The Center Problem for Homogeneous Case of Polynomial Maps

Abstract

We study the center problem for polynomial maps $y=f\left(x\right)=-x-{\displaystyle \sum _{n=1}^{\infty }}{a}_n{x}^{n+1},$ arising from homogeneous algebraic curves $x+y+{\displaystyle \sum _{i=0}^n}{\alpha }_{n-i,i}{x}^{n-i}{y}^i=x+y+H_n(x,y)=0.$ While explicit conditions were previously known only for low even degrees $n=2,4,6,8,10$, their general structure remained conjectural. In this paper we resolve the case $n=12$ and prove that for all even degrees $n=2k$, the center condition is completely characterized by two families of algebraic relations: mirror symmetry conditions and alternating-sum conditions. The proof combines algebraic methods with a direct structural argument. In particular, the necessity part is established without relying on explicit formulas for focus quantities, instead, we make use of the involutive property of the associated map and analyze the symmetric difference $H_n(x,f\left(x\right))-H_n(f\left(x\right),x),$ which leads to a simple and rigorous characterization of the center condition. This provides a complete and conceptually transparent solution of the homogeneous center problem for polynomial maps.

1. Introduction

The center problem is a classical topic in the theory of dynamical systems, originating from the study of planar polynomial differential equations and the characterization of periodic orbits. In the continuous case, the problem reduces to determining when a singular point is surrounded by closed trajectories, which is closely related to the vanishing of focus quantities and the structure of the Bautin ideal.

Over time, the concepts of center and focus have been extended beyond the continuous case of differential equations to discrete dynamical systems [1,2]. In discrete systems, such as polynomial maps arising from algebraic curves, iterations can be locally conjugate to rotations and thus exhibit center-type behavior without spiraling. In this context, the center condition is characterized by the identity $f\left(f\right(x\left)\right)=x$. This provides a natural discrete analogue of the classical center problem and leads to a rich interplay between algebraic geometry and dynamical systems. This discrete analogue has been extensively studied in connection with classical models, including discrete Lotka–Volterra systems [3,4].

Several methodological approaches have been developed to tackle the center problem. On the analytical side, works by Ilyashenko and Yakovenko clarified the structure of analytic differential equations [5]. On the computational side, algebraic geometry and computer algebra have become indispensable: in computer algebra system Singular [6], using its library primdec.lib [7], we can compute the decomposition of the variety of the ideal, generated by first few focus quantities. In practice, one considers the chain of ideals

$$I_k=\langle p_2,p_3,\dots ,p_k\rangle$$

which stabilizes by Hilbert’s Basis Theorem, incrementally reduces new relations modulo a chosen Gröbner basis of the current ideal, and uses a radical test to check whether new information is already contained in $\sqrt{I_k}$. This yields an algorithmic description of the center variety, consistent with the algebraic viewpoint of ideals and varieties [8,9].

Within this broad context, the present paper focuses on the homogeneous case of polynomial maps. Previous work has addressed specific values of the degree, such as $n=2,4,6,8,$ and 10, leading to a conjectural picture for general even n [1,2,10]. In this article, we resolve the open case $n=12$ and prove that the conjecture holds for arbitrary even degrees. Our results contribute both to the algebraic theory of dynamical systems and to the computational study of discrete maps, as they provide a complete characterization of the center condition in this case. We show that the center occurs if and only if one of two algebraic conditions is satisfied: the mirror symmetry condition $\left(S_{2k}\right)$ or the alternating-sum condition $\left(T_{2k}\right)$ (see

Table 1. Conditions for the existence of a center at the origin for Equation (3). Results for $n=2,4,6,8$ and 10 are known from [10,15], the case $n=12$ is obtained in this work.

