@Kjetil, unfortunately I don't have many other details, I am just surfing the literature about mixture models, and seems that nobody cares about the absence of diagonal terms, so from this my question..

@Suvrit and Federico, do you have a reference for your comments? I would be great!

@ Benjamin Steinberg The algorithm, theorethically, should take a system that is known to have roots and produce approximations of one of the roots.

@ Steven Stadnick Actually, I am trying to solve a big system, described at high level here mathoverflow.net/questions/234254/… What I am experiencing is that most of the solvers spend a lot of time in counting the roots, while the time spent in calculating the various individual roots is much smaller. From this, I am trying to understand if it makes sense to modify the existing methods to make them calculate only one root, or if this is a lost battle (ie the problem is NP hard).

Thanks for your comment. Encoding 3-sat we prove that is NP hard to understand the solvability of a system, ie if a system has solutions. But if we know by construction that a system has solution (and also that is unique), things change? what can be said about finding that precise solution (also numerically..)?

Yes, multivariate polynomials over the real field

I am trying with a very standard starting system: if $F$ is my original system, then I use $G = F-F(xo)$ where $x0$ is a random value, so I know a zero of G. What I see is that, even for very small values of $t$, $F*t + (1-t)*G$ is high even in $x0$, because of the 4th degree equations. Another element that probably blocks everything is that tipically the system is non squared, so I can not use most of the solvers (eg Newton Rhapson) to get the zeros of the perturbed system $F*t + (1-t)*G$ starting from a zero of $G$. I should look for solvers of non squared systems in that case.