CENTER PROBLEM FOR SYSTEMS WITH TWO MONOMIAL NONLINEARITIES

. We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our ﬁrst result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.

1. Introduction and statement of the main results. The center-focus problem consists in distinguishing whether a monodromic singular point is a center or a focus. For singular points with imaginary eigenvalues, usually called nondegenerate singular points, this problem was already solved by Poincaré and Lyapunov, see [19,21,22]. The solution consists in computing several quantities called commonly the Poincaré-Lyapunov constants, and study whether they are zero or not. There are different methods to compute them, for a brief survey of these methods see [4,10,12,13,20] and the references therein.
Despite the existence of many methods, the solution of the center-focus problem for simple families, like for instance the complete cubic systems or the quartic systems with homogeneous nonlinearities, has resisted all the attempts, see for instance [11,18,24]. For this reason, in this paper and following [17], we propose to push on this question in another direction. We study this problem for a natural family of differential systems with four real (two complex) free parameters but arbitrary degree. Before introducing this "simple" family of differential systems we recall the formulation of the center-focus problem for nondegenerate singular points in complex coordinates.
A real analytic planar differential system with a weak-focus can always be written asẋ = −y + P(x, y) = −y + j≥2 P j (x, y),ẏ = x + Q(x, y) = x + j≥2 Q j (x, y), where P j and Q j are real homogeneous polynomials of degree j. Equivalently, in complex notation, it writes as the equatioṅ where z = x + iy and f k, are complex numbers obtained from the coefficients of the polynomials P j and Q j . The center-focus problem for equation (1) consists in finding necessary and sufficient conditions on the coefficients f k, to distinguish if the origin is a center or a focus. Complex notation has often been used in several works, see for instance [9,14,18,24,25].
In this paper we study this problem for the family of differential equationṡ z = iz + Az kz + Bz mzn (2) where k + ≤ m + n, (k, ) = (m, n) and A, B ∈ C. As we will see, the integer values will play a key role in our study. One of the reasons for this special role of both numbers is that when α = 0 (resp. β = 0) the monomial z kz (resp. z mzn ) appears as a resonant monomial in the Poincaré normal form of the complex differential equation.
Our first result lists the centers that we have found in family (2). To the best of our knowledge the centers presented in statement (d) when k = m ≥ 1 are not known. They belong to a bigger new class of Darboux centers presented in Theorem 2.1. In fact, as we will see in Section 2, this family contains some new persistent and weakly persistent centers, see [4] for a first study of this class of centers.
Theorem 1.1. The origin of equation (2) is a center when one of the following (nonexclusive) conditions hold: We remark that when in our work we say reversible center we refer to a center that has a line of symmetry in the sense already introduced by Poincaré. There is a more general notion of reversible center (with respect to general curves) which is not used in this paper. In this broader sense, it can be seen by using normal form theory, that every analytic nondegenerate center is reversible with respect to an analytic curve.
We also remark that in the above theorem, the names between brackets classifying the centers are only for orientation and they neither give an exclusive classification. For instance, in case (d) the Hamiltonian systems are the ones satisfying that k = m = 0 but in (c) there are reversible centers that are Hamiltonian as well, see Proposition 3.3(a).
The main goal of this paper is to investigate if the above list of centers is complete. We prove: Theorem 1.2. In the following cases the list of centers given in Theorem 1.1 is complete: When k, , m and n satisfy p α + q β = (k + − 1)Q − (m + n − 1)P = 0, for some P, Q, p and q, where P ≤ Q and N (P, Q) are given in Table 1 and (p, q) ∈ N×Z are such that pP + |q|Q ≤ N (P, Q). (e) When the nonlinearities are homogeneous (k + = m + n = d) and either d is even and d ≤ 34 or d is odd and d ≤ 57. (f ) When 4 ≤ k + + m + n ≤ 36. 1 2  3 4 5  6   1  8 10 13 13 15 15  2  --19 -19 -3 ---23 23 - Table 1. Values of N (P, Q) for P ≤ Q and coprime P and Q.

P \Q
In cases (a) and (b) of Theorem 1.2 the result is already known, see [4,17]. We provide short and different proofs that do not need the computation of Poincaré-Lyapunov constants, see Sections 3.2 and 3.3, respectively. Observe that all cases but (a) define varieties in the discrete space of the exponents, (k, , m, n) ∈ N 4 . Specifically, the dimension of cases (b), (c), (d), (e) and (f) are 3, 3, 2, 0 and 0, respectively. In other words, while cases (e) and (f) refer to fixed degree equations, all the other cases deal with unbounded degree families.