n	Conditions
2	$S_2$: ${\alpha }_{2,0}-{\alpha }_{0,2}=0$
	$T_2$: ${\sum }_{k=0}^2{(-1)}^k{\alpha }_{2-k,k}=0$
4	$S_4$: ${\alpha }_{4,0}-{\alpha }_{0,4}={\alpha }_{3,1}-{\alpha }_{1,3}=0$
	$T_4$: ${\sum }_{k=0}^4{(-1)}^k{\alpha }_{4-k,k}=0$
6	$S_6$: ${\alpha }_{6,0}-{\alpha }_{0,6}={\alpha }_{5,1}-{\alpha }_{1,5}={\alpha }_{4,2}-{\alpha }_{2,4}=0$
	$T_6$: ${\sum }_{k=0}^6{(-1)}^k{\alpha }_{6-k,k}=0$
8	$S_8$: ${\alpha }_{8,0}-{\alpha }_{0,8}={\alpha }_{7,1}-{\alpha }_{1,7}={\alpha }_{6,2}-{\alpha }_{2,6}={\alpha }_{5,3}-{\alpha }_{3,5}=0$
	$T_8$: ${\sum }_{k=0}^8{(-1)}^k{\alpha }_{8-k,k}=0$
10	$S_{10}$: ${\alpha }_{10,0}-{\alpha }_{0,10}={\alpha }_{9,1}-{\alpha }_{1,9}={\alpha }_{8,2}-{\alpha }_{2,8}={\alpha }_{7,3}-{\alpha }_{3,7}={\alpha }_{6,4}-{\alpha }_{4,6}=0$
	$T_{10}$: ${\sum }_{k=0}^{10}{(-1)}^k{\alpha }_{10-k,k}=0$

). These conditions define two natural components of the center variety. In particular they are consistent with and extend the center varieties obtained in all previously studied low-dimensional even degree cases. They also complement related developments on bifurcation phenomena and planar vector fields [11,12,13,14,15].

We now turn to the specific case of discrete polynomial maps. With this in mind, we now consider maps of the form

$$f\left(x\right)=-x-{\displaystyle \sum _{n=1}^{\infty }}{a}_n{x}^{n+1},$$

(1)

where $f:\mathbb{R}\to \mathbb{R}$ denotes an irreducible component of an algebraic curve defined by

$$x+y+{\displaystyle \sum _{i+k=2}^n}{\alpha }_{i,k}{x}^i{y}^k=0.$$

(2)

Such equations have been studied in [1,2,10,16] and provide a natural discrete analogue of planar differential systems. If one solves (2) for y and interprets the result as a map $f\left(x\right)$, then the dynamics of its iterations describe a discrete dynamical system with a nontrivial (nonhyperbolic) singular point at $x=0$. While a number of results are known for low-degree cases, the structure of the center problem for higher degrees is more subtle. Building on earlier studies for $n=2,4,6,8,$ and 10 [10,15], we investigate the existence of a center at the origin for (2) when $n=12$ and extend the conclusions to all even $n\in \mathbb{N}$.

2. Preliminary Results for Center of Maps (1)

The aim of this section is to reformulate the center problem in an algebraic framework that allows effective computation. The key idea is to replace the dynamical condition defining a center with a system of polynomial equations in the coefficients of the map.

This is achieved through the Poincaré return map and the associated focus quantities, whose vanishing characterizes the center. We briefly recall these constructions and explain how they lead to an algebraic description of the center variety. Here we provide an overview of key definitions and results related to the dynamics of maps defined by Equation (1). For more details, see [1,2,10].

If one solves Equation (2) for y and interprets the solution as a function $f\left(x\right)$, then the dynamics of its iterations describe a discrete dynamical system possessing a nontrivial (nonhyperbolic) singular point at $x=0$. The special situation where Equation (2) takes the form

$$x+y+{\displaystyle \sum _{i=0}^n}{\alpha }_{n-i,i}{x}^{n-i}{y}^i=x+y+H_n(x,y)=0,$$

(3)

where $H_n$ is a homogeneous polynomial of degree n. We refer to this as the homogeneous case of degree n.

