Center problem for a class of degenerate quartic systems

This paper, using pseudo-division algorithm, introduces a method for computing resonant focus numbers of a class of complex polynomial differential systems, establishes the necessary and sufficient conditions for existence of a center for a class of complex quartic systems with a degenerate resonant singular point.


Introduction
In the qualitative theory of real planar differential systems, focal values and saddle values are two important detection quantities.In [2], the authors introduce a new efficient computational method, which combines the computation of focal values and saddle values into a unified calculation of singular point quantities for a class of complex planar differential systems.Using pseudo-divisions, Wang [21] gives an improved formal power series method for computing focal values of a class of polynomial differential systems.Using a perturbation technique based on multiple time scales, Yu [25] presents an efficient method for computing focal values of some classes of differential systems.
For system (1.1), we can derive a formal power series of the form (1.2) with B s(p+q),sp = 0, s = 2, 3, . . ., such that dF dt (1.1) where W n are called the n-th order p : −q resonant focus numbers.For some computational methods of such quantities, see [14,19].For large n, the computation of W n is very complicated, which is the main reason of slow progress in the center problems.
The only way to get the necessary conditions for a center is to compute the p : −q resonant focus numbers.Before presenting a new algorithm, we start with a precise definition of pseudo-remainder of polynomials.For more details, see [6,[22][23][24].
Let K[x 1 , x 2 , . . ., x n ] denote the ring of polynomials in indeterminates x 1 , x 2 , . . ., x n with coefficients in a field K of characteristic 0. Consider a fixed ordering on the set of indeterminates: is the maximum index such that f has a positive degree in x i .The class of elements of K is zero.If f is of class i the coefficient of the x i of the maximum degree is said to be the initial of polynomial f and is denoted by In( f ).
If f and g are two polynomials of class respectively i and j, with i < j, or such that i = j and the degree in x i of f is less than the degree of g, then it is possible, using the Euclidean algorithm over K(x 1 , x 2 , . . ., x i−1 )[x i ] to find polynomials q and r with deg The polynomial r is called the pseudo-remainder of g with respect to f , and it is denoted by prem(g, f ).This operation is called pseudodivision.

Definition 1.3 ([22]
).Consider a triangular set AS = [ f 1 , f 2 , . . ., f r ], and a polynomial g ∈ K[x 1 , x 2 , . . . ,x r ].Let us pseudo-divide g by f r , f r−1 , . . ., f 1 successively as polynomials in x c r , . . . ,x c 1 , c i = class( f i ), and denote the final remainder by R. Then we shall get an expression of the form: where I i is the initial of f i , s i assumes the smallest possible power achievable.R is called the pseudo-remainder of g with respect to AS, denoted as R = Prem(g, AS).
Now we are in a position to develop the algorithm for computing W n in (1.3).Grouping the like terms in the second expression of (1.3), we get When computing the n-th order resonant focus number W n , the coefficients f l,j , f (p+q)(n+1),j have to be zero.Thus in order to eliminate indeterminates B k,j from V n , we use successive pseudo-divisions: first choosing a suitable variable order of B k,j ; secondly, rearranging some polynomials f l,j , f (p+q)(n+1),j to get a triangular set TS n ; finally, performing successive pseudodivision of V n + v by TS n to get the pseudo-remainder R n , then the n-th order p : −q resonant focus number can be written as in the polynomial R n , and v is a new variable.
To illustrate the main idea of the algorithm, we compute the second order 1 : −2 resonant focus number W 2 of the family ( Let and using the same notations as described in the algorithm, we have where Under the variable ordering the following sequence of polynomials B. Sang is a triangular set, where By computing pseudo-remainder of V 2 + v by TS 2 , one gets Hence the second order 1 : −2 resonant focus number can be written as A general purposed Maple package Myvalue based on our algorithm is developed in Maple V.18 on Intel Core 2 Quad CPU Q8400, 4G RAM, and such Maple package is available for noncommercial purpose via email to: sangbo_76@163.com.Another Maple package Liuc based on the method [14] is also developed by us using the same computing platform.For technical comparison of these two packages, let us consider a class of cubic differential systems Center problem for a class of degenerate quartic systems 5 in C 2 , and Computing the first eight 1 : −1 resonant focus numbers W j , 1 ≤ j ≤ 8 by Myvalue and Liuc respectively, we find that the outputs (in expanded form) are the same for these two methods, and get the following experimental results on efficiency, see For computing W n with n large, it is worth noting that the expansion of long polynomials in the last stage of package Liuc is pretty time-consuming, whereas the package Myvalue does not need any expansions before generating its outputs.
Consider the system of differential equations where The polynomial h is called a cofactor.If h ≡ 0 then f (x, y) = const is a first integral of system (1.5).
Mattei and Moussu [16] proved the next result for all isolated singularities.
Lemma 1.6.Assume that system (1.1) with an isolated singularity at the origin has a formal first integral F(x, y) ∈ R[[x, y]] around it.Then, there exists an analytic first integral around the singularity.
Another mechanism to prove the integrability of system (1.5) is time-reversibility.From [20], we have the following result.

B. Sang
Lemma 1.7.System (1.5) is time-reversible with respect to a transformation where γ is a nonzero scalar, if and only if,

Main result
In the qualitative theory of planar differential systems, there are few works about degenerate singular point.Most of the work focuses on the center problem of the system where P, Q are polynomials in x and y with degree no less than two, see [1,7,9,17].
Let us consider the real analytic system where U(u, v), V(u, v) are analytic in a sufficiently small neighborhood of the origin, U k (u, v), V k (u, v) are homogeneous polynomials of degree k, and n ≥ 0. Because the singularity (u, v) = (0, 0) of system (2.1) has no characteristic directions, it is a center or a focus.Under the transformation u = r cos(θ), v = r sin(θ), system (2.1) becomes It can be written as where the function on the right side of (2.3) is convergent in the range θ ∈ [−4π, 4π], r < r 0 , and For sufficient small h, let be the Poincaré successor function and the solution of (2.3) satisfying the initial value condition r| θ=0 = h.

Lemma 2.2 ([14]
). System (2.9) has a complex center at the origin if and only if there exists a non-zero real number s and a first integral of the form where fm(2n+3) are homogeneous polynomials of degree m(2n + 3).The power series in (2.12) has a non-zero convergence radius.

Definition 2.3 ([13-15]
).For any positive integer m, the number µ m is called the m-th singular point value of system (2.7) at the origin.And v 2m+1 (2π) ∼ iπµ m is called the m-th focal value of system (2.1) at the origin.
Lemma 2.5.The origin of system (2.7) is a complex center if and only if the origin of system (2.9) is a complex center.
Proof.Necessity.Suppose that system (2.7) has a complex center at the origin, then for all m, µ m = 0. Hence system (2.9) has a formal first integral F(x, y) of the form (2.10), so by Lemma 1.6, it has a complex center at the origin.Sufficiency.Suppose that system (2.9) has a complex center at the origin , then by Lemma 2.2 it has an analytic first integral F(x, y) of the form (2.12).Thus it also has an analytic first integral of the form F(x, y) = [ F(x, y)] 1 s , which implies µ m = 0 for all m by Lemma 2.1, and so that the origin of system (2.7) is a complex center.
Because system (2.9) is integrable at the origin if and only if the origin of it is a complex center, we have the following theorem.
Theorem 2.6.The origin of system (2.7) is a complex center if and only if system (2.9) is integrable at the origin.
As a consequence of Theorem 2.6, we have the following corollary.
Corollary 2.7.The origin of the real system (2.1) is a center if and only if system (2.9) is integrable at the origin.
The authors of [26] obtain the center conditions of the following system: In this paper, we consider the center problem of a class of complex quartic systems where Using the non-linear change (2.8) for n = 1, system (2.13) becomes (2.15) Applying our method to compute the first thirty 1 : −1 resonant focus numbers, we get W 1 , W 2 , . . . ,W 30 , where the quantity W k is reduced w.r.t. the Gröbner basis of W j : j < k .
and W 25 , W 30 are very complicated so we do not present these polynomials here, but the interested reader can easily compute them using any computer algebra system.If condition (2.14) holds, by applying suitable non-degenerate similarity transformation and time scaling, system (2.13) becomes one of the two forms: (2.16) (2.17) where Theorem 2.9.If condition (2.18) holds, system (2.17) has a complex center at the origin if and only if one of the following conditions holds: 3 Proof of the Theorems 2.8 and 2.9 Using the transformation (2.8) for n = 1, system (2.16) becomes Proof.Necessity.Let W 1 , W 2 , . . ., W 30 be the first thirty 1 : −1 resonant focus numbers of system (2.15).By substituting a 2 = 1, b 2 = −1 into these numbers respectively, one gets the first thirty 1 : −1 resonant focus numbers W 1 , W 2 , . . ., W 30 of system (3.1).Computing a Gröbner basis of the ideal W 1 , W 2 , . . ., W 30 with respect to the graded reverse lexicographical order with b 1 a 3 b 3 a 1 , we obtain a list of polynomials Sufficiency.In the case a 1 = −b 3 , a 3 = −b 1 , system (3.1)takes the form From Lemma 1.7, we know that system (3.2) is time-reversible w.r.t. the transformation x → y, y → x.So by the symmetry principle, the origin of such system is a resonant center, and hence system (3.2) is integrable at the origin.Using the transformation (2.8) for n = 1, system (2.17) becomes 3) is integrable at the origin if and only if one of the following conditions holds: ( Proof.Necessity.Let W 1 , W 2 , . . ., W 30 be the first thirty 1 : −1 resonant focus numbers of system (2.15).By substituting a 2 = b 2 = 0 into these numbers respectively, one gets the first thirty 1 : −1 resonant focus numbers W 1 , W 2 , . . . ,W 30 of system (3.3).