To the best of our knowledge the characterization of the centers for case (c) is new and it constitutes one of the main results of this paper, see Theorem 3.4. Also the results given in (d), (e) and (f) are new. A key point for our proof for cases (c) and (d), see Sections 3.4 and 3.5, respectively, has been a reparametrization of the degrees k, , m, n in terms of some P, Q that provides a compact expression of equation (2) in polar coordinates, see equation (9) below.
Our proof of (e) and (f) covers the low degree cases and includes all the particular ones solved in [17]. We use either the algorithm developed in [10], that we recall for completeness in Section 4, or the well-known classical approach developed by Lyapunov for constructing suitable Lyapunov functions, see [19,26].
Statements (e) and (f) of Theorem 1.2 are proved then with a case by case study, computing all the necessary Poincaré-Lyapunov constants to solve the center-focus problem. In Sections 3.6 and 3.7, we detail some of these results. For instance, the results for cases (d), (e), and (f) needed several weeks of CPU time. We have used MAPLE 18 in a Xeon computer (CPU E5-450, 3.0 GHz, RAM 32 Gb) with GNU Linux for all the computations. Along the paper we include some remarks about the computational difficulties.
In general, to predict the exact number of Poincaré-Lyapunov constants needed to solve the center-focus problem or, equivalently, to know which is the highest order weak-focus in a given family is a difficult and intriguing question. Our study for proving statement (e), see Section 3.6, suggests an answer in terms of k, , m and n for the homogeneous nonlinearities case. As a consequence of our results we also give the highest known order for weak-foci of polynomial systems of odd degree d when d ≤ 89, see Proposition 3.6. In Section 3.7 we also detail the values that we have found for the case (f) and we explain why such a prediction is complicated for the nonhomogeneous family. In particular, the highest order weak-focus found for this last case has been 304, it corresponds to the Poincaré-Lyapunov constant V 609 , and it happens when (k, , m, n) ∈ { (16,0,1,19), (0, 16, 20, 0), (0, 16,1,19)}.
Notice that from statements (a) and (b) of Theorem 1.2 the center-focus problem for equation (2) is totally solved when αβ = 0 or AB = 0. Therefore, as we will see in Section 3.4, instead of dealing with equation (2) we can reduce our study by consideringż with k + ≤ m + n, (k, ) = (m, n), αβ = 0 and 0 = C ∈ C. We also remark that for equation (3) the characterization of the reversible centers given in Theorem 1.1(c) reduces to C |q| + (−1) p+|q|+1C |q| = 0, where (p, q) ∈ N × Z are the coprime values such that pα + qβ = 0. The results of this paper suggest the following challenging question: Is the list of centers of equation (3) presented in Theorem 1.1 exhaustive? All our attempts seem to indicate that the answer is yes.
In the particular case of homogeneous nonlinearities the above question reduces to: Is it true that when k + = m + n ≥ 3 all the centers of equation (3) are reversible?
2. Sufficient center conditions. It is said that the origin of (1) is a persistent center if it is center forż = iz + λF (z,z) for all λ ∈ C and the origin is a weakly persistent center if it is a center forż = iz + µF (z,z) for all µ ∈ R, see [4]. The following theorem gives a new and large family of planar differential systems with persistent or weakly persistent centers at the origin.
and F starts at the origin at least with second degree terms. Then the origin is a center if and only if either k ∈ {0, 1} or k > 1 and Re(f k−1 ) = 0. Indeed, in all cases the origin is a weakly persistent center, Hamiltonian when k = 0 and of Darboux type when k ≥ 1. Moreover, it is a persistent center when k ∈ {0, 1} or k > 1 and f k−1 = 0.

CENTER PROBLEM FOR TWO MONOMIAL SYSTEMS 581
We start proving that the function U −k (z,z) = (zz) −k , constructed from this invariant algebraic curve, is an integrating factor of the differential equation. This is precisely the definition of a Darboux integrable equation, see for instance the survey [15].
It is easy to see that if X is the vector field associated to a differential equatioṅ z = G(z,z) then div(X) = 2 Re ∂ ∂z G(z,z) .
