , θ)∂ r be the vector field defined by (14) and v(r , θ) given by (16) where W is defined in Theorem 2.1. Then 1η k (ω, ν)x 2k and is defined in (15). Proof of Theorem 2.2 Let r 0 > 0 and r 0 > 0 both be sufficiently small and (θ; r 0 ) the solution of equation (14) with initial condition (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  v(r , θ) belonging to the set of T -periodic functions of θ . Taking the positive anti-clock-wise orientation in ∂ D, we shall formally use Green's Theorem in the domain D to the scaled vector field X/v with associated differential Taking also into account that v(r , 0) = v(r , T ), equation (25) becomes where P(r ) is a primitive of 1/v(r , 0). Regarding the right-hand side of (26) we note that From (15) it follows that with 0 (θ ) = 1 + ω Cs 2 θ Sn θ > 0. We consider the series W (x, y) = x 4 + ωx 2 y + 2y 2 + · · · of Theorem 2.1 and now we define the function v(r , θ) via the relation (16) v(r , θ) = W (r Cs θ, r 2 Sn θ) Remark 4.5 Notice that ifη j = 0 for any j ∈ N then v(r , θ) is an inverse integrating factor of X, that is div(X/v) ≡ 0, hence the fundamental relation (17) can be applied and both limits in (27) give the same result. Thus (26) is a trivial equality.

Proof of Theorem 2.17
Proof Reversing the time t → − t and denoting ω = − k family (7) 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 ∈ defined in (4) as it must be. The following definition will be used from now on.

Definition 4.6
We defineη j =η j mod I 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 the polynomial 21−2ω 2 has no roots on , it is clear that In order to achieve parts (i) and (ii) we shall use Theorem 2.15. The center variety only has one irreducible component C = {a = c = 0} of codim(C) = 2. Moreover the differential of the map It follows that rank(d P G 2 ) = 2 since ω ∈ and in consequence statement (i) holds by Theorem 2.15.
Using the notation of Definition 2.7 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 any member of family (8) 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. (8) is a particular case of systeṁ

Remark 4.7 System
with P 3 (0) = 0. Performing the (analytic and tangent to the identity) Cherkas change of variables (x, y) → (x, y/ξ(x)) with and rescaling the time (dividing the vector field by P 3 (x)ξ(x)), system (36) is transformed into the Liénard systeṁ When (36) comes from (8) we have P 3 (x) = − 1 + Ax, P 0 (x) = x 3 + Dx 4 , P 1 (x) = ωx + E x 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 [22,Theorem 2.6] the authors make a generalization of a result of Cherkas [9] that characterize the non-degenerate 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 (37). In our context their result is stated as follows: The origin of the analytic Liénard system with f and g as in (37) 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 of Remark 4.7 to family (8) when the parameters lie on and the solution of (38) is If (ω, ν) ∈ C 2 we take D = 2F/3, A = − F and E = ωF. Then and the solution of (38) is Let (ω, ν) ∈ C 3 so that F = 2(A + D) and E = ωD. Then We note that system (38) is written locally around (x, z) = (0, 0) as the analytic equations 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 fails 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) stand 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 − x + O(x 2 ), hence C 3 ⊂ V C .
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.
Each irreducible component C j of the center variety is defined by The claim is proved because the differential of F j has maximal rank at any point (ω, ν) ∈ R 5 for all subindexes j. Now we define the map (ω, ν) → G k (ω, ν) = (η 1 (ω, ν), . . . ,η k (ω, ν)) and we compute, for k ∈ {3, 2}, the rank of its Jacobian at arbitrary points P j = (ω, ν) on each component C j . Taking into account that ω ∈ we obtain the following results: • If P j ∈ C j with j = 3 then rank(d P j G 3 ) = 3 if P j = (ω, 0, 0, 0, 0) and rank(d P j G 3 ) = 1 otherwise.
The above implies statements (i) and (ii) from Theorem 2.15. 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 remain open. Statement (iii) analyzes the first case. We start the proof of (iii). Arbitrary perturbations within family (8) of the integrable center at the origin when (ω, ν) = (ω, 0, 0, 0, 0) can be analyzed as follows. The unperturbed systemẋ = − y,ẏ = x 3 + ωx y has a global center possessing the inverse integrating factor V 0 (x, y; ω) = x 4 + 2y 2 + ω x 2 y from which we obtain its first integral We consider an arbitrary deformationẋ = − y + ε P(x, y; ν, ε),ẏ = x 3 + ωx y + ε Q(x, y; ν, ε), with P(x, y; ν, ε) = a(ε)x y and Q(x, y; D, E, F). Taking a transversal to the perturbed trajectories and parameterizing it with the values of H , it is known from Poincaré that the expansion of the displacement map near ε = 0 is δ(h; ω, ν, ε) = M(h; ω, ν)ε + O(ε 2 ) and the first Melnikov function has the expression of a generalized Abelian integral where γ h are the family of ovals surrounding the origin contained in the level curves { H (x, y; ω) = h : h ∈ R + }. The reader can consult the book [11] for a modern exposition. Assuming M ≡ 0 (otherwise higher order bifurcations may occur) and recalling that we are only interested in the limit cycle bifurcations from the origin (not in the entire period annulus), we need to study the number of isolated zeroes of the function M(h; ω, ν) for h > 0 sufficiently small which corresponds with small amplitude limit cycles. One can use the variables ( The linear expression of M( ·; ω, ν) finishes the proof. Remark 4. 8 We are going to apply Theorem 2.14 to analyze family (8). We know that is true or not we will use Proposition 2.11 and Remark 2.12 to check that the equality V C (G * ) = V C (G * 4 ) holds where G * 4 = g 1 (ω * , ν), . . . , g 4 (ω * , ν) in the ring R[ν] and ω is fixed to any value ω * ∈ .

