I got a cubic equation system that contains 3 cubic equations with 3 variables.
I want to find the number of solutions and the solutions themselves (as a numerical approximation).
Do you know good algorithms and/or theorems that apply to this problem?
One could use the Buchberger algorithm, to obtain algebraic solutions. Since I do not know your equations, we may consider the following cubic equations $$ (x_1+x_2)^3=x_3 $$
$$ (x_2+x_3)^3=x_1 $$
$$ (x_3+x_1)^3=x_2 $$
Then Buchberger's algorithm yields the solutions, e.g., $\{x_1,x_2,x_3\}$ means $(x_1,x_2,x_3)=(0,0,0)$, and $\{x_1,x_2 + x_3,x_3^2 + 1\}$ denotes the solutions $(x_1,x_2,x_3)$ with $x_1=0$, $x_3=-x_2$ and $x_3^2=-1$ (obviously only possible over the complex numbers). We have $\{x_1 - 8x_3^3, x_2 - x_3, 64x_3^6 + 32x_3^4 + 7x_3^2 + 1\}$, etc. It is possible to list all solutions in this way.