Therefore, if X is the vector field associated toż = iz as we wanted to show. Let us compute a real first integral of our differential equation. From the equation we obtain that where g satisfies g (u) = f (u)u −k . Therefore the real function is a candidate to be a first integral of the equation in C \ {0}. Notice that, since the origin is a monodromic critical point, to prove that it is a center it suffices to construct a smooth first integral ofż = iz + z k f (z) that is continuous at the origin. When k = 0 we have that where g(u) = u 0 f (s) ds is a smooth first integral at the origin. Therefore, in this case we are done. When k = 1, consider the first integral H(z,z) = e 2H1(z,z) = zz e 2 Im(g(z)) , It is smooth at the origin because by hypotheses f (0) = 0. Hence, for k = 1 we have also proved that the origin is a center. Finally consider k > 1. In this case we define To ensure that the above function is not multivaluated in C we need to impose that Im(f k−1 logz) is not multivaluated. This forces that Re(f k−1 ) = 0. Under this hypothesis the function H(z,z) is well defined in C and moreover it is continuous at the origin, because lim z→0 (zz) k−1 Im(g(z)) = 0.
Hence, when k > 1 and Re(f k−1 ) = 0 the origin of our differential equation has a center, as we wanted to prove. When, Re(f k−1 ) = 0 the same expression of H implies that the origin is not a center. Therefore, the characterization of the centers follows.
It is clear that when Re(f k−1 ) = 0 the centers are weakly persistent and when f k−1 = 0 they are persistent. Hence the theorem is proved.
We will also use the following well-known proposition to characterize the reversible and the holomorphic centers, see [3,Prop. 7] and [8]. For completeness we also include its proof. (1) has a center at the origin when one of the following conditions holds:

Proposition 2.2. Equation
Proof. (a) Assume that f k, = −f k, e i(k− −1)ϕ holds for all k and . Let us prove that if z = Z(t) is a solution of the differential equation (1), then z = e −iϕZ (−t) is a solution as well. To do this, consider the transformation w = e −iϕz , s = −t.
as we wanted to prove. Fix a small enough neighborhood of the origin. In this neighborhood, let Γ + = {Z(t) : 0 ≤ t ≤ t 1 > 0} be a piece of the solution of (1) between two consecutive cuts of Z(t) with the straight line with slope θ = −ϕ/2. These cuts exist because the origin is a monodromic weak-focus. Therefore, is another piece of the solution of (1), and moreover By the uniqueness of solutions, joining both pieces Γ + and Γ − we obtain that the solution passing through Z(0) is periodic, with period 2t 1 . As a consequence, the origin of equation (1) is a center, as we wanted to show. (b) When F (z,z) ≡ F (z) we can write equation (1) aṡ with G a holomorphic function such that G(0) = i. Following [8] we consider the holomorphic map Notice that it is well defined and invertible in a neighborhood of z = 0 because Φ(0) = 0 and Φ (0) = 0. In fact, let us see that the local change of variables w = Φ(z) is a local holomorphic conjugacy betweenż = F (z) and z = iz, or in other words, a holomorphic linearization of the differential equation. In fact, if w = Φ(z) we havė Therefore the origin of equation (1) is an (isochronous) center, as we wanted to prove.
Proof of Theorem 1.1. Equation (2)  For case (d), the significant center conditions are obtained not working directly with the differential equation in polar coordinates but using a preliminary simplification that, as we will see, will also constitute the key point for studying case (c). The Poincaré-Lyapunov constants for case (e) are obtained by using the method developed in [10] that is briefly summarized in Section 4. The case (f) is studied by using the classical Lyapunov approach, see [19,26].
3.1. Poincaré-Lyapunov constants. Differential equation (1) can be written in polar coordinates, z = r e iθ , aṡ Then, in a neighborhood of r = 0, it can be studied through the differential equation Denote by r(θ; η) the solution of (5) such that r = η ≥ 0 when θ = 0. For r small enough, we can write with v j (0) = 0 for j ≥ 2. The Poincaré return map is defined as (1). It is well known that the first N such that V N = 0 is always odd, see [1, p. 243]. The number N 0 , where N = 2N 0 + 1, is called the order of the weak-focus.
3.2. The case AB = 0. As we have already said, the result in this case is well known, see [4,17]. We provide a proof for the sake of completeness. It uses the next general result that gives necessary conditions for a system to have a center.
Lemma 3.1. Letż = G(z,z) andż = H(z,z) be two smooth differential equations with a critical point at the origin. If one of the equations has a center at the origin and then a necessary condition for the other equation to have a center at the origin is γ = 0.
