Main theorem of elimination theory (original) (raw)
In algebraic geometry, the main theorem of elimination theory states that every projective scheme is 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, denote by the n-dimensional projective space over k. The main theorem of elimination theory is the statement that for any n and any algebraic variety V defined over k, the projection map sends Zariski-closed subsets to Zariski-closed subsets.
| Property | Value |
|---|---|
| dbo:abstract | In algebraic geometry, the main theorem of elimination theory states that every projective scheme is 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, denote by the n-dimensional projective space over k. The main theorem of elimination theory is the statement that for any n and any algebraic variety V defined over k, the projection map sends Zariski-closed subsets to Zariski-closed subsets. The main theorem of elimination theory is a corollary and a generalization of Macaulay's theory of multivariate resultant. The resultant of n homogeneous polynomials in n variables is the value of a polynomial function of the coefficients, which takes the value zero if and only if the polynomials have a common non-trivial zero over some field containing the coefficients. This belongs to elimination theory, as computing the resultant amounts to eliminate variables between polynomial equations. In fact, given a system of polynomial equations, which is homogeneous in some variables, the resultant eliminates these homogeneous variables by providing an equation in the other variables, which has, as solutions, the values of these other variables in the solutions of the original system. (en) |
| dbo:wikiPageID | 42753316 (xsd:integer) |
| dbo:wikiPageLength | 9133 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 973491547 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Prime_ideal dbr:Principal_ideal dbr:Projective_space dbr:Elimination_theory dbr:Monomial dbr:Monomial_basis dbr:Macaulay's_resultant dbr:Generic_property dbr:Noetherian_ring dbr:Closed_map dbr:Projective_scheme dbr:Commutative_algebra dbr:Zariski-closed dbr:Francis_Sowerby_Macaulay dbr:Algebraic_geometry dbr:Algebraic_variety dbr:Field_(mathematics) dbr:Direct_product dbr:Gröbner_basis dbr:Hilbert's_Nullstellensatz dbr:Hyperbola dbc:Algebraic_geometry dbr:Affine_plane dbr:System_of_polynomial_equations dbr:Homogeneous_polynomial dbr:Polynomial_ring dbr:Scheme_theory dbr:Scheme_(mathematics) dbr:Zariski_topology dbr:Multivariate_resultant dbr:Integer_polynomial dbr:Ring_homomorphism dbr:Proper_scheme dbr:Elimination_of_quantifiers dbr:Projective_completion dbr:Homogeneous_ideal |
| dbp:wikiPageUsesTemplate | dbt:Cite_book dbt:Math dbt:Mvar dbt:Short_description |
| dcterms:subject | dbc:Algebraic_geometry |
| rdfs:comment | In algebraic geometry, the main theorem of elimination theory states that every projective scheme is 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, denote by the n-dimensional projective space over k. The main theorem of elimination theory is the statement that for any n and any algebraic variety V defined over k, the projection map sends Zariski-closed subsets to Zariski-closed subsets. (en) |
| rdfs:label | Main theorem of elimination theory (en) |
| owl:sameAs | wikidata:Main theorem of elimination theory https://global.dbpedia.org/id/775Sn |
| prov:wasDerivedFrom | wikipedia-en:Main_theorem_of_elimination_theory?oldid=973491547&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Main_theorem_of_elimination_theory |
| is dbo:wikiPageWikiLink of | dbr:Elimination_theory dbr:Projective_variety dbr:List_of_theorems_called_fundamental |
| is foaf:primaryTopic of | wikipedia-en:Main_theorem_of_elimination_theory |