On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables

Question

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:

1. 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.

Answer 1

The solutions to system $S_2$ is formed by the union of the solutions to $2^n$ systems indexed by $I\subseteq [n]$, in which $f_i=0$ for $i\in I$ and $x_j=0$ for all $j\in [n]\setminus I$. This includes $I=[n]$ corresponding to system $S_1$.

As a starting point for information about systems of polynomial equations, you can check this Wikipedia article.