Selberg trace formula: Difference between revisions

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In the [[Mathematics|mathematical]] field of [[algebraic geometry]], a '''singular point of an [[algebraic variety]]''' ''V'' is a point ''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 reals, this notion generalizes the notion of non-[[local flatness]]. A point of an algebraic variety which is not singular is said to be '''regular'''. An algebraic variety which has no singular point is said to be '''non singular''' or '''smooth'''.


For example, the [[plane algebraic curve]] (a [[cubic curve]]) of equation
:''y''<sup>2</sup> - ''x''<sup>2</sup>(''x'' + 1) = 0,
which is plotted below, crosses itself at the origin (0,0) and the origin is thus a [[double point]] of the curve. It is ''singular'' because a single [[tangent]] may not be correctly defined there.


[[Image:Singularptfn.JPG]]
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More generally a plane curve defined by an [[implicit equation]]
:''F''(''x'',''y'') = 0,
where ''F'' is a [[smooth function]] is said to be ''singular'' at point if the [[Taylor series]] of ''F'' has [[Power series#Order of a power series|order]] at least 2 at this point.
 
The reason for that is that, in [[differential calculus]], the tangent at the point (''x''<sub>0</sub>, ''y''<sub>0</sub>) of such a curve is defined by the equation
:<math>(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0),</math>
whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exists or a special definition must be provided.
 
In general for a [[hypersurface]]
 
:''F''(''x'', ''y'', ''z'', ...) = 0
 
the '''singular points''' are those at which all the [[partial derivative]]s simultaneously vanish. A general [[algebraic variety]] ''V'' being defined as the common zeros of several [[polynomial]]s, the condition on a point ''P'' of ''V'' to be singular point is that the [[Jacobian matrix]] of the first order partial derivatives of the polynomials has a [[rank of a matrix|rank]] at ''P'' that is lower than the rank at other points of the variety.
 
Points of ''V'' that are not singular are called '''non-singular''' or '''regular'''. It is always true that most points are non-singular in the sense that the non-singular points form a set that is both [[open set|open]] and non-empty.<ref>Hartshorne, ''Algebraic Geometry'', page 33</ref>
 
In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a [[manifold]] near every regular point. But it is important to note that a real variety may be a manifold and have singular points. For example the equation <math>y^3 + 2 x^2 y - x^4 = 0</math> defines a real [[analytic manifold]] but has a singular point at the origin.<ref>Milnor, pp. 12–13</ref> This may be explained by saying that the curve has two [[complex conjugate]] [[branch (complex analysis)|branch]]es that cut the real branch at the origin.
 
==Singular points of smooth mappings==
As the notion of singular points is a purely local property, the above definition can be extended to cover the wider class of [[smooth function|smooth]] mappings, (functions from ''M'' to '''R'''<sup>n</sup> where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the [[Jet (mathematics)|jets]] of the mapping. The ''k''-th jet is the [[Taylor series]] of the mapping truncated at degree  ''k'' and deleting the [[constant term]].
 
== See also ==
*[[Singular point of a curve]]
*[[Singularity theory]]
 
==References==
{{reflist}}
{{refimprove|date=September 2008}}
* {{cite book | title=Singular Points of Complex Hypersurfaces | author=John Milnor | authorlink=John Milnor | series=Annals of Mathematics Studies | volume=61 | publisher=[[Princeton University Press]] | year=1969 | isbn=0-691-08065-8 }}
 
[[Category:Algebraic varieties]]
[[Category:Singularity theory]]

Revision as of 22:27, 3 March 2014


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