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?)



Resultants and elimination theory
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 $n$.


