Asked and answered

Why do I need homotopy continuation when I have Newton's method?

Newton's methods (a.k.a. Newton-Raphson method) is a simple and quick to solve a polynomial system. However, in general Newton's method only converges when a good initial point is used, and even then this only yields a single solution. But if you are interested in all possible solutions you are usually out of luck.

How should I cite HomotopyContinuation.jl when I use it?

First of all, HomotopyContinuation.jl is absolutely free for personal or commercial use. It is licensed under the MIT license. If you use it in your work we would be grateful, if you could cite our extended abstract. A preprint of it is freely available at arXiv:1711.10911.

Are there other homotopy continuation solvers available?

Yes! HomotopyContinuation.jl is only the newest of a couple of implementations established through academic research. Others that must be mentioned are Bertini, PHCpack and HOM4PS. We wish to stress that by using those programs we learned a lot about homotopy continuation and numerical algebraic geometry in general. Without those programs the development of HomotopyContinuation.jl wouldn't have been possible. There are very simple and unofficial Julia wrappers for Bertini and for PHCpack available.