Computing a Gröbner basis of the ideal W 1 , W 2 , . . ., W 30 with respect to the graded reverse lexicographical order with b 1 a 3 b 3 a 1 and we get a list of polynomials The vanishing of G gives rise to six cases in the Lemma.Sufficiency.When condition (1) holds, system (3.3) is of the form , so by the symmetry principle, the origin of it is a resonant center, and hence system (3.4) is integrable.
Hence, we have proved that system (3.5)admits a formal first integral of the form F(x, y) = ∑ ∞ n=1 v n (y)x n .Consequently it has an analytic first integral in some neighborhood of the origin.
If condition (3) holds, system (3.3) is reduced to (3.8) We will show that for system (3.8)there exists a formal first integral in the form F(x, y) = ∑ ∞ n=1 v n (x)y n , where functions v n (x) should satisfy the first-order linear differential equation Solving this equation, we obtain taking the integration constants equal to 1.We will show by induction that the functions v n (x) are polynomials of degree n.Hence, we assume that for k = 1, 2, . . ., n − 1 there exist k-th degree polynomials v k (x) satisfying (3.9).We then solve the linear differential equation (3.9) for k = n and obtain where taking the integration constant equals to 1. Using the induction hypothesis that we find that the degree of g n−3 (x) is at most n − 3. Now, we must study whether the integral can give any logarithmic terms.Therefore, we must prove that terms involving x −1 do not appear in the integrand of (3.10).Since the exponents that can appear in the integrand are of the form −(s + 4), s = 0, 1, 2, . . ., n − 3, there can be no logarithmic terms in (3.10) and v n (x) is an n-th degree polynomial in x.
Hence, we have proved that system (3.8)admits a formal first integral of the form (3.12) We will show that for system (3.12)there exists a formal first integral in the form F(x, y) = ∑ ∞ n=1 v n (y)x n , where functions v n (y) should satisfy the first-order linear differential equation  taking the integration constants equal to 1.We will show by induction that the functions v n (y) are polynomials of degree n.Hence, we assume that for k = 1, 2, . . ., n − 1 there exist k-th degree polynomials v k (y) satisfying (3.13).We then solve the linear differential equation we find that the degree of g n−3 (y) is n − 3. Now, we must study whether the integral can give any logarithmic terms.Therefore, we must prove that terms involving y −1 do not appear in the integrand of (3.14).Since the exponents that can appear in the integrand are of the form −(s + 4), s = 0, 1, 2, . . ., n − 3, there can be no logarithmic terms in (3.14) and v n (y) is an n-th degree polynomial in y.
Hence, we have proved that system (3.12)admits a formal first integral of the form F(x, y) = ∑ ∞ n=1 v n (y)x n .Consequently it has an analytic first integral around the origin.If condition (6) holds, system (3.(3.15) By the transformation x → y, y → x, t → −t, system (3.15) can be transformed into the form of (3.5), and therefore system (3.15) is integrable at the origin.

2 B. Sang Definition 1 . 1 .
j x k−j y j , Y m (x, y) = m ∑ k=2 k ∑ j=0 b k,j x k−j y j .Email: sangbo_76@163.comSystem (1.1) is said to have a p : −q resonant center at the origin if it admits a local first integral of the form F(x, y) = x q y p +

From
x)y n .Consequently it has an analytic first integral around the origin.If condition (4) holds, system (3.3) is of the form Lemma 1.7, we know that system (3.11) is time-reversible w.r.t. the transformationx → γ 0 y, y → γ 0 −1 x, where γ 0 = 5 −a 3 b 1 ,B.Sang so by the symmetry principle, the origin of system is a resonant center, and hence system (3.11) is integrable.If condition (5) holds, system (3.3) is of the form

Table 1 .
1: Computing times (in CPU seconds) for the first eight resonant focus numbers If condition (2.18) holds, system (2.16) has a complex center at the origin if and only if a 1