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{{refimprove|date=September 2008}}
In the [[Mathematics|mathematical]] field of [[algebraic geometry]], a '''singular point of an [[algebraic variety]]''' {{math|''V''}} is a point {{math|''P''}} that is 'special' (so, singular), in the geometric sense that at this point the [[tangent space]] at the variety may not be regularly defined. In case of varieties defined over the [[real number|real]]s, this notion generalizes the notion of [[local flatness|local non-flatness]]. A point of an algebraic variety that is not singular is said to be '''regular'''. An algebraic variety that has no singular point is said to be '''non-singular''' or '''smooth'''. The concept is generalized to [[smooth scheme]]s in the modern language of [[scheme theory]].
[[File:Newtonsche Knoten.png|thumb|The [[plane algebraic curve]] (a [[cubic curve]]) of equation
{{math|1=''y''<sup>2</sup> − ''x''<sup>2</sup>(''x'' + 1) = 0}} crosses itself at the origin {{math|(0, 0)}}. The origin is a [[double point]] of this curve. It is ''singular'' because a single [[tangent]] may not be correctly defined there.]]
== Definition ==
A [[plane curve]] defined by an [[implicit equation]]
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