I have very large systems (>20) of polynomial (max degree 3) equations that I would like to find a solution to. I'm not interested in all solutions as presumably there are too many (a huge number based on Bézout's bound), but I would like to find a single solution. I haven't been able to achieve this using FindInstance (program hangs) or FindRoot (singular jacobian message). I suspect the latter problem is due to not being able to guess a good initial starting point for the root. It is known that the Homotopy continuation method avoids the problem of having to choose a starting point for the initial root.
My question is: does FindInstance use Homotopy continuation method?
If it does then I'm out of ideas. If it doesn't then I will try to implement the method in another way. All I've been able to find is the following statement https://www.wolfram.com/mathematica/new-in-10/enhanced-algebraic-computation/high-performance-numeric-solution-of-polynomial-sy.html
saying that the method is "selected when appropriate" but I'm not sure if its implemented for FindInstance.
Thanks for your help.