Proof. Notice that Im(G(z,z)H(z,z)) gives the scalar product between the vector field associated toż = G(z,z) and the orthogonal of the vector field associated tȯ z = H(z,z). Therefore, if γ = 0, the above equality implies that, in a neighborhood of the origin, the level curves of the solutions of the equation having a center are without contact for the flow associated to the other equation, giving the impossibility of having a center for the second equation. Hence γ = 0 is a necessary condition to have a center for the other differential equation.   Proof. The three conditions are included in Theorem 1.1(c), therefore their sufficiency is proved. To show that they are necessary we will apply again Lemma 3.1.
Therefore, Re(A) = 0 is a first necessary center condition. In case (b) we are done. In case (a), then m = n + 1 and therefore Im(iB)z n+1zm = Re(B)(zz) n+1 and Re(A) = Re(B) = 0 are the center conditions, as we wanted to prove.
(c) Here we take G(z,z) = iz + Az kz + Im(B)i z n+1zn and H(z,z) = iz + Az kz + Bz n+1zn . By Theorem 1.1(c) the origin ofż = G(z,z) is a center. Direct computations lead to Hence, Re(B) = 0 is the necessary condition to have a center.
3.4. The case (α + β)(α − β) = 0. This section proves Theorem 1.2(c). Here we will assume that αβ = 0 and AB = 0 because the cases αβ = 0 or AB = 0 are already studied in the previous sections. Under these conditions there exists λ ∈ C such that the change of variables w = λz writes equation (2) aṡ for some C = c 1 + ic 2 ∈ C \ {0}. Consider the following reparametrization with P = Q = 1 when (6) has homogeneous nonlinearities or P < Q and coprime otherwise. Then equation (6) writeṡ This equation in polar coordinates, z = r e iθ , is where a P (θ) = 1 2 (e iαθ + e −iαθ ), Doing the change of variables r = R 1/M the above equation writes as the following differential equation where A P = M a P , A Q = M a Q , B P = b P and B Q = b Q . The characterization of the centers of equation (7) is equivalent to find conditions that imply that u(2π; ρ) ≡ ρ, for ρ small enough, where u(θ; ρ) is the solution of equation (9) such that u(0; ρ) = ρ.
In fact, let us see that if u(2π; ρ) = ρ + U ρ S + · · · , U = 0, then the first significant Poincaré-Lyapunov constant for equation (1) is Let r(θ, η) be the solution of (8) satisfying r(0, η) = η. Since R = r M , it holds that as we wanted to see.  (6) is Moreover, the origin is a center if and only if c 2 = 0 (reversible) or k = m (Hamiltonian or Darboux).
Proof of Theorem 3.4. A first step for proving the theorem will be to show that where u(θ; ρ) denotes the solution of equation (9) such that u(0; ρ) = ρ. Developing the right hand side of equation (9) in power series of R up to order P + Q + 1 we obtain where K is the maximal natural number satisfying P (K − 1) ≤ Q.
Notice that when K = 2 the expressions (16) and (17) are not well defined and have to be reinterpreted: equations (16) do not appear and equation (17) is u 1 (θ) = A Q (θ)+A P (θ). In fact, K = 2 corresponds to k+ = m+n and P = Q = 1, i.e. the case of homogeneous nonlinearities in equation (6).

3.5.
Case (d) in Theorem 1.2. In this section we will describe the method that we use to prove that all the centers of the 2-parameter families given in Theorem 1.2(d) are the ones listed in Theorem 1.1. We fix integer values of P, Q, p, and q satisfying the restrictions of the theorem. Let k, , m, n be such that pα + βq = 0 and (k + − 1)Q − (m + n − 1)P = 0. Notice that the latter condition implies that k + − 1 = P M and m + n − 1 = QM for any M ∈ N, because in Table 1 it is always satisfied that gcd(P, Q) = 1.
For these values, using the procedure described in Section 3.4, we get that the first significant center condition is of the form for a nonzero number D ∈ Q and a polynomial E L of degree L with integer coefficients and such that E L (0) = 0. Recall, that using (10), we get that this center condition corresponds to the Poincaré-Lyapunov constant V p(k+ −1)+|q|(m+n−1)+1 . When C |q| + (−1) p+|q|+1C |q| = 0 the system always has a reversible center at the origin (see (4) and Theorem 1.1(c)) and we are done. Otherwise, E L (M/β) = 0. Since β and M are integer numbers, each rational root s j ∈ Q, of E L , gives rise to a condition of the form β = M/s j , which provides a new candidate to be a center. Fixing each one of these rational roots we have got either a holomorphic center (Theorem 1.1(b)), or a Darboux center (Theorem 1.1(d)), or that we need to continue computing the next significant center condition. In this latter case, for all the studied equations, we have arrived to a reversible center at the origin.
