Elimination theory

Remarks

$I_l$ is an ideal of $k[x_{l+1}, \ldots, x_n]$; it consists of the consequences of the equations of $I$ in which $x_1, \ldots, x_l$ do not occur. Elimination theorem: if $G$ is a Gröbner basis of $I$ for the lexicographic order with $x_1 > \cdots > x_n$, then $G \cap k[x_{l+1}, \ldots, x_n]$ is a Gröbner basis of $I_l$. Closure theorem: let $k$ be algebraically closed, and project the zero set of $I$ to the last $n - l$ coordinates. Then the zero set of $I_l$ in $k^{n-l}$ is the Zariski closure of this projection.

Example: in $k[t, x, y]$ with the lexicographic order $t > x > y$, let $I = (x - t, \; y - t^2)$. Since $y - t^2 = (y - x^2) - (t - x)(t + x)$, one gets $I = (t - x, \; x^2 - y)$. These two generators have coprime leading terms $t$ and $x^2$, so they form a Gröbner basis, and the first elimination ideal is $I_1 = (x^2 - y)$: eliminating $t$ gives the parabola $y = x^2$.

Non-example: $x - t$ lies in $I$ but not in $I_1$, because it involves the eliminated variable $t$.