en>John of Reading: /* Application at a permeable wall */Typo fixing, replaced: that is has → that it has using AWB

2013-04-18T15:18:47Z

Application at a permeable wall<ref name="zas1"/>: Typo fixing, replaced: that is has → that it has using AWB

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	~~Democrats also blocked GOP sought curbs on~~ the ~~Environmental Protection Agency~~'~~s power to regulate utilities~~' ~~greenhouse gas emissions~~.~~<br>~~The ~~price limit~~ is ~~meant partly to ensure~~ that the ~~sidewalk stands don't evolve into minimarts or convenience stores~~; ~~they also are barred from selling clothes~~, ~~handbags and some other items~~. ~~The New York State Association of Convenience Stores had no immediate comment on Councilwoman Karen Koslowitz' price cap proposal~~, ~~which~~ is ~~due for~~ a ~~formal introduction in the council later this month~~.<br>~~http:~~//~~www~~.~~shlgrouptr~~.~~com~~/~~warm~~/~~?p=382~~ <br /> ~~http~~:~~//www~~.~~shlgrouptr~~.~~com/warm/?p~~=~~375~~ <br /> ~~http:~~/~~/www~~.~~shlgrouptr~~.~~com/warm/?p~~=~~258~~ <br /> ~~http~~:/~~/www~~.~~shlgrouptr~~.~~com/warm/?p~~=~~193~~ <br /> ~~http:~~//~~www.shlgrouptr.com/warm/?p=25~~ <br /><br><br>~~If you want~~ to ~~find out more info about~~ [~~http://www~~.~~bendtrapclub~~.~~com/cheap/ugg~~.~~asp Cheap Uggs Boots~~] ~~have a look at the webpage.~~		In [[mathematics]], the '''Tsen rank''' of a [[field (mathematics)\|field]] describes conditions under which a system of [[polynomial equations]] must have a solution in the [[Field (mathematics)\|field]]. The concept is named for [[C. C. Tsen]], who introduced their study in 1936.

			We consider a system of ''m'' polynomial equations in ''n'' variables over a field ''F''. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that ''F'' is a '''T'''<sub>'''''i'''''</sub>-'''field''' if every such system, of degrees ''d''<sub>1</sub>, ..., ''d''<sub>''m''</sub> has a common non-zero solution whenever

			:<math>n > d_1^i + \cdots + d_m^i. \, </math>

			The ''Tsen rank'' of ''F'' is the smallest ''i'' such that ''F'' is a T<sub>''i''</sub>-field. We say that the Tsen rank of ''F'' is infinite if it is not a T<sub>''i''</sub>-field for any ''i'' (for example, if it is [[formally real field\|formally real]]).

			==Properties==
			* A field has Tsen rank zero if and only if it is [[Algebraically closed field\|algebraically closed]].
			* A finite field has Tsen rank 1: this is the [[Chevalley–Warning theorem]].
			* If ''F'' is algebraically closed then rational function field ''F''(''X'') has Tsen rank 1.
			* If ''F'' has Tsen rank ''i'', then the rational function field ''F''(''X'') has Tsen rank at most ''i'' + 1.
			* If ''F'' has Tsen rank ''i'', then an algebraic extension of ''F'' has Tsen rank at most ''i''.
			* If ''F'' has Tsen rank ''i'', then an extension of ''F'' of [[transcendence degree]] ''k'' has Tsen rank at most ''i'' + ''k''.
			* There exist fields of Tsen rank ''i'' for every integer ''i'' ≥ 0.

			==Norm form==
			We define a ''norm form of level i'' on a field ''F'' to be a homogeneous polynomial of degree ''d'' in ''n''=''d''<sup>''i''</sup> variables with only the trivial zero over ''F'' (we exclude the case ''n''=''d''=1). The existence of a norm form on level ''i'' on ''F'' implies that ''F'' is of Tsen rank at least ''i'' − 1. If ''E'' is an extension of ''F'' of finite degree ''n'' > 1, then the field [[norm form]] for ''E''/''F'' is a norm form of level 1. If ''F'' admits a norm form of level ''i'' then the rational function field ''F''(''X'') admits a norm form of level ''i'' + 1. This allows us to demonstrate the existence of fields of any given Tsen rank.

			==Diophantine dimension==
			The '''Diophantine dimension''' of a field is the smallest natural number ''k'', if it exists, such that the field of is class C<sub>''k''</sub>: that is, such that any homogeneous polynomial of degree ''d'' in ''N'' variables has a non-trivial zero whenever ''N'' >  ''d''<sup>''k''</sup>. Algebraically closed fields are of Diophantine dimension 0; [[quasi-algebraically closed field]]s of dimension 1.<ref name=NSW361>{{cite book \| title=Cohomology of Number Fields \| volume=323 \| series=Grundlehren der Mathematischen Wissenschaften \| first1=Jürgen \| last1=Neukirch \| first2=Alexander \| last2=Schmidt \| first3=Kay \| last3=Wingberg \| edition=2nd \| publisher=[[Springer-Verlag]] \| year=2008 \| isbn=3-540-37888-X \| page=361}}</ref>

			Clearly if a field is T<sub>''i''</sub> then it is C<sub>''i''</sub>, and T<sub>0</sub> and C<sub>0</sub> are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.

			==See also==
			* [[Tsen's theorem]]

			== References ==
			{{reflist}}
			* {{cite journal \| first=C. \| last=Tsen \| authorlink=C. C. Tsen \| title=Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper \| journal=J. Chinese Math. Soc. \| volume=171 \| year=1936 \| pages=81–92 \| zbl=0015.38803 }}
			* {{cite book \| first=Falko \| last=Lorenz \| title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics \| year=2008 \| publisher=Springer \| isbn=978-0-387-72487-4 }}

			[[Category:Field theory]]
			[[Category:Diophantine geometry]]

en>Luksaz at 05:25, 4 July 2012

2012-07-04T05:25:20Z

New page

Democrats also blocked GOP sought curbs on the Environmental Protection Agency's power to regulate utilities' greenhouse gas emissions. The price limit is meant partly to ensure that the sidewalk stands don't evolve into minimarts or convenience stores; they also are barred from selling clothes, handbags and some other items. The New York State Association of Convenience Stores had no immediate comment on Councilwoman Karen Koslowitz' price cap proposal, which is due for a formal introduction in the council later this month. http://www.shlgrouptr.com/warm/?p=382 http://www.shlgrouptr.com/warm/?p=375 http://www.shlgrouptr.com/warm/?p=258 http://www.shlgrouptr.com/warm/?p=193 http://www.shlgrouptr.com/warm/?p=25 If you want to find out more info about [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] have a look at the webpage.

Henry adsorption constant - Revision history

en>John of Reading: /* Application at a permeable wall */Typo fixing, replaced: that is has → that it has using AWB

en>Luksaz at 05:25, 4 July 2012