Hi all,
I have a system of 6th-order polynomial equations in 4 variables $q_1, q_2, q_3, q_4$ (i.e. polynomials with all the terms such as $q_1^6, q_2^6, q_2^4 q_3^2$):
$P_k(q_1, q_2, q_3, q_4) = 0$ with $k=2,\dots,N$
I don't have any good guess of the q_i. So, Newton method and its variant won't work because they need a good starting point to converge to the right solution.
My question is whether exists any numerical method to find the solution of that system of equations.
Any help is greatly appreciated.
p/s: If you want to know further background information, you may want to check my previous post. Systems of polynomial equations
Best Answer
It really depends on the kind of system you have: do you have any reasons, for instance, to believe that the number of solutions is finite?
If the system is zero-dimensional (which essentially means that you have a finite number of solutions over the complexes), then the rational univariate representation means that you can essentially reduce to the univariate case. You would still have to figure out which solution is of interest to you, but the numerical aspect is essentially the same as in the one variable case.
Now, Groebner bases are not the only tool at your disposal in this case. You might want to look at Homotopy Continuation Methods which are entirely numerical. I believe that Jan Verschelde, whose homepage I've linked to, has readily usable software from his webpage.
Good luck, and let me know how it works for you!