Applications of algebraic methods in solving the center-focus problem

Abstract

The nonlinear differential system $\dot{x}={\sum }_{i=0}^{\ell }{P}_{m_i}(x,y),\dot{y}={\sum }_{i=0}^{\ell }{Q}_{m_i}(x,y)$ is considered, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i\ge 1$ in $x$ and $y$, $m_0=1$. The set $\{1,m_i{\}}_{i=1}^{\ell }$ consists of a finite number $(\ell <\infty )$ of distinct natural numbers. It is shown that the maximal number of algebraically independent focal quantities that take part in solving the center-focus problem for the given differential system with $m_0=1$, having at the origin of coordinates a singular point of the second type (center or focus), does not exceed $\varrho =2({\sum }_{i=1}^{\ell }{m}_i+\ell )+3.$ We make an assumption that the number $\omega$ of essential conditions for center which solve the center-focus problem for this differential system does not exceed $\varrho$, i.\,e. $\omega \le \varrho$.