For the map (1), we denote the p-th iterate of f by $f^p\left(x\right)$. The singular point $x=0$ of the map f is then classified as:

• a stable focus, if there exists $\epsilon >0$ such that for all $\left|x\right|<\epsilon$, ${lim}_{n\to \infty }{f}^n\left(x\right)=0$,
• an unstable focus, if it is a stable focus for the inverse map $f^{-1}$,
• a center, if there exists $\epsilon >0$ such that for all $\left|x\right|<\epsilon$, $f^2\left(x\right)=x$.

These notions mirror the classification of equilibrium points in continuous dynamical systems, but in the discrete case, they are expressed through the iterates of the map rather than through trajectories of a flow. We emphasize that the center condition $f^2\left(x\right)=x$ does not, in general, coincide with the symmetry condition $F(x,y)=F(y,x)$, $\forall x,y\in \mathbb{R}$. The latter represents only one possible mechanism producing a center.

2.1. Poincaré Map and Focus Quantities

To make this classification effective, one needs algebraic tools that detect whether the origin is a center or a focus. The key idea is to characterize the center condition using the Poincaré return map and its associated focus quantities.

Let us examine Equation (2), where the coefficients ${\alpha }_{i,k}\in \mathbb{R}$. This equation admits a unique analytic solution expressible in the form of (1). Based on the analysis of such solutions, the following question arises: how can one characterize, within the space of coefficients $\left\{{\alpha }_{i,k}\right\}$, the manifold on which the associated map f has a center at the origin?

In order to study that, we define the Poincaré return map

$$\mathcal{R}\left(x\right)=f^2\left(x\right)=x+p_2{x}^3+p_3{x}^4+···$$

(4)

and the difference map

$$\mathcal{P}\left(x\right)=f^2\left(x\right)-x=p_2{x}^3+p_3{x}^4+···$$

(5)

By definition, a map f has a center at the origin if $\mathcal{P}\left(x\right)=0$ holds for all $\left|x\right|<\epsilon$, where the coefficient $p_m$ is called the m-th focus quantity. All focus quantities of the Poincaré map are polynomials in the coefficients $\left\{{\alpha }_{i,k}\right\}$ of (2), and for a center at the origin all must vanish simultaneously. The importance of the focus quantities lies in the fact that they translate a dynamical property (the existence of a center) into algebraic conditions. In practice, instead of studying the iterates of the map directly, one computes the coefficients $p_m$ and imposes their vanishing.

In this formulation, the coefficients $p_m$ play the same role as Lyapunov quantities in continuous systems: their vanishing characterizes the presence of a center.

2.2. Algebraic Reformulation via Ideals

Since an infinite number of focus quantities are in principle involved, it is natural to rephrase the problem in terms of commutative algebra, where ideals and varieties encode the common vanishing of polynomial conditions.

To find conditions for the center at the origin we introduce

$$I=〈p_m:m\in \mathbb{N},m>1〉,$$

the ideal generated by all focus quantities, and its variety $\mathbf{V}\left(I\right)$, which consists of all common zeros of polynomials in I. Since I is generated by infinitely many focus quantities, it cannot be computed directly. We therefore define truncated ideals

$$I_k=〈p_2,p_3,\dots ,p_k〉.$$

If $m\ge k$, then $I_k\subseteq I_m$, and we obtain an ascending chain of ideals. By Hilbert’s basis theorem [8], this chain stabilizes, meaning there exists $n\in \mathbb{N}$ such that $I_k=I_n$ for all $k\ge n$.

If $J\subset k[x_1,x_2,\dots ,x_n]$ is an ideal, the radical of J is defined by

$$\sqrt{J}=\{f\in k[x_1,x_2,\dots ,x_n]:\exists p\in \mathbb{N},f^p\in J\}.$$

The radical ideal $\sqrt{J}$ determines the same variety as J, so $\mathbf{V}\left(\sqrt{J}\right)=\mathbf{V}\left(J\right)$. This reduction from infinitely many focus quantities to a finite generating set is a key structural feature of the problem and underlies the possibility of obtaining explicit center conditions. This algebraic reformulation is crucial for determination of the center variety.

