Main theorem of elimination theory: Difference between revisions

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In [[algebraic geometry]], the '''main theorem of elimination theory''' states that anyevery [[projective scheme]] is [[proper scheme|proper]]. A version of this theorem predates the existence of [[scheme theory]]. It can be stated, proved, and applied in the following more classical setting. Let ''k'' be a [[field (mathematics)|field]], denote by <math>\mathbb{P}_k^n</math> the ''n''-dimensional [[projective space]] over ''k''. The main theorem of elimination theory is the statement that for any ''n'' and ''m'' the projection map <math>\mathbb{P}_k^n \times \mathbb{P}_k^m \to \mathbb{P}_k^m</math> sends [[Zariski-closed]] subsets to Zariski-closed subsets. Since Zariski-closed subsets in projective spaces are related to [[homogeneous polynomial]]s, it's possible to state the theorem in that language directly, as was customary in the [[elimination theory]].

===A simple motivating example===

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The theorem is: there is an ideal {{mvar|r}} in {{mvar|R}}, uniquely determined by {{mvar|I}}

such that, for every ring homomorphism <math>\varphi</math> from {{mvar|R}} into a field {{mvar|K}}, the homogeneous polynomials <math>\varphi(f_1),\ldots, \varphi(f_k)</math> have a nontrivial common zero (in an algebraic closure of {{mvar|K}}) if and only if <math>\phivarphi(r)=\{0\}.</math>

Moreover, {{math|1=''r'' = 0}} if {{math|1=''k'' < ''n''}}, and {{mvar|r}} is [[principal ideal|principal]] if {{math|1=''k'' = ''n''}}. In this latter case, a generator of {{mvar|r}} is called the [[Macaulay's resultant|resultant]] of <math>f_1,\ldots, f_k.</math>