turkmath.org

We discuss two problems related to the theory of polynomial plane differential  systems, that is,  systems of the form
        \begin{equation} \label{1}
            \frac{dx}{dt}=P_{n}(x,y), \ \ \
            \frac{dy}{dt}=Q_{n}(x,y),
        \end{equation}                                                                
        where $P_{n}(x,y), Q_{n}(x,y)$ are polynomials of degree $n$, $x$ and $y$ are real unknown functions.
        
The first one  is the problem of local integrability, that is, the problem of  finding local analytic integrals in a neighborhood of singular points of system (1).  We present a computational approach to  find integrable systems within given parametric families of  systems and describe some mechanisms of integrability.

The second problem is called the cyclicity problem, or the local 16th Hilbert problem,  and is related to the estimation of the number of limit cycles arising in system (1) after perturbations of integrable systems. The approach is algorithmic and is based on algorithms of computational commutative algebra relying on  the Groebner bases theory.