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

1. Is there a way to represent $I_1$ as $\langle r_1,\dots,r_l\langle$, 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?)

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?)