For instance, when P = 1, Q = 3, according to Table 1, we have that N (1, 3) = 13. This means that our study covers all (p, q) ∈ N × Z such that p + 3|q| ≤ 13. Our computations when p + 3|q| = 13 needed around one hour of CPU time with the software and computer characteristics explained in the introduction.
Next, to illustrate in more detail the procedure, we study the cases where P = 1, Q = 3 and p + 3|q| = 6, that is p = 3 and q = ±1. When q = 1, the first significant center condition is When C −C = 0 we obtain a reversible center. Assume now that C −C = 0. When 2β + 3M = 0 the center is of Darboux type. In the other two cases, β + 3M = 0 and β + 6M = 0, the centers are reversible because the next significant center conditions respectively. Similarly, when q = −1, we obtain When C −C = 0 we obtain again a reversible center. When C −C = 0, the case β − 3M = 0 gives a holomorphic center, the case β + 3M = 0 is a Darboux center, and when β − 6M = 0 we have once more a reversible center because the next significant center condition is Notice that the second significant center condition depends on the relation between β and M . In this case, it is either u 9 (2π) or u 13 (2π).
To have an idea of the computational effort needed to solve the center-focus problem in this case (d) we show in Table 2 the time needed for each fixed couple P ≤ Q, gcd(P, Q) = 1 when (p, q) ∈ N × Z are such that pP + |q|Q ≤ N (P, Q).  Table 2. Values of N (P, Q) and, in brackets, the total CPU time needed in each case to solve the center-focus problem.
In all the studied cases the polynomial E L has some rational roots. In most of the cases, all them are rational and simple. Anyway, in very few cases there is a multiple rational root, for instance when P = 1, Q = 2, 2α − 3β = 0 or when P = 3, Q = 5, α − 3β = 0. Sometimes E L has an irreducible factor without rational roots, for example when P = 1, Q = 2, 4α ± 3β = 0 or when P = 1, Q = 3, 4α ± 3β = 0.
As we have already explained, when the polynomial E L vanishes for some rational value it is necessary to go further in the computation of the center conditions. This second condition associated to a given rational root always has been enough to solve the center-focus problem. But, a priori, we do not know how far we need to go to reach this second significant center condition. In all the cases given in Table 1 the second significant condition is either u (p+2)P +|q|Q+1 (2π) or u pP +(|q|+2)Q+1 (2π).
Nevertheless, we present two cases not covered by Table 1, for which the second center condition is none of both conditions, illustrating that the question of knowing a priori how far we need to go to get all the significant center conditions is very intricate.
The first case is for P = 3, Q = 14 and 2α + 3β = 0. Following the procedure described in Section 3.4 we obtain, using p = 2, q = 3, that the first significant center condition is The first root, 5β + 22M = 0, corresponds with a Darboux case. The other three roots correspond with reversible cases because the next center conditions are respectively, but notice that (p + 2)P + |q|Q + 1 = 55 and pP + (|q| + 2)Q + 1 = 77. When M = 1 this family corresponds with the exponents (k, , m, n) = (4, 0, 7, 8) and is also considered in the next section. The total CPU time needed to finish this special family has been of 11 hours. The second case corresponds to P = 5, Q = 11 and α + 5β = 0. Then, the first significant center condition is Fixing the exponents (k, , m, n) = (1, 5, 7, 5) we are in the above situation. For these values we have that u 61 (2π) = 0 because β = M = 1, situation that corresponds with the last root of E L . The next center condition is This last example neither satisfies our first guess about which is the next significant center condition because (p + 2)P + |q|Q + 1 = 71 and pP + (|q| + 2)Q + 1 = 83.
3.6. Case (e) in Theorem 1.2. As in Section 3.4 it suffices to study the center problem for equation (6),ż = iz + z kz + Cz mzn , for some 0 = C ∈ C, with αβ = 0 and d = k + = m + n. We start computing the first significant Poincaré-Lyapunov constant. If this constant decides the center-focus problem we stop our computations. Otherwise we compute the second significant constant. As in case (d), for all the k, , m and n that we have considered these two constants decide the center-focus problem. Moreover, in this case, they always give a reversible center.
