I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions.

I want to solve this system numerically, but if I plug it in an algebraic computing system (Macaulay2 or Bertini) and try to solve it with homotopy continuation, the PC just remains blocked, and I can't find any solution.

I am trying to understand why this happens, with the following consideration: ALL the solutions are in some way equivalent (I mean that if $(x_1,..x_n)$, than the other solutions are just permutations of this array), so I am interested in computing just 1 solution, not all the possible solutions.

Here is my question

- Where is the bottleneck of the homotopy method, in the number of solutions or in the number of variables?
- The fact that I am interested only in one solution, may speed up the method just analyzing a single homotopy path?
- Is there a way in Macaulay2 to calculate only a single solution, and not all of them?

To give more details about the system, if I have $k+n$ unknowns ($k<n$), then I have $(n+1) + {n \choose 2} +{n \choose 3}$ equations and $k!$ equivalent solutions. The degree of the system is always 4, and I have one equation of degree 1, $n$ of degree 2, ${n \choose 2}$ of degree 3 and ${n \choose 3}$ of degree 4. The system can be solved easily with Groebner basis when $k=2$ and $k = 3$, but also computing Groebner basis for higher values of $k$ becomes very slow..

Thanks!

bottleneck=a problem that delays progress; from here: dictionary.cambridge.org/fr/dictionnaire/anglais/bottleneck $\endgroup$ – YCor Mar 22 '16 at 17:21homotopy continuation methodin your favorite search engine! $\endgroup$ – Moritz Firsching Mar 23 '16 at 7:50