Configuration (geometry) - Revision history

en>Double sharp at 15:54, 19 April 2014

2014-04-19T15:54:10Z

← Older revision		Revision as of 17:54, 19 April 2014
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	~~In [[mathematics]], a '''Pfister form'''~~ is a particular kind of [[quadratic form]] over a [[field (mathematics)\|field]] ''F'' (whose [[characteristic (algebra)\|characteristic]] is usually assumed to be not 2), introduced by [[Albrecht Pfister (mathematician)\|Albrecht Pfister]] in 1965. ~~A Pfister form~~ is ~~in 2<sup>''n''</sup> variables, for some natural number ''n'' (also called an '''n-Pfister form'''), and may be written as a [[tensor product of quadratic forms]] as:~~		Greetings! I am Myrtle Shroyer. Bookkeeping is what I do. To gather badges is what her family members and her appreciate. Her husband and her reside in Puerto Rico but she will have to move one working day or another.<br><br>My web blog; [http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw]

	~~:<math>\langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle \cong \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle,</math>~~

	~~for ''a<sub>i</sub>'' elements of the field ''F''. An ''n''-Pfister form may also be constructed inductively from an (''n''-1)-Pfister form ''q''~~ and ~~an element ''a'' of ''F'', as <math>q \oplus (a)q</math>~~.

	~~So all 1-Pfister forms~~ and ~~2-Pfister forms look like:~~

	~~:<math>\langle\!\langle a\rangle\!\rangle\cong \langle 1, a \rangle \cong x^2 + ay^2</math>.~~
	~~:<math>\langle\!\langle a,b\rangle\!\rangle\cong \langle 1, a, b, ab \rangle \cong x^2 + ay^2 +bz^2 +abw^2.</math>~~

	~~For n ≤ 3 the ''n''-Pfister forms are [[norm form]]s of [[composition algebra]]s.<ref name=Lam316>Lam (2005) p.316</ref> In fact,~~ in ~~this case, two ''n''-Pfister forms are [[Isometry\|isometric]] if and only if the corresponding composition algebras are [[isomorphic]].~~

	The Pfister forms are generators for the torsion in the [[Witt group]].<ref name=Lam395>Lam (2005) p.395</ref> The ''n''-fold forms additively generate the ''n''-th power ''I''<sup>''n''</sup> of the fundamental ideal of the Witt ring.<ref name=Lam316>Lam (2005) p.316</ref>

	~~==Characterisation==~~
	~~We define a quadratic form ''q'' over a field ''F''~~ to be '''multiplicative''' if when '''x''' and '''y''' are vectors of indeterminates, then ''q''('''x''').''q''('''y''') = ''q''('''z''') where '''z''' is a vector of [[rational function]]s in the '''x''' and '''y''' over ''F''. [[Isotropic quadratic form]]s are multiplicative.<ref name=Lam324>Lam (2005) p.~~324~~<~~/ref~~> ~~For [[anisotropic quadratic form]]s, Pfister forms are multiplicative and conversely.~~<~~ref name=Lam325~~>~~Lam (2005) p.325</ref><ref name=Raj164>Rajwade (1993) p.164</ref>~~

	~~==Connection with K-theory==~~
	~~Let ''k''<sub>''n''</sub>(''F'') be the ''n''-th group in~~ [~~[Milnor K-theory]] modulo 2. There are homomorphisms from ''k''<sub>''n''</sub>(''F'') to the Witt ring by taking the symbol~~

	:~~<math> \{a_1,\ldots,a_n\} \mapsto \langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle ,<~~/~~math>~~

	~~where the image is an ''n''-fold Pfister form<~~/~~sup>~~.~~<ref name=Lam366>Lam (2005) p~~.~~366<~~/~~ref> The image can be taken as ''I''<sup>''n''<~~/~~sup>~~/~~''I''<sup>''n''+1<~~/~~sup> and the map is surjective since the Pfister forms additively generate ''I''<sup>''n''</sup>~~. ~~The [[Milnor conjecture]] can be interpreted as stating that these maps are isomorphisms~~.~~<ref name=Lam366~~/>

	~~==Pfister neighbours==~~
	~~A '''Pfister neighbour''' is a form (''W'',σ) such that (''W'',σ) is similar to a subspace of a space with Pfister form (''V'',φ) where dim.''V'' < 2 dim.''W''.<ref name=Lam339>Lam (2005) p.339<~~/ref> The associated Pfister form φ is uniquely determined by σ. Any ternary form is a Pfister neighbour; a quaternary form is a Pfister neighbour if and only if its discriminant is a square.<ref name=Lam341>Lam (2005) p.341</ref> A degree five form is a Pfister neighbour if and only if the underlying field is a [[linked field]].<ref name=Lam342>Lam (2005) p.342</ref>

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{Citation \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 \| mr = 2104929 }}, Ch. 10
	* {{citation \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. R. \| last=Rajwade \| publisher=[[Cambridge University Press]] \| year=1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}

	~~==Further reading==~~
	* {{citation \| last1=Knebusch \| first1=Manfred \| last2=Scharlau \| first2=Winfried \| title=Algebraic theory of quadratic forms. Generic methods and Pfister forms \| others=Notes taken by Heisook Lee \| series=DMV Seminar \| volume=1 \| location=Boston - Basel - Stuttgart \| publisher=Birkhäuser Verlag \| year=1980 \| isbn=3-7643-1206-8 \| zbl=0439.10011 }}

	~~[[Category:Quadratic forms]~~]

en>David Eppstein: fix {{harv|Hilbert|Cohn-Vossen|1952}} link (authors need to be separated with vbars)

2013-11-21T03:31:37Z

fix {{harv|Hilbert|Cohn-Vossen|1952}} link (authors need to be separated with vbars)

