Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:

**System 1 ($S_1$):**

$f_1(x_1,\dots,x_n) = 0$,

$f_2(x_1,\dots,x_n) = 0$,

$\vdots$

$f_n(x_1,\dots,x_n) = 0$ and

**System 2 ($S_2$):**

$x_1 f_1(x_1,\dots,x_n) = 0$,

$x_2 f_2(x_1,\dots,x_n) = 0$,

$\vdots$

$x_n f_n(x_1,\dots,x_n) = 0$.

**System 3 ($S_3$):**

$x_1^2 f_1(x_1,\dots,x_n) = 0$,

$x_2^2 f_2(x_1,\dots,x_n) = 0$,

$\vdots$

$x_n^2 f_n(x_1,\dots,x_n) = 0$.

I am interested in understanding the relation between the zeros of systems 1,2 and 3. I have the following questions:

- Is there a way to calculate the complete solution of the systems $S_2$ and $S_3$? Also, how they are related to the solutions of the system $S_1$?

Thanks for your time and consideration. Have a good day.