2.3. Constructing the Center Variety

To determine the center variety associated with the focus quantities of the Poincaré map for (2), we proceed iteratively. For illustration, in low-degree cases one computes the first few focus quantities explicitly and observes that they factor into simple algebraic expressions. These expressions suggest the general form of the center conditions, which we later confirm for arbitrary even degree. We compute the first non-zero focus quantity, denoted by $p_{k_1}$, and set $G_1=\left\{p_{k_1}\right\}$. We then compute the next non-zero quantity, $p_{k_2}$, reduce it modulo $\langle p_{k_1}\rangle$ (see Definition 1.2.15 in [9]), and apply the Radical Membership Test [9] to check whether there exists $k\in \mathbb{N}$ with ${\tilde{p}}_{k_2}\in {\langle p_{k_1}\rangle }^k$. Here ${\tilde{p}}_{k_2}$ denotes the reduced form of $p_{k_2}$.

If ${\tilde{p}}_{k_2}\notin \sqrt{\langle p_{k_1}\rangle }$, we add it to the generating set, so $G_2=\{p_{k_1},{\tilde{p}}_{k_2}\}$. We then compute the next non-zero quantity $p_{k_3}$, reduce it modulo the Gröbner basis of $\langle p_{k_1},{\tilde{p}}_{k_2}\rangle$, and repeat the procedure. Continuing this way, we arrive at the smallest index n such that

$${\tilde{p}}_{k_{n+1}}\in \sqrt{G_n},G_n=\{p_{k_1},{\tilde{p}}_{k_2},\dots ,{\tilde{p}}_{k_n}\}.$$

This produces a nested chain of radical ideals

$$\sqrt{G_1}\subset \sqrt{G_2}\subset \cdots \subset \sqrt{G_n},$$

from which the center variety can be computed.

Finally, one needs to prove that $\mathbf{V}\left(I\right)=\mathbf{V}\left(G_n\right)$. It is sufficient to note that $\mathbf{V}\left(I\right)\supset \mathbf{V}\left(G_n\right)$, since the reverse inclusion is obvious. By the theorem [8], the variety $\mathbf{V}\left(G_n\right)$ decomposes as $V_1\cup V_2\cup \dots \cup V_n$, where each $V_i$ is irreducible. For each component, to ensure its sufficiency we prove, that the corresponding map has the center at the origin.

The outcome of this procedure is a finite set of generating conditions whose common zeros define the center variety. This reduction from infinitely many to finitely many conditions is the key step that makes the problem tractable. These definitions and constructions thus provide the formal basis for the analysis of the center problem in homogeneous polynomial maps. In the following section we build on this framework to derive explicit conditions for the existence of a center, first for the case $n=12$ and then, using the involutive structure of the associated map, for arbitrary even degrees.

3. Main Results

Building on the framework introduced in the previous section, we now derive explicit conditions for the existence of a center in the homogeneous case. We first recall the known results for small values of n. These cases illustrate the recurring algebraic patterns that will guide our treatment of the next open case $n=12$.

While several authors have characterized the center conditions for low-degree cases, these results reveal a recurring algebraic structure but not a complete description.

In particular, two distinct algebraic patterns emerge: mirror symmetries in the coefficients, denoted by $\left(S_n\right)$, and alternating-sum conditions, denoted by $\left(T_n\right)$. These features are evident in the known results for $n=2,4,6,8,$ and 10, which are summarized in Table 1.

The next unresolved case in this sequence is $n=12$. Its resolution confirms that the two known families of conditions, $\left(S_n\right)$ and $\left(T_n\right)$, are indeed exhaustive and, at the same time, provides the final step toward a general characterization valid for all even degrees.

In the sequel, we first consider the equation

$$x+y+{\displaystyle \sum _{k=0}^{12}}{\alpha }_{12-k,k}{x}^{12-k}{y}^k=0.$$

(6)

Theorem 1. 

The Equation (6) defines a center at the origin if and only if one of the following conditions holds:

1.

