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Journal cites, added 1 DOI using AWB (9904)

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	~~More specifically there are two closely related types~~ of ~~quartic surface: affine~~ and ~~projective~~. ~~An ''affine'' quartic surface is the solution set of an equation of the form~~
	:<~~math~~>~~f(x,y,z)=0\~~ <~~/math~~>
	~~where ''f''~~ is a ~~polynomial~~ of ~~degree 4~~, ~~such as ''f''(''x'',''y'',''z'') = ''x''~~<~~sup~~>4<~~/sup~~> ~~+ ''y''~~<~~sup~~>4<~~/sup~~> ~~+ ''xyz'' + ''z''~~<~~sup~~>2</~~sup> − 1~~. ~~This is a surface in~~ [~~[affine space~~]].

	~~On the other hand~~, a ~~projective quartic surface is a surface in [[projective space]] '''P'''~~<~~sup~~>3<~~/sup~~> of the ~~same form, but now ''f'' is a ''homogeneous'' polynomial~~ of ~~4 variables~~ of ~~degree 4, so for example ''f''(''x''~~,~~''y''~~,'~~'z'',''w'') = ''x''~~<~~sup~~>4<~~/sup~~> ~~+ ''y~~''<~~sup~~>4<~~/sup~~> ~~+ ''xyzw'' + ''z''~~<~~sup~~>2<~~/sup~~>~~''w''~~<~~sup~~>2<~~/sup~~> ~~− ''w''~~<~~sup~~>4<~~/sup~~>.

	~~If the base field in '''R''' or '''C''' the surface~~ is ~~said~~ to ~~be ''real'' or ''complex''~~. ~~If on~~ the ~~other hand the base field is finite~~, ~~then~~ it is ~~said~~ to ~~be an ''arithmetic quartic surface''~~.

	==~~Special quartic surfaces==~~
	* [[Dupin cyclide]]s
	* The [[Fermat quartic]], ~~given by ''x''~~<~~sup~~>4<~~/sup~~> ~~+ ''y''~~<~~sup~~>4<~~/sup~~> ~~+ ''z''<sup>4</sup> + ''w''<sup>4</sup> =0 (an example~~ of ~~a K3 surface) – and tiled by 12 octagons~~, in the ~~[[Dyck tiling]] (named after [[Walther von Dyck]])~~.
	* [[K3 surface]]s
	* [[Klein quartic]]
	* [[Kummer surface]]
	* [[Plücker surface]]
	* [[Weddle surface]]

	~~==See also==~~
	*[[Quadric surface]] (The union of two quadric surfaces is a ~~special case of~~ a ~~quartic surface)~~
	*[[Cubic surface]] (The union of a cubic surface and a ~~plane is another particular type~~ of ~~quartic surface)~~

	~~==References==~~
	*{{Citation \| authorlink=R. W. H. T. ~~Hudson \| last1=Hudson \| first1=R~~. W. H. ~~T. \| title=Kummer~~'~~s quartic surface \| publisher=~~[~~[Cambridge University Press]] \| series=Cambridge Mathematical Library \| isbn=978-0-521-39790-2 \| id={{MathSciNet \| id = 1097176}} \| year=1990\|url=~~http://~~www~~.~~archive~~.~~org~~/~~details~~/~~184605691}}~~
	*{{Citation \| last1=Jessop \| first1=C. M. \| title=Quartic surfaces with ~~singular points \| url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=04290002 \| publisher=Cornell University Library \| isbn=978~~-1-~~4297~~-~~0393~~-~~2 \| year=1916}}~~
	~~{{geometry~~-~~stub}}~~

	~~[[Category:Complex surfaces]]~~
	~~[[Category:Algebraic surfaces]~~]

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Your syѕtem has vital needs in order to stay in good condition. Disregardіng nourishment might have dire implications. Begin using these ways to feel much better!~~		More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surface is the solution set of an equation of the form
			:<math>f(x,y,z)=0\ </math>
			where ''f'' is a polynomial of degree 4, such as ''f''(''x'',''y'',''z'') = ''x''<sup>4</sup> + ''y''<sup>4</sup> + ''xyz'' + ''z''<sup>2</sup> − 1. This is a surface in [[affine space]].

			On the other hand, a projective quartic surface is a surface in [[projective space]] '''P'''<sup>3</sup> of the same form, but now ''f'' is a ''homogeneous'' polynomial of 4 variables of degree 4, so for example ''f''(''x'',''y'',''z'',''w'') = ''x''<sup>4</sup> + ''y''<sup>4</sup> + ''xyzw'' + ''z''<sup>2</sup>''w''<sup>2</sup> − ''w''<sup>4</sup>.

			If the base field in '''R''' or '''C''' the surface is said to be ''real'' or ''complex''. If on the other hand the base field is finite, then it is said to be an ''arithmetic quartic surface''.

			==Special quartic surfaces==
			* [[Dupin cyclide]]s
			* The [[Fermat quartic]], given by ''x''<sup>4</sup> + ''y''<sup>4</sup> + ''z''<sup>4</sup> + ''w''<sup>4</sup> =0 (an example of a K3 surface) – and tiled by 12 octagons, in the [[Dyck tiling]] (named after [[Walther von Dyck]]).
			* [[K3 surface]]s
			* [[Klein quartic]]
			* [[Kummer surface]]
			* [[Plücker surface]]
			* [[Weddle surface]]

			==See also==
			*[[Quadric surface]] (The union of two quadric surfaces is a special case of a quartic surface)
			*[[Cubic surface]] (The union of a cubic surface and a plane is another particular type of quartic surface)

			==References==
			*{{Citation \| authorlink=R. W. H. T. Hudson \| last1=Hudson \| first1=R. W. H. T. \| title=Kummer's quartic surface \| publisher=[[Cambridge University Press]] \| series=Cambridge Mathematical Library \| isbn=978-0-521-39790-2 \| id={{MathSciNet \| id = 1097176}} \| year=1990\|url=http://www.archive.org/details/184605691}}
			*{{Citation \| last1=Jessop \| first1=C. M. \| title=Quartic surfaces with singular points \| url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=04290002 \| publisher=Cornell University Library \| isbn=978-1-4297-0393-2 \| year=1916}}
			{{geometry-stub}}

			[[Category:Complex surfaces]]
			[[Category:Algebraic surfaces]]

en>RjwilmsiBot: /* References */fixing page range dashes using AWB (7840)

2011-09-24T10:17:15Z

References: fixing page range dashes using AWB (7840)

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