Let a system of $n$ polynomial equations of degree $d$ with $m$ variables.
I'm interested in a sparse system with $d = 3$, $n \sim 2000000$, $m \sim 50000$ and integer coefficients.
What techniques can I use from numerical analysis to show the existence of a solution? What numerical methods exist for finding a solution? What are the complexities? In others words, do these methods compute a solution to my problem in a reasonable time if we implement it on an efficient computer?
For more details about my system (it's a pentagon equation), see this post.