Algebarski varijeteti — разлика између измена

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[[File:Twisted cubic curve.png|200px|thumb|[[twisted cubic|Upleteni kubni]] objekat je projektivni algebarski varijetet.]]
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'''Algebarski varijeteti''' su centralni objekti izučavanja u [[algebraic geometry|algebarskoj geometriji]]. Klasično, an algebraic variety is defined as the [[solution set|set of solutions]] of a [[system of polynomial equations]] over the [[real number|real]] or [[complex number]]s. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.{{r|Hartshorne|page1=58}}
 
'''Algebarski varijeteti''' su centralni objekti izučavanja u [[algebraic geometry|algebarskoj geometriji]]. Klasično, analgebarski algebraicvarijetet varietyje isdefinisan defined as thekao [[solution set|setskup of solutionsrešenja]] of asistema [[system of polynomial equations|polinomskih jednačina]] over thenad [[real number|realrealnim]] orili [[complex number|kompleksnim brojevima]]s. ModernSavremene definitionsdefinicije generalizegenerališu thisovaj conceptkoncept inna severalnekoliko differentrazličitih waysnačina, while attempting to preservepokušavajući theda geometricsačuvaju intuitiongeometrijsku behindintuiciju theiza originalprvobitne definitiondefinicije.{{r|Hartshorne|page1=58}}
Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the [[Zariski topology]]. Under this definition, non-irreducible algebraic varieties are called '''algebraic sets'''. Other conventions do not require irreducibility.
 
Konvencije o definiciji algebarskog varijeteta neznatno se razlikuju. Na primer, neke definicije zahtevaju da je algebarski varijetet nereduktivan, što znači da nije unija dva manja skupa koja su zatvorena u [[Zariski topology|Zariskovoj topologiji]]. Pod ovom definicijom, algebarski varijeteti koji se mogu redukovati nazivaju se ''algebarske grupe''. Druge konvencije ne zahtevaju nereduktivnost.
The [[fundamental theorem of algebra]] establishes a link between [[algebra]] and [[geometry]] by showing that a [[monic polynomial]] (an algebraic object) in one variable with complex number coefficients is determined by the set of its [[Zero of a function|roots]] (a geometric object) in the [[complex plane]]. Generalizing this result, [[Hilbert's Nullstellensatz]] provides a fundamental correspondence between ideals of [[polynomial ring]]s and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of [[ring theory]]. This correspondence is a defining feature of algebraic geometry.
 
[[Основна теорема алгебре|Fundamentalna teorema algebre]] uspostavlja vezu između [[algebra|algebre]] i [[geometry|geometrije]], pokazujući da je [[monic polynomial|monski polinom]] (algebarski objekat) u jednoj promenljivoj sa kompleksnim brojevima kao koeficijenatima određen setom njegovih [[Нула функције|korena]] (geometrijski objekt) u [[complex plane|kompleksnoj ravni]]. Generalizirajući ovaj rezultat, [[Hilbert's Nullstellensatz|Hilbertova teorema nula]] daje fundamentalnu korespondenciju između ideala [[polynomial ring|polinomskih prstenova]] i algebarskih skupova. Koristeći teoremu nula i srodne rezultate, matematičari su uspostavili čvrstu korespondenciju između pitanja o algebarskim skupovima i pitanja [[ring theory|teorije prstena]]. Ova korespondencija je definišuća karakteristika algebarske geometrije.
Many algebraic varieties are [[manifold]]s, but an algebraic variety may have [[singular point of an algebraic variety|singular points]] while a manifold cannot. Algebraic varieties can be characterized by their [[dimension of an algebraic variety|dimension]]. Algebraic varieties of dimension one are called [[algebraic curve]]s and algebraic varieties of dimension two are called [[algebraic surface]]s.
 
Mnogi algebarski varijeteti su [[mnogostrukost]]i, ali algebarski varijetet može da ima [[singular point of an algebraic variety|singularne tačke]] dok mnogostrukost ne može. Algebarski varijeteti se mogu karakterisati njihovom [[dimension of an algebraic variety|dimenzijom]]. Algebarski varijeteti dimenzije jedan se nazivaju [[algebraic curve|algebarskim krivama]], a algebarski varijeteti dimenzije dva se nazivaju [[algebraic surface|algebarskim površima]].
 
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