I met with a very elementary question on linear algebra which turns out to be quite resistant when I work on it. Any comments or references on either solutions or the possible value of the question will be greatly appreciated.

**Initial problem**: Given $n$ homogeneous polynomials $ F_1,\dots, F_n $ in $R[x_1,\dots, x_m]$ such that $\deg(F_1)= \dots= \deg(F_n)=d$. For a fixed positive integer $k < n$, set $$P_k=\{F_{i_1}F_{i_2} \cdots F_{i_k} \,\vert\, 1\leq i_1 < i_2 < \cdots < i_k \leq n \}$$
If $ F_1,\dots, F_n $ are linearly independent, that is, $P_1$ is a linearly independent set, then is $P_k$ a linearly independent set for $k > 1$?

The question is trivial for $d=1$. In this case monomials in $F_1,\dots,F_n$ are always linearly independent as we can assume $n \le m$ and $F_i = x_i$ up to a coordinate change. On the other hand, the question has a negative answer when $d > 1$ if we do not impose any additional conditions. For instance, set $$F_1=x_1^2, F_2=x_1x_2,F_3=x_1x_3,F_4=x_2x_3,$$ then $F_1,F_2,F_3,F_4$ are independent. However, $P_2$ is dependent since $F_1F_4=F_2F_3$.

Now the question becomes:

**Question 1.** Assume $d > 1$. Under what condition will $P_k$ be linearly independent?

A first thought is that we restrict to homogeneous polynomials that are powers of linear polynomials. But this seems to be complicated as well. For instance, the question for $P_2$ and $d = 2$ goes as follows:

**Question 2.** Let $F_1,\dots, F_n \in R[x_1,\dots, x_m]$ be homogeneous polynomials of degree one such that $F_1^2, F_2^2,\dots, F_n^2$ are linearly independent ($F_1,\dots, F_n$ might be linearly dependent in this case). Under what condition will $F_1^2F_2^2, F_1^2F_3^2,\cdots, F_{n -1}^2F_n^2$ be linearly independent?