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Solve a "big" polynomial system numerically

I would like to solve numerically a polynomial system with around 10^5 variables and 10^7 equations. Each equation is of degree 4 with integer coefficients and contains around 10 variables, so that the jacobian of the system is a very "hollow" matrix with integer entries.

Let X=[x_0,x_1, x_2,...] and E=[eq_0,eq_1, eq_2...] the list of variables and equations :

Is there a SAGE function f(E,X) which find numerically solutions for such a system?

Is SAGE relevant for such intensive computation ?

I think the code needs to be 100% cython, for not losing time.

My computer has a hard disk of 980 Go, 4Go of RAM and its processor "Intel Core i5 CPU 760 @ 2.80GHz × 4" is around 10^9 FLOPS.

I guess my computer is not enough powerful for doing such computation in a reasonable time :
The jacobian J is a (10^5)x(10^7) matrix, so for example, the Gauss-Newton algorithm would require around 1000 Go of RAM and 10^16 operations.

I know very few things in numerical analysis, so some advices would be welcome.