Linear combination of homogeneous polynomials

Linear combination of homogeneous polynomials

Let $f_1=1,f_2,\cdots, f_k$ be homogeneous polynomials and $F$ be product
of linear forms in $\mathbb{R}[x_1,\cdots, x_n]$ such that
$0=deg(f_1)=d_1<deg(f_2)=d_2,\cdots,deg(f_k)=d_k, deg(F)=d$. What is
condition of $f_1,\cdots, f_k$ so that there exists homogeneous
polynomials $g_1,\cdots, g_k$ with $deg(g_i)=d-d_i$ satisfying :
$$\sum_{i=1}^k g_i.f_i=F ? $$
For example, if $f_1=1, f_2=y, F=(x+y)(x+y+z)$ then $g_1=x(x+y+z),
g_2=x+y+z.$