← Older revision		Revision as of 05:31, 21 November 2013
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	~~I am Oscar and I totally dig that title~~. ~~California~~ is ~~where her std testing at home~~ ([~~http~~://~~theoffshoremanual~~.~~com~~/~~?testimonial~~=~~know~~-~~facing~~-~~candida~~-~~albicans My Page]~~) ~~is but she needs~~ to ~~transfer because~~ of ~~her family~~. ~~To collect coins~~ is a ~~factor that I~~'~~m completely addicted~~ to. ~~Since she was eighteen she~~'~~s been working~~ as a ~~meter reader but she~~'~~s usually needed her own business~~.		In [[mathematics]], a '''Pfister form''' is a particular kind of [[quadratic form]] over a [[field (mathematics)\|field]] ''F'' (whose [[characteristic (algebra)\|characteristic]] is usually assumed to be not 2), introduced by [[Albrecht Pfister (mathematician)\|Albrecht Pfister]] in 1965. A Pfister form is in 2<sup>''n''</sup> variables, for some natural number ''n'' (also called an '''n-Pfister form'''), and may be written as a [[tensor product of quadratic forms]] as:

			:<math>\langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle \cong \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle,</math>

			for ''a<sub>i</sub>'' elements of the field ''F''. An ''n''-Pfister form may also be constructed inductively from an (''n''-1)-Pfister form ''q'' and an element ''a'' of ''F'', as <math>q \oplus (a)q</math>.

			So all 1-Pfister forms and 2-Pfister forms look like:

			:<math>\langle\!\langle a\rangle\!\rangle\cong \langle 1, a \rangle \cong x^2 + ay^2</math>.
			:<math>\langle\!\langle a,b\rangle\!\rangle\cong \langle 1, a, b, ab \rangle \cong x^2 + ay^2 +bz^2 +abw^2.</math>

			For n ≤ 3 the ''n''-Pfister forms are [[norm form]]s of [[composition algebra]]s.<ref name=Lam316>Lam (2005) p.316</ref> In fact, in this case, two ''n''-Pfister forms are [[Isometry\|isometric]] if and only if the corresponding composition algebras are [[isomorphic]].

			The Pfister forms are generators for the torsion in the [[Witt group]].<ref name=Lam395>Lam (2005) p.395</ref> The ''n''-fold forms additively generate the ''n''-th power ''I''<sup>''n''</sup> of the fundamental ideal of the Witt ring.<ref name=Lam316>Lam (2005) p.316</ref>

			==Characterisation==
			We define a quadratic form ''q'' over a field ''F'' to be '''multiplicative''' if when '''x''' and '''y''' are vectors of indeterminates, then ''q''('''x''').''q''('''y''') = ''q''('''z''') where '''z''' is a vector of [[rational function]]s in the '''x''' and '''y''' over ''F''. [[Isotropic quadratic form]]s are multiplicative.<ref name=Lam324>Lam (2005) p.324</ref> For [[anisotropic quadratic form]]s, Pfister forms are multiplicative and conversely.<ref name=Lam325>Lam (2005) p.325</ref><ref name=Raj164>Rajwade (1993) p.164</ref>

			==Connection with K-theory==
			Let ''k''<sub>''n''</sub>(''F'') be the ''n''-th group in [[Milnor K-theory]] modulo 2. There are homomorphisms from ''k''<sub>''n''</sub>(''F'') to the Witt ring by taking the symbol

			:<math> \{a_1,\ldots,a_n\} \mapsto \langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle ,</math>

			where the image is an ''n''-fold Pfister form</sup>.<ref name=Lam366>Lam (2005) p.366</ref> The image can be taken as ''I''<sup>''n''</sup>/''I''<sup>''n''+1</sup> and the map is surjective since the Pfister forms additively generate ''I''<sup>''n''</sup>. The [[Milnor conjecture]] can be interpreted as stating that these maps are isomorphisms.<ref name=Lam366/>

			==Pfister neighbours==
			A '''Pfister neighbour''' is a form (''W'',σ) such that (''W'',σ) is similar to a subspace of a space with Pfister form (''V'',φ) where dim.''V'' < 2 dim.''W''.<ref name=Lam339>Lam (2005) p.339</ref> The associated Pfister form φ is uniquely determined by σ. Any ternary form is a Pfister neighbour; a quaternary form is a Pfister neighbour if and only if its discriminant is a square.<ref name=Lam341>Lam (2005) p.341</ref> A degree five form is a Pfister neighbour if and only if the underlying field is a [[linked field]].<ref name=Lam342>Lam (2005) p.342</ref>

			==Notes==
			{{reflist}}

			==References==
			* {{Citation \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 \| mr = 2104929 }}, Ch. 10
			* {{citation \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. R. \| last=Rajwade \| publisher=[[Cambridge University Press]] \| year=1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}

			==Further reading==
			* {{citation \| last1=Knebusch \| first1=Manfred \| last2=Scharlau \| first2=Winfried \| title=Algebraic theory of quadratic forms. Generic methods and Pfister forms \| others=Notes taken by Heisook Lee \| series=DMV Seminar \| volume=1 \| location=Boston - Basel - Stuttgart \| publisher=Birkhäuser Verlag \| year=1980 \| isbn=3-7643-1206-8 \| zbl=0439.10011 }}

			[[Category:Quadratic forms]]

en>David Eppstein: biregular

2012-09-02T00:23:34Z

biregular

New page

I am Oscar and I totally dig that title. California is where her std testing at home ([http://theoffshoremanual.com/?testimonial=know-facing-candida-albicans My Page]) is but she needs to transfer because of her family. To collect coins is a factor that I'm completely addicted to. Since she was eighteen she's been working as a meter reader but she's usually needed her own business.