The center problem for a1:−4resonant quadratic system

On the maximal saddle order of <inline-formula><tex-math id="M1">\begin{document}$ p:-q $\end{document}</tex-math></inline-formula> resonant saddle

“…where A k = (a k,mj ) is a (n + 1) × (n + 2) matrix and a k,mj is the coefficient of term x qmn1−n−1+j y pmn1+2−j of C k im f im−k g k . For A 0 , from equation (11) x n−1−δm y δm + x n−1−δ 2+p y δ 2+p , where δ m = j m − (p + 4 − m) for 1 ≤ m ≤ 1 + p and δ 2+p = j 2+p − 1. It is easy to see that 0 < δ m < N 1 .…”

A hybrid symbolic-numerical approach to the center-focus problem

“…From the computational point of view, the biggest obstacle in solving the center-focus problem for a specific system is the determination of the irreducible components of the variety (i.e., solution set) defined by a certain number of focus quantities. The most common approach [2,32,24] is the application of computer algebra algorithms for computing the primary decomposition of the ideal generated by the focus quantities such as Gianni-Trager-Zacharias (GTZ) [31] or Shimoyama-Yokoyama (SY) [58], which have been implemented in various symbolic packages (e.g. Singular [35], or Macaulay [34]).…”

Linearizability of 2:−3 Resonant Systems with Quadratic Nonlinearities

“…In [6], system (1) is considered for p = 1 and q = 2 and with P(x, y) and Q(x, y) being quadratic polynomials. The case when P(x, y) and Q(x, y) are quadratic polynomials has been studied also by several other authors; in particular, the solution of the 1:−3 resonant center problem can be found in [7] and that of the 1:−4 resonant center problem can be found in [8]. Some results are also obtained for P(x, y) and Q(x, y) for cubic, (homogeneous) quartic, and quintic polynomials.…”