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 $m_i\geq 1$ 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. $\omega\leq\varrho$.