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## Applications of algebraic methods in solving the center-focus problem

Authors: Popa Mihail, Pricop Victor

### 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 Pmi and Qmi are homogeneous polynomials of degree mi>=1 in x and y, m0=1. The set {1,mi}li=1 consists of a finite number (l<∞) of distinct integer 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 m0=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. $\omega\leq\varrho$.

Institute of Mathematics and Computer Science