Homotopy continuation methods to solve polynomial systems run in two stages. Symbolic homotopy methods construct a family of systems, connecting the system to be solved to a generic problem with known solutions. Numerical Continuation methods track the solution paths starting at the known solutions and ending at the solutions of the given system. A numerical irreducible decomposition represents not only all isolated solutions, but also all positive dimensional solution sets of a system. What it means to solve a polynomial system depends as much on the technology as on the algorithms.
NUMA Seminar, 1 February 2019, K.U. Leuven, Belgium