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

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.

• S2 and S3 have the same solution set. A point is a solution of S2 if and only if it is a solution of S3. (There might be higher multiplicity when considering S3; more precisely the solutions of S2 and S3 may have different scheme structure. But ignoring multiplicities, the sets of solutions are the same.) Commented Feb 16 at 13:34
• @ZachTeitler But I want to understand these multiplicities. Any idea how they change when we go from S1 to S2 and S2 to S3? Thanks. Commented Feb 16 at 13:35

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 $$S_1$$.