I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real solutions. The equations were derived as the gradients of a sum-of-squares cost function, which I am attempting to find all global optima of. I believe there are a finite number of real solutions but I have not confirmed this yet. I have floating point coefficients and I'm looking for numerical solutions (as opposed to symbolic solutions).
Which software packages (and which functions specifically) are generally most promising to solve such a problem?
I am aware of various functions in Maple, Matlab, and Mathematica that can solve systems of polynomial equations but there are a large number of options in each software package and I'm interested in advice on where I should be looking first for problems of this scale.
A numerical dump of the cost function is here: