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Guangfeng Dong

Publications and source records attributed to Guangfeng Dong.

6 recordsLinked to original sources
arXiv2025

On the holomorphic foliations admitting a common invariant algebraic set

In this paper, we study the holomorphic foliations admitting a common invariant algebraic set $C$ defined by a polynomial $f$ in $ \mathbb{K}[x_1,x_2,...,x_n]$ over any characteristic $0$ subfield $\mathbb{K}\subseteq\mathbb{C}$. For the $\mathbb{K}[x_1,x_2,...,x_n]$-module $V_f$ of vector fields generating foliations that admit $C$ as an invariant set, we provide several conditions under which the module $V_f$ can be freely generated by a minimal generating set. In particular, when $n=2$ and $f$ is a weakly tame polynomial, we show that the $\mathbb{K}[x,y]$-module $V_f$ is freely generated by two polynomial vector fields, one of which is the Hamiltonian vector field induced by $f$, if and only if, $f$ belongs to the Jacobian ideal $\langle f_x, f_y\rangle$ in $\mathbb{K}[x,y]$. Our proof employs a purely elementary method.

math.DS↗
arXiv2024

The topology and isochronicity on complex Hamiltonian systems with homogeneous nonlinearities

In this paper, we study the Hamiltonian differential systems with homogeneous nonlinearity parts on $\mathbb{C}^2$. Firstly, we present a series of topological properties of polynomial Hamiltonian functions, with a particular focus on the characteristics of critical points and non-trivial cycles that vanish at infinity. Secondly, we use these topological properties to derive a complete set of necessary and sufficient conditions for isochronous centers in this class of systems of any degree. Our method avoids tedious computation of the coefficients of normalization occurring in the usual tools to deal with the isochronicity problem.

math.DS↗
arXiv2023

Topological properties on isochronous centers of polynomial Hamiltonian differential systems

In this paper, we study the topological properties of complex polynomial Hamiltonian differential systems of degree $n$ having an isochronous center. Firstly, we prove that if the critical level curve possessing an isochronous center contains only a single singular point, and the period $1$-form does not have poles with zero residue at infinity on level curves sufficiently close to the critical curve, then the vanishing cycle associated to this center is trivial in the 1-dimensional homology group of the projective closure of a generic level curve. Our result provides a positive answer to a question asked by L. Gavrilov under relatively simple conditions and can be applied to achieve an equivalent description of the Jacobian conjecture on $\mathbb{C}^2$. Secondly, we obtain a very simple but useful necessary condition for isochronicity of Hamiltonian systems, which is that the $(n+1)$-degree part of the Hamiltonian function must have a factor with multiplicity no less than $(n+1)/2$. Thirdly, we show a relation between Gavrilov's question and the conjecture proposed by X. Jarque and J. Villadelprat on the non-isochronicity of real Hamiltonian systems of even degree $n$.

math.DS↗
arXiv2019

On the transformations linearizing isochronous centers of Hamiltonian systems

In this paper we study the transformations linearizing isochronous centers of planar Hamiltonian differential systems with polynomial Hamiltonian functions $H(x,y)$ having only isolated singularities. Assuming the origin is an isochronous center lying on the level curve $L_0$ defined by $H(x,y)=0$, we prove that, there exists a canonical linearizing transformation analytic on a simply-connected open set $Ω$ with closure $\overlineΩ=\mathbb{R}^2$, if and only if, $L_0$ consists of only isolated points; furthermore, if the origin is the unique center, then the condition that $L_0$ consists of only isolated points implies that the corresponding canonical linearizing transformation can be analytically defined on the whole plane.

math.CA↗
arXiv2019

On the maximal saddle order of p:-q resonant saddle

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree $n$. Firstly, we prove that, for any given resonance $p:-q$, $(p, q)=1$, and sufficiently big integer $n$, the maximal saddle order can grow at least as rapidly as $n^2$. Secondly, we show that there exists an integer $k_0$, which grows at least as rapidly as $3n^2/2$, such that $L_{k_0}$ does not belong to the ideal generated by the first $k_0-1$ saddle values $L_1, L_2, \cdots, L_{k_0-1}$, where $L_{k}$ means the $k$-th saddle value of the given system. In particular, if $p=1$ (or $q=1$), we obtain a sharper result that $k_0$ can grow at least as rapidly as $2 n^2$.

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