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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]], let $S = k[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], and let $I$ be an [[ideal|096Z9MP2ME]] of $S$.\nFor $0 \\le l \u003c n$, the *$l$-th elimination ideal* of $I$ is\n$$I_l = I \\cap k[x_{l+1}, \\ldots, x_n].$$\nElimination theory is the study of the elimination ideals of ideals and of their zero sets.\n\n## Remarks\n\n$I_l$ 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.\nElimination theorem: if $G$ is a [[Gröbner basis|C83KPT47R0]] of $I$ for the [[lexicographic order|0B68KNPGCG]] with $x_1 \u003e \\cdots \u003e x_n$, then $G \\cap k[x_{l+1}, \\ldots, x_n]$ is a Gröbner basis of $I_l$.\nClosure theorem: let $k$ be [[algebraically closed|0PF6TM8TWT]], and project the zero set of $I$ to the last $n - l$ coordinates.\nThen the zero set of $I_l$ in $k^{n-l}$ is the Zariski closure of this projection.\n\nExample: in $k[t, x, y]$ with the lexicographic order $t \u003e x \u003e y$, let $I = (x - t, \\; y - t^2)$.\nSince $y - t^2 = (y - x^2) - (t - x)(t + x)$, one gets $I = (t - x, \\; x^2 - y)$.\nThese 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 $y = x^2$.\n\nNon-example: $x - t$ lies in $I$ but not in $I_1$, because it involves the eliminated variable $t$.\n",
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