($S_{12}$): ${\alpha }_{12,0}-{\alpha }_{0,12}={\alpha }_{11,1}-{\alpha }_{1,11}={\alpha }_{10,2}-{\alpha }_{2,10}={\alpha }_{9,3}-{\alpha }_{3,9}={\alpha }_{8,4}-{\alpha }_{4,8}={\alpha }_{7,5}-{\alpha }_{5,7}=0$.

2.

($T_{12}$): ${\displaystyle {\displaystyle \sum _{k=0}^{12}}{(-1)}^k{\alpha }_{12-k,k}=0}$

Proof. 

We consider the case $n=12$ using an approach based on the Poincaré map (4).

Let f be a function of the form (1). It has been proven that Equation (2) has a center at the origin if and only if the solution y is of the form f, which itself has a center at the origin. From this we see that the coefficients $a_i$ of (1) are polynomials in the parameters

$$\{{\alpha }_{12,0},{\alpha }_{11,1},{\alpha }_{10,2},{\alpha }_{9,3},{\alpha }_{8,4},{\alpha }_{7,5},{\alpha }_{6,6},{\alpha }_{5,7},{\alpha }_{4,8},{\alpha }_{3,9},{\alpha }_{2,10},{\alpha }_{1,11},{\alpha }_{0,12}\}.$$

(7)

This means that the focus quantities of the Poincaré map are also polynomials in the same parameters. We see that every coefficient $a_{11m}$ ($m\in \mathbb{N}$) is non-zero, i.e., $a_{11}\ne 0$, $a_{22}\ne 0$, $a_{33}\ne 0$, etc.

We obtain

$$\begin{array}{cc}\hfill a_{11}& ={\alpha }_{12,0}-{\alpha }_{11,1}+{\alpha }_{10,2}-{\alpha }_{9,3}+{\alpha }_{8,4}-{\alpha }_{7,5}+{\alpha }_{6,6}-{\alpha }_{5,7}+{\alpha }_{4,8}-{\alpha }_{3,9}+{\alpha }_{2,10}-{\alpha }_{1,11}\hfill \\ \hfill & + {\alpha }_{0,12}\hfill \\ \hfill a_{22}& =(12{\alpha }_{0,12}-11{\alpha }_{1,11}+10{\alpha }_{2,10}-9{\alpha }_{3,9}+8{\alpha }_{4,8}-7{\alpha }_{5,7}+6{\alpha }_{6,6}-5{\alpha }_{7,5}+4{\alpha }_{8,4}-3{\alpha }_{9,3}\hfill \\ \hfill & + 2{\alpha }_{10,2}-{\alpha }_{11,1}\left)\right({\alpha }_{0,12}-{\alpha }_{1,11}+{\alpha }_{2,10}-{\alpha }_{3,9}+{\alpha }_{4,8}-{\alpha }_{5,7}+{\alpha }_{6,6}-{\alpha }_{7,5}+{\alpha }_{8,4}-{\alpha }_{9,3} +\hfill \\ \hfill & {\alpha }_{10,2}-{\alpha }_{11,1}+{\alpha }_{12,0})\hfill \\ \hfill & ⋮\hfill \end{array}$$

The computation of the first focus quantities provides insight into the structure of the center conditions. These computations are used as motivation, and the general result will be proved by a different method. We compute the first few focus quantities of the Poincaré map and we obtain that $p_{22},p_{44},\dots$ are non-zero. As described in the previous section, we reduce $p_{44}$ modulo the Gröbner basis of the ideal $\langle p_{22}\rangle$ and denote the remainder by ${\tilde{p}}_{44}$. So we obtain

