Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The method of Puiseux series and invariant algebraic curves   An explicit expression for the cofactor related to an irreducible invariant
algebraic curve of a polynomial dynamical system in the plane is derived. A
sufficient condition for a polynomial dynamical system in the plane to have a
finite number of irreducible invariant algebraic curves is obtained. All these
results are applied to Li\'enard dynamical systems $x_t=y$, $y_t=-f(x)y-g(x)$
with $\text{deg}\, f<\text{deg}\,g<2\,\text{deg}\,f+1$. The general structure
of their irreducible invariant algebraic curves and cofactors is found. It is
shown that Li\'enard dynamical systems with $\text{deg}\, f<\text{deg}\,
g<2\,\text{deg}\, f+1$ can have at most two distinct irreducible invariant
algebraic curves simultaneously and consequently are not integrable with a
rational first integral.

17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph