Neville's algorithm - Revision history

en>Abhinav: Undid revision 610075578 by 129.12.103.201 (talk)

2014-05-25T14:30:17Z

Undid revision 610075578 by 129.12.103.201 (talk)

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	~~In mathematics, '''Birch's theorem'''~~,~~<ref>B~~. ~~J. Birch, ''Homogeneous forms of odd degree in a large number of variables'', Mathematika, '''4''~~'~~, pages 102–105 (1957)</ref> named for [[Bryan John Birch]], is~~ a ~~statement about the representability of zero by odd degree forms.~~		Nice to meet you, I am Marvella Shryock. What I adore doing is doing ceramics but I haven't made a dime with it. Years in over the counter std test ([http://www.smithandkenworld.com/activity-posted-by-2455/ click through the following website page]) past we moved to Puerto Rico and my family enjoys it. Bookkeeping is her day job now.

	~~==Statement of Birch's theorem==~~
	Let ''K'' be an [[algebraic number field]], ''k'', ''l'' and ''n'' be natural numbers, ''r''<sub>1</sub>, . . . ,''r''<sub>''k''</sub> be odd natural numbers, and ''f''<sub>1</sub>, . . . ,''f''<sub>''k''</sub> be homogeneous polynomials with ~~coefficients in ''K'' of degrees ''r''<sub>1</sub>, ~~.~~ . . ,''r''<sub>''k''</sub> respectively~~ in ~~''n'' variables, then there exists a number ψ~~(~~''r''<sub>1<~~/~~sub>, . ~~.~~ ~~.~~ ,''r''<sub>''k''<~~/~~sub>,''l'',''K'') such that~~
	~~:<math>n\ge\psi(r_1,\ldots,r_k,l,K)</math>~~
	~~implies that there exists an ''l''~~-~~dimensional vector subspace ''V'' of ''K''<sup>''n''</sup> such that~~
	~~:<math>f_1(x)=\cdots = f_k(x)=0,\quad\forall x\in V.<~~/~~math>~~

	~~==Remarks==~~
	~~The proof of~~ the ~~theorem is by [[mathematical induction\|induction~~]~~] over the maximal degree of the forms ''f''<sub>1</sub>, . . . ,''f''<sub>''k''</sub>. Essential~~ to ~~the proof is a special case, which can be proved by an application of the [[Hardy–Littlewood circle method]], of the theorem which states that if ''n'' is sufficiently large~~ and ~~''r'' is odd, then the equation~~

	~~:<math>c_1x_1^r+\cdots+c_nx_n^r=0,\quad c_i\in\mathbb{Z}, i=1,\ldots,n</math>~~

	~~has a solution in integers ''x''<sub>1</sub>, . . . ,''x''<sub>''n''</sub>, not all of which are 0~~.

	~~The restriction to odd ''r''~~ is ~~necessary, since even-degree forms, such as [[Quadratic_form#Definiteness_of_a_quadratic_form\|positive definite quadratic form]]s, may take the value 0 only at the origin~~.

	~~==References==~~
	~~<references/>~~

	~~[[Category:Diophantine equations]]~~
	~~[[Category:Analytic number theory]]~~
	~~[[Category:Theorems in number theory]]~~

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7884545

2013-04-28T05:09:31Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7884545

← Older revision		Revision as of 07:09, 28 April 2013
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	~~Nice to meet you~~, ~~my name is Figures Held although I don~~'~~t truly like being known as like that~~. ~~My day job~~ is a ~~meter reader~~. ~~California is exactly where I~~'~~ve always been living~~ and ~~I love each day residing here~~. ~~One~~ of ~~the issues he loves most is ice skating but he is having difficulties to discover time for it~~.<br><br>~~My webpage~~ ... ~~home std test kit~~, ~~[http~~://~~btcsoc~~.~~com~~/~~index~~.~~php?do~~=/~~profile-4621~~/~~info~~/ go to ~~website~~],		In mathematics, '''Birch's theorem''',<ref>B. J. Birch, ''Homogeneous forms of odd degree in a large number of variables'', Mathematika, '''4''', pages 102–105 (1957)</ref> named for [[Bryan John Birch]], is a statement about the representability of zero by odd degree forms.

			==Statement of Birch's theorem==
			Let ''K'' be an [[algebraic number field]], ''k'', ''l'' and ''n'' be natural numbers, ''r''<sub>1</sub>, . . . ,''r''<sub>''k''</sub> be odd natural numbers, and ''f''<sub>1</sub>, . . . ,''f''<sub>''k''</sub> be homogeneous polynomials with coefficients in ''K'' of degrees ''r''<sub>1</sub>, . . . ,''r''<sub>''k''</sub> respectively in ''n'' variables, then there exists a number ψ(''r''<sub>1</sub>, . . . ,''r''<sub>''k''</sub>,''l'',''K'') such that
			:<math>n\ge\psi(r_1,\ldots,r_k,l,K)</math>
			implies that there exists an ''l''-dimensional vector subspace ''V'' of ''K''<sup>''n''</sup> such that
			:<math>f_1(x)=\cdots = f_k(x)=0,\quad\forall x\in V.</math>

			==Remarks==
			The proof of the theorem is by [[mathematical induction\|induction]] over the maximal degree of the forms ''f''<sub>1</sub>, . . . ,''f''<sub>''k''</sub>. Essential to the proof is a special case, which can be proved by an application of the [[Hardy–Littlewood circle method]], of the theorem which states that if ''n'' is sufficiently large and ''r'' is odd, then the equation

			:<math>c_1x_1^r+\cdots+c_nx_n^r=0,\quad c_i\in\mathbb{Z}, i=1,\ldots,n</math>

			has a solution in integers ''x''<sub>1</sub>, . . . ,''x''<sub>''n''</sub>, not all of which are 0.

			The restriction to odd ''r'' is necessary, since even-degree forms, such as [[Quadratic_form#Definiteness_of_a_quadratic_form\|positive definite quadratic form]]s, may take the value 0 only at the origin.

			==References==
			<references/>

			[[Category:Diophantine equations]]
			[[Category:Analytic number theory]]
			[[Category:Theorems in number theory]]

72.28.178.83: /* External links */

2012-07-22T14:57:16Z

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Nice to meet you, my name is Figures Held although I don't truly like being known as like that. My day job is a meter reader. California is exactly where I've always been living and I love each day residing here. One of the issues he loves most is ice skating but he is having difficulties to discover