# What are the numerical methods for huge polynomial systems?

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.

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• Some insights are contained in mathoverflow.net/questions/2506/… – Avitus Jul 27 '14 at 11:55
• @Avitus: thank you, it's interesting, but it does not answer the question about complexity and workability. – Sebastien Palcoux Jul 27 '14 at 12:05
• @Paul: Thank you for the edits. I'm interested in at least showing the existence of a solution, and finding one would be nice. Yes I've tried to use Newton's method, more precisely Gauss-Newton's method (because the Jacobian matrix is non-square), but my system seems too huge, and this method doesn't work on my desk computer. Could this method be workable on a supercomputer (or something like that)? I don't know, I'm not at all an expert... – Sebastien Palcoux Jul 30 '14 at 16:23
• Assuming a solution exists, a non-linear system of equations with approximately 50,000 unknowns is certainly within the reach of parallel computers and even some desktops. – Paul Jul 30 '14 at 18:21
• Many links and some general thoughts on a related problem are contained in cs.stackexchange.com/questions/11112/… – Thomas Klimpel Jul 31 '14 at 7:12