Main theorem of elimination theory: Difference between revisions

Revision as of 09:24, 14 July 2018

Revision as of 16:42, 11 February 2019

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==Classical formulation==

For stating the theorem in terms of [[commutative algebra]], one has to consider a [[polynomial ring]] <math>R[\mathbf x]=R[x_1, \ldots, x_n]</math> over a commutative [[Noetherian ring]] {{mvar|R}}, and ana [[homogeneous ideal]] {{mvar|I}} generated by [[homogeneous polynomial]]s <math>f_1,\ldots, f_k.</math> (In the original proof by [[Francis Sowerby Macaulay|Macaulay]], {{mvar|k}} was equal to {{mvar|n}}, and {{mvar|R}} was a polynomial ring over the integers, whose indeterminates were all the coefficients of the<math>f_i\mathrm s.</math>)

Any [[ring homomorphism]] <math>\varphi</math> from {{mvar|R}} into a field {{mvar|K}}, defines a ring homomorphism <math>R[\mathbf x] \to K[\mathbf x]</math> (also denoted <math>\varphi</math>), by applying <math>\varphi</math> to the coefficients of the polynomials.

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*{{cite book|last=Eisenbud|first=David|title=Commutative Algebra: with a View Toward Algebraic Geometry|publisher=Springer|year=2013|isbn=9781461253501|author-link=David Eisenbud}}

*{{cite book|last=Milne|first=James S.|title=The Abel Prize 2008&ndash;2012|chapter=The Work of John Tate|publisher=Springer|year=2014|isbn=9783642394492|author-link=James Milne (mathematician)}}

{{algebraic-geometry-stub}}

[[Category:Algebraic geometry]]