Proof of Theorem 2.19
Proof The first integrability focus quantities, up to positive multiplicative constants, areη Now we reduceη j =η j mod I j−1 in the ring R(ω)[ν] according to Definition 4.6 and obtaiñ Computing the first polynomial ideals G k of Definition 2.7 we check that g 4 / ∈ √ G 3 and √ G 4 = G 4 . Using again Singular [12] we obtain the prime decomposition 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) / ∈ . Therefore, V C = V R (J) and 4 ) ∩ 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 three and there are examples in [5] with 3 limit cycles proving statement (iii). Now we are going to prove statements (i), (ii) by using Theorem 2.15. Notice that the center variety only has one irreducible component C = { A = B = D = 0} with codim( C) = 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) ∈ C is Taking into account that 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 2.15.

Remark 4.10
In the sequel we prepare the use of Theorem 2.14 to analyze family (9). We recall that V R (G) ∩ E = V R (G 4 ) ∩ E from the proof of Theorem 2.19. 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 2.11 and Remark 2.12, 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 * ∈ . To apply Proposition 2.11 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 ) with 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 )\ where = V C (G 4 )∩σ = V C (K) with 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 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 )\ = V C (G 4 ). Therefore, for any fixed value ω = ω * ∈ we have by Proposition 2.11 that V C (G * 4 ) = V C (G * ) as claimed. We cannot apply Theorem 2.14 (i) because √ G 4 = G 4 . Thus we try to apply Theorem 2.14 (ii) with a fixed (but arbitrary) value 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 √ N = B, C, D, 5A − ω * B . These generators lead to In summary, since the minimal basis of G 4 has cardinality 4, Theorem 2.14 (ii) tells us that the cyclicity of the center at the origin, only allowing perturbations inside family (9) keeping 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 2.19 which is stronger than the former claim.

Proof of Theorem 2.20
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 4.7 taking into account that f is linear. Therefore the center variety V C is (11). 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, arẽ η j (ω, ν) = ων 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 (40) 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 (10) by means of Theorem 2.15. 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 (40) 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 2.15 (ii).

Remark 4.11
For family (10), {η 1 (ω, ν), . . . ,η m (ω, ν)} is a minimal basis of G m of cardinality m since in this example theη j lie in R[ω, ν ] (look at (40)) 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 2.20. At first glance it seems that family (10) is a good candidate to apply Theorem 2.14 (i) because the monomial ideal G m is radical, i.e., G m = √ G m . But first we need to rely on Proposition 2.11 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 2.20 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 )\ where = {(ω, ν) ∈ C d−2 : ω = 0} V C (G m ) is a hyperplane in C d−2 . Unfortunately, in this example is a complete irreducible component of the variety V C (G m ) because V C (G m ) = ∪ V C . Hence the Zariski closure V C (G m )\ = V C = V C (G m ) and we cannot apply Proposition 2.11.

Proof of Theorem 2.21
Proof Taking into account that (12) is time-reversible with respect to the symmetry (x, y, t) → (− x, y, − t) when the parameters are in C sym it follows that (12) 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 2.15 (ii).