Which method for solving system of nonlinear polynomials equations?

Hello. I have a research project and need to solve the system of nonlinear polynomial equations like that:

1.The polynomial equations have degrees 2, 4, 6, 8 respectively...........

2.The system is positive-dimensional.

For example:

(Of course, my system is biger than that very much). I only need one of its solution, which satisfied condition . The system is very unstable, it can not be solved by any numerical method, such as Newton method, Homotopy method or so... and needs a "arbitrary precision arithmetic" environment to calculate.

I use Maple, command **Triangularize** - package **RegularChains** - to transform this system to triangular form. That command gave me nice solution, but it only works for system up to 4 equations. For system consists of 6 equations, Maple used ~2Gigabytes of RAM and gave up. (Crying)

So, do you know how to solve this problem? Which method is efficient to my system? Thank you very much for your help (Smirk)