As an example, Table 3 shows the significant constants when k + = m + n = 5. Notice that, by symmetry we can assume without loss of generality, that k > m.
To have an idea of the difficulty of the computations we comment that the CPU time used to study all the cases with d ≤ 34, even, or d ≤ 57, odd, is similar and it is of several days. In fact, the necessary time to finish each degree seems to increase exponentially.
The reason why we can go further in the odd case is well understood: the significant Poincaré-Lyapunov constants are V d+j(d−1) with j ∈ N ∪ {0} when d is odd and j ∈ 2N when d is even. See also Corollary 4.6 in the appendix. Therefore, in general to have the same number of significant constants in the even case we need to go further in the computations. In particular the time to finish all the odd degrees smaller or equal than 31 is 1.4 hours but we need around 6 days for finishing the even degrees smaller or equal than 30.
As it can be seen in Table 3 the maximum order of a weak-focus depends on k, , m and n. Using the notions introduced in Section 3.4, the homogeneous case corresponds to P = Q = 1 and there exist two coprime numbers p and q, (p, q) ∈ N×Z, such that pα+qβ = 0. Then the maximum order of a weak-focus corresponds to j = p + |q| − 1 or j = p + |q| + 3 in all the above expressions of V N . This fact has been checked in all the homogeneous systems presented in our study, but we can not predict, a priori, neither which of both situations appears, nor if there are other possible values for N . Moreover, in almost all cases, with only one Poincaré-Lyapunov constant the procedure finishes and then j = p + |q| − 1. But, sometimes two are necessary and, when this happens, j = p + |q| + 3.
The computations done in this section provide concrete examples of simple polynomial systems of degree d with a weak-focus of very high order. This problem has been already studied in [2,16,23]. For general even degree d, systems with a weak-focus of order d 2 − d and d 2 − 1 are given in [2] and [16,23], respectively. When the degree d is odd, there are examples of systems with weak-foci of order (d 2 − 1)/2, see again [16] and [23]. We notice that in the first paper the equation has nonhomogeneous nonlinearities while in the second one the nonlinearities are homogeneous. Moreover, examples with a nonlinearity formed by three complex monomials and providing weak-foci of a higher order, (d 2 + d − 2)/2, are presented in [23] when d is odd and d ≤ 19.
Studying all the families of statement (e) in Theorem 1.2 we select the two concrete families with higher order weak-focus to extend our computations only for them up to odd degree d ≤ 89 using several days of CPU time. All these computations are summarized in the next result.
Proposition 3.6. Consider the differential equationṡ Then there exist values of C 1 and C 2 such that the origin is a weak-focus of order (d 2 + d − 2)/2 for any odd 5 ≤ d ≤ 89 and of order d 2 − d for any even d ≤ 34.
As far as we know, our result for d odd and 21 ≤ d ≤ 89 provides the highest known orders for weak-foci of planar polynomial equation of any of these degrees. The orders of the weak-foci obtained when d is even coincide with the ones given in [2] and are lower than the ones obtained in [16,23]. 3.7. Case (f ) in Theorem 1.2. As in the previous section, we can restrict our attention to solve the center problem for equation (6): for some 0 = C ∈ C and αβ = 0. Moreover, because of the results of previous section, we only need to consider the nonhomogeneous nonlinearities case: 5 ≤ Z ≤ 36, with Z = k + + m + n, k + < m + n, and fix k, , m, and n.
We follow the next procedure. If the fixed values k, , m, and n satisfy one of the conditions: (i) Case = n = 0 (holomorphic center); or (ii) Case k = m and α = 0 (possible Hamiltonian or Darboux center); there is nothing to be computed, because we already know the centers by using the previous results. Otherwise, we start computing the first significant Poincaré-Lyapunov constant. In all cases of our study this constant decides the center-focus problem and gives a reversible center.
To illustrate the procedure, in Table 4 we show the results for the simplest case Z = 5, k + = 2 < 3 = m + n. We comment that the CPU time needed to complete this study is of several weeks.
