Symbolic computation and the cyclicity problem for singularities

We show how methods of computational commutative algebra are employed to investigate the local 16th Hilbert Problem, which is to find an upper bound on the number of limit cycles that can bifurcate from singularities in families of polynomial systems of differential equations on R²${\mathbb{R}}^2$, and is one step in a program for solving the full 16th Hilbert Problem. We discuss an extension of a well-known theorem, and illustrate the concepts and methods with concrete examples.