$$\begin{array}{cc}\hfill p_{22}& =2(6{\alpha }_{0,12}-5{\alpha }_{1,11}+4{\alpha }_{2,10}-3{\alpha }_{3,9}+2{\alpha }_{4,8}-{\alpha }_{5,7}+{\alpha }_{7,5}-2{\alpha }_{8,4}+3{\alpha }_{9,3}-4{\alpha }_{10,2} +\hfill \\ \hfill & 5{\alpha }_{11,1}-6{\alpha }_{12,0}\left)\right({\alpha }_{0,12}-{\alpha }_{1,11}+{\alpha }_{2,10}-{\alpha }_{3,9}+{\alpha }_{4,8}-{\alpha }_{5,7}+{\alpha }_{6,6}-{\alpha }_{7,5}+{\alpha }_{8,4}-{\alpha }_{9,3}\hfill \\ \hfill & + {\alpha }_{10,2}-{\alpha }_{11,1}+{\alpha }_{12,0})\hfill \\ \hfill {\tilde{p}}_{44}& ={\displaystyle \frac{1}{108}}(55{\alpha }_{1,11}-80{\alpha }_{2,10}+81{\alpha }_{3,9}-64{\alpha }_{4,8}+35{\alpha }_{5,7}-35{\alpha }_{7,5}+64{\alpha }_{8,4}-81{\alpha }_{9,3} +\hfill \\ \hfill & 80{\alpha }_{10,2}-55{\alpha }_{11,1}\left)\right(-12{\alpha }_{0,12}+11{\alpha }_{1,11}-10{\alpha }_{2,10}+9{\alpha }_{3,9}-8{\alpha }_{4,8}+7{\alpha }_{5,7}-6{\alpha }_{6,6}+5{\alpha }_{7,5}\hfill \\ \hfill & -4{\alpha }_{8,4}+3{\alpha }_{9,3}-2{\alpha }_{10,2}+{\alpha }_{11,1}{)}^2({\alpha }_{0,12}-{\alpha }_{1,11}+{\alpha }_{2,10}-{\alpha }_{3,9}+{\alpha }_{4,8}-{\alpha }_{5,7}+{\alpha }_{6,6}-{\alpha }_{7,5}\hfill \\ \hfill & + {\alpha }_{8,4}-{\alpha }_{9,3}+{\alpha }_{10,2}-{\alpha }_{11,1}+{\alpha }_{12,0}).\hfill \end{array}$$

If we compute ${\tilde{p}}_{66}$ and ${\tilde{p}}_{88}$, we see that ${\tilde{p}}_{66},{\tilde{p}}_{88}\in \sqrt{\langle p_{22},{\tilde{p}}_{44}\rangle }$. By computing the decomposition of the variety $V\left(\langle p_{22},{\tilde{p}}_{44}\rangle \right)$ we obtain both conditions of Theorem 1. Therefore, these two conditions are necessary for the Equation (6) to have a center at the origin.

Finally, we prove that the two conditions are also sufficient. Since $\left(S_{12}\right)$ corresponds to the mirror symmetry of (6), it yields a center at the origin.

We consider the equation

$$x+y+{\displaystyle \sum _{k=0}^{12}}{\alpha }_{k,12-k}{x}^k{y}^{12-k}=(x+y)\left(1+\sum _{i+j=11}{c}_{ij}{x}^i{y}^j\right).$$

Equating coefficients of the same monomials on the left and right-hand side and eliminating parameters $c_{ij}$ gives condition $\left(T_{12}\right)$, which means symmetry with respect to the line $y=-x$ and is equivalent to $f^2\left(x\right)=x$, yielding a center at the origin. □

The case $n=12$ thus confirms that the two algebraic families $\left(S_n\right)$ and $\left(T_n\right)$ remain both necessary and sufficient for the existence of a center. This naturally raises the question of whether the same pattern persists for all even degrees. But the passage from the case $n=12$ to arbitrary even degree is not immediate. To better understand the algebraic structure suggested by the case $n=12$, we analyze the coefficients of the expansion $y=f\left(x\right)$.

Proposition 1. 

Let us consider the Equation (3) for $n=2k$. Then the coefficients of the expansion ${\displaystyle y=f\left(x\right)=-x-{\displaystyle \sum _{n=1}^{\infty }}{a}_n{x}^{n+1}}$ admit a structured factorization of the form

$$a_{n-1}=A,a_{2n-2}=AB,a_{3n-3}=A{B}^2+A^{2}C,$$

where A corresponds to the alternating-sum condition $\left(T_n\right)$, while B and C are linear combinations of the symmetry expressions defining $\left(S_n\right)$ and A.

