Resultants and elimination theory

Question

Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$. Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$. For any two polynomials $f$ and $g$ in $I$, the resultant $\operatorname{Res}(f,g,x_1)$ belongs to the first elimination ideal $I_1$.

1. Is there a way to represent $I_1$ as $\langle r_1,\dotsc,r_l\rangle$, where the basis $r_j$ are obtained as resultants of polynomials in $I$ with respect to $x_1$?
2. Is there a way to represent each elimination ideal in this way? (I imagine by iterating the possible representation of point 1?)

Answer 1

A bit long for a comment, so posted as an answer, although this is really a comment.

For 1) I would rather try the following. Assuming the $f$'s have the same degree in $x_1$, introduce formal variables $s_1,\dotsc,s_n$ and $t_1,\dotsc,t_n$, and take the binary resultant $\operatorname{Res}(s_1f_1+\dotsb+s_nf_n,t_1f_1+\dotsb+t_nf_n,x_1)$. This is a monstrous polynomial in the $s,t$ variables. I think the coefficients of this polynomial should be in $I_1$ and may give a basis. If the $f$'s don't have the same degree, you may have to make two copies of them, multiply one copy by powers of $x_1$ and the other copy by powers of $1-x_1$, so they all have the same degree, then do the above with $2n$ instead of $n$.