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. http://mathoverflow.net/questions/55190/systems-of-polynomial-equations