The total number of cases satisfying 5 ≤ Z ≤ 36 and k + < m + n is 42 448. Moreover, 37 631 of them (88.7%) turn out to be of reversible type. These are the cases for which the computation of the Poincaré-Lyapunov constant is necessary. Using the classical algorithm introduced by Lyapunov [19,26], adapted to complex coordinates, we have found that the first significant Poincaré-Lyapunov constant, V N , corresponds to N ∈ {N 1 , N 1 + 2P M, N 1 + 2QM }, where N 1 = pP M + |q|QM + 1 (19) in 86.57%, 11.89% and 1.52% of the cases, respectively. Only six cases (0.02%) do not satisfy (19). They are listed in Table 5.
The second case corresponds to α = 3, β = −2, P = 3, Q = 14, p = 2, q = 3, and M = 1. Here, looking at (19), the expected values of N would be 49, 55, 77 but the good one is 61, as we have also explained in Section 3.5. The other four special cases can be studied similarly.
The highest computational difficulties have to different origins. The first one is the big amount of cases to be studied for a fixed value of Z. The second one is due to some special cases that need a lot of CPU time. Usually, these are the cases with = 0 and m = 0, 1. For example, for Z ≤ 20, the cases that have needed the biggest CPU time have been (k, , m, n) = (7, 0, 0, 12) and (k, , m, n) = (8, 0, 1, 11). For the first we have got that where α = q = 6, β = −p = −13, P = 6, Q = 11, and M = 1. For the second one, where α = q = 7, β = −p = −11, P = 7, Q = 11, and M = 1. It turns out that this last example is also one of the cases with the highest order weak-focus when Z ≤ 20. For 21 ≤ Z ≤ 36, due to their huge expressions, we do not give the Poincaré-Lyapunov constants (V 609 ) corresponding to the cases with highest order weak-foci. Nevertheless, it is worth mentioning that in our study, for a given total degree d, the order of the highest order weak-focus obtained when the nonlinearities are homogeneous (case (e)) is bigger than the corresponding one obtained in this case, where the nonlinearities are nonhomogeneous.
In all the presented cases, the significant Poincaré-Lyapunov constant V N needed to finish the study satisfies N ≤ pP + (|q| + 2)Q + 1, value that corresponds to the maximum of the three values of (19). Moreover, it writes as for some non zero rational constant D and s ∈ {0, 1}. This result supports once more the fact that all centers are as we expect, see (4).

Appendix. Computation of the Poincaré-Lyapunov constants.
Apart of the well-known method introduced by Lyapunov, see [19,26], the other algorithm that we use for proving cases (e) and (f) of Theorem 1.2 relies on the following theoretical result, that is given in [10], and which in turn is based on the results of [7]. This second algorithm is also used by the software P4 (Polynomial Planar Phase Portraits), introduced in [6, Sec. 9-10].
Theorem 4.1. Consider the differential equatioṅ or the equivalent expression dH(z,z) + ω 1 (z,z) + ω 2 (z,z) + · · · = 0, where H(z,z) = 1 2 zz and ω j (z,z) = 2 Im(F j+1 (z,z)dz), for all j ∈ N. If V 2 = V 3 = · · · = V N −1 = 0 then its N -th Poincaré-Lyapunov constant is To apply the above result for obtaining V N we need to compute the given integral and to obtain the functions h j . The first goal is solved by the following lemma. where coeff(f, z kzk ) denotes the coefficient of the monomial z kzk of f for any k.
To achieve the second goal and to facilitate the implementation of Theorem 4.1 we introduce some operators. Definition 4.3. Let P be the set of all complex polynomials in z,z variables and set G(z,z) = g k, z kz ∈ P of degree n. We consider, for each j ≥ 2, the following operators G, F, F j : P −→ P and H j : P −→ R, where F j are given in (20). Notice that they are not defined on the whole space P.
Next lemma shows how to find the functions h j .
In [5], the Poincaré-Lyapunov constants are written in term of some "words". The procedure suggested by Theorem 4.1 gives a different expression of the Poincaré-Lyapunov constants in terms of words, where the "words" are given by the homogeneous components of the differential equation (20). Next two results are the ones implemented to find the centers of families (e) and (f) of Theorem 1.2, respectively. In the particular case that the nonlinearities of (20) are homogeneous, next corollary simplifies the expression of the V N given above.
Corollary 4.6. Let F d (z,z) be a homogeneous polynomial of degree d. Then, the only significant Poincaré-Lyapunov constants of the equationż = iz + F d (z,z) are V d+j(d−1) = H d (F d (F d ( j · · ·(F d (1))))) := H d (F j d (1)), where j is any natural number (resp. any odd natural number) when d is odd, (resp. even).