Proof. 

The statement follows from the recursive determination of the coefficients obtained by substituting the formal power series expansion of f into ${\displaystyle F(x,-x-{\displaystyle \sum _{n=1}^{\infty }}{a}_n{x}^{n+1})=0}$, $\forall x\in \mathbb{R},$ and comparing coefficients of like powers of x (on both sides of the identity). The straight forward computation shows, that the first non-zero term in the expansion of $f\left(x\right)$ is the monomial $x^n$ with coefficient

$$a_{n-1}={\displaystyle \sum _{i=0}^n}{(-1)}^i{\alpha }_{n-i,i}=:A=H_n(1,-1),$$

which is exactly the alternating-sum expression defining $\left(T_n\right)$. Note that the homogeneity of $H_n$ yields $H(x,-x)=x^{n}H(1,-1)$, $\forall x\in \mathbb{R}$.

The same recursive comparison for the next non-zero monomial $x^{2n-1}$ shows that

$$a_{2n-2}=AB,$$

where ${\displaystyle B={\displaystyle \sum _{\ell =2}^k}{(-1)}^{\ell -1}(\ell -1)b_{\ell }+\left(k\right)(-b_1+b_2+{\displaystyle \sum _{\ell =3}^k}{(-1)}^{\ell }{b}_{\ell }+A)}$ and $b_1={\alpha }_{n,0}-{\alpha }_{0,n}$, $b_2={\alpha }_{n-1,1}-{\alpha }_{1,n-1}$, …, $b_k={\alpha }_{k+1,k-1}-{\alpha }_{k-1,k+1}$.

The monomial $x^{3n-2}$ reveals that the next non-zero coefficient in the expansion of the function $y=f\left(x\right)$ is

$$a_{3n-3}=A{B}^2+A^{2}C.$$

In the above expression, C is to lengthy to be written out explicitly. However, as stated in the proposition, C turns out to be a linear combination of the symmetry expressions defining $\left(S_n\right)$ and A. □

Despite the strong computational evidence from focus quantities and their algebraic structure, this approach alone does not yield a complete proof for arbitrary even degrees. In particular, while the factorization patterns observed in low-degree cases and formalized in Proposition 1 suggest that the center conditions depend on a small number of fundamental algebraic components, extending these observations to a full inductive argument becomes increasingly difficult and technically involved.

For this reason, we adopt an approach based on the involutive property $f\left(f\right(x\left)\right)=x$, which provides an intrinsic algebraic characterization of the center condition and leads to a complete proof.

Theorem 2. 

For all even degrees n ($n=2k$), the center condition of the polynomial (3) is completely characterized by the two families:

1.

($S_n$): ${\alpha }_{n,0}-{\alpha }_{0,n}={\alpha }_{n-1,1}-{\alpha }_{1,n-1}={\alpha }_{n-2,2}-{\alpha }_{2,n-2}=\dots =0$

2.

($T_n$): ${\displaystyle {\displaystyle \sum _{i=0}^n}{(-1)}^i{\alpha }_{n-i,i}=0}$.

Proof. 

We prove the necessity and sufficiency separately.

To prove the necessity, assume that the polynomial (3) defines a center at the origin. By definition, the associated map f satisfies

$$f\left(f\right(x\left)\right)=x.$$

Let the defining relation be written in the form

$$F(x,y)=x+y+H_n(x,y)=0.$$

Since $y=f\left(x\right)$ satisfies the equation, we have

$$F(x,f(x\left)\right)=0.$$

Since f is an involution, it also holds that

$$F\left(f\right(x),x)=0.$$

Subtracting these two identities yields

$$H_n(x,f\left(x\right))-H_n(f\left(x\right),x)=0.$$

Define

$$D(x,y)=H_n(x,y)-H_n(y,x).$$

Then D is a homogeneous polynomial of degree n, and we obtain

$$D(x,f(x\left)\right)=0,\forall x\in \mathbb{R}.$$

Now we write the expansion

$$f\left(x\right)=-x-\tau x^n-···,$$

where $\tau =a_{n-1}$ is the first non-zero coefficient after $-x$ and we distinguish two cases.

