Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Applications of algebraic methods in solving the center-focus problem   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. $\omega\leq\varrho$.

34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int