Last month the latest stable version (1.10) of Macaulay2 was released. This includes the first stable version of the package MonodromySolver, developed by some of us at Georgia Tech (Anton, Cvetelina, Kisun, Tim), as well as our collaborators Jeff Sommars and Anders Jensen. The corresponding paper has been on the arXiv for about a year.
People in algebraic geometry have used monodromy groups for a long time—in particular, use of monodromy group to solve polynomial systems has appeared elsewhere recently (see here, and here) and is certainly implicit in earlier algorithms. In fact, the possible variations on this idea seem limitless. Our implementation allows one to compare various approaches and unifies them in a general framework.
The package currently works best for the following problem: we are given a family of square polynomial systems in variables over the complex numbers, where the parameter space is a finite-dimensional affine space, such that, for a Zariski open subset of there exists such that However, the various methods are modular enough to incorporate them into solving more general systems—see here for a great example. Understanding what users want in more general settings will help us improve the package in the future, so try it out and let us know what you think!