• Case 1: $\tau \ne 0$.
• We obtain

$${\displaystyle \frac{f\left(x\right)}{x}}=-1-\tau x^{n-1}-···,$$

which is not constant.

• Since D is homogeneous of degree n, we have

$$D(x,f\left(x\right))=x^{n}D\left(1,{\displaystyle \frac{f\left(x\right)}{x}}\right).$$

Since $D(x,f(x\left)\right)=0$, it follows that

$$D\left(1,{\displaystyle \frac{f\left(x\right)}{x}}\right)=0.$$

The function ${\displaystyle \frac{f\left(x\right)}{x}}$ takes infinitely many values near $x=0$, hence the polynomial $z\mapsto D(1,z)$ has infinitely many zeros. Therefore, since $D(1,z)$ is a polynomial and vanishes on an infinite set,

$$D(1,z)=0,$$

and by homogeneity,

$$D(x,y)=0.$$

Thus

$$H_n(x,y)=H_n(y,x),$$

which is exactly the symmetry condition $\left(S_n\right)$.

• Case 2: $\tau =0$.
• Substituting ${\displaystyle f\left(x\right)=-x-{\displaystyle \sum _{n=1}^{\infty }}{a}_n{x}^{n+1}}$ into $F(x,f(x\left)\right)=0$ and using the homogeneity of $H_n$, the lowest-order monomial is

$$H_n(x,-x).$$

Hence

$$H_n(x,-x)=0,\forall x\in \mathbb{R}.$$

By homogeneity,

$$H_n(x,-x)\equiv x^n{H}_n(1,-1),$$

so

$$H_n(1,-1)=0,$$

as already indicated in the proof of Proposition 1. This gives

$${\displaystyle \sum _{i=0}^n}{(-1)}^i{\alpha }_{n-i,i}=0,$$

which is the condition $\left(T_n\right)$.

• Therefore,

$$\mathrm{center}\Rightarrow \left(S_n\right)\mathrm{or}\left(T_n\right).$$

This exhausts all possible cases, completing the necessity.

• If $\left(S_n\right)$ holds, then $H_n(x,y)\equiv H_n(y,x)$, hence $F(x,y)\equiv F(y,x)$. This implies that if $y=f\left(x\right)$, then also $x=f\left(y\right)$, so $f^2\left(x\right)\equiv x$ and the origin is a center.

If $\left(T_n\right)$ holds, then $H_n(x,-x)\equiv 0$, which implies that the curve has a branch $y=-x$. Therefore, locally $f\left(x\right)=-x$, and hence $f^2\left(x\right)\equiv x$. Consequently, the origin is a center, which proves the sufficiency of the statement.

• This completes the proof. □

4. Conclusions

In this paper, we have characterized the necessary and sufficient conditions for the center problem in discrete dynamical systems of the polynomial maps of the form (1). The approach developed here combines computational insight from low-degree cases with a direct algebraic argument based on symmetry. A key feature of this method is that it relies on the involutive property $f\left(f\right(x\left)\right)=x$, which allows one to bypass the explicit computation of focus quantities and derive the center conditions in a uniform way for all even degrees.

This perspective leads to a significant conceptual simplification of the problem. Rather than working with increasingly complex algebraic relations arising from focus quantities, the center condition is obtained directly from the intrinsic symmetry of the defining equation. This combination of computational intuition and structural reasoning appears to be a promising strategy for further extensions to mixed-degree systems, where new interactions between different homogeneous components may arise.

One such mixed case was already considered in [1,16], where the combination of degree two and three terms was shown to yield a center at the origin.

Looking ahead, applying this framework to mixed-degree systems has the potential to deepen our understanding of limit cycle bifurcations and to reveal the intricate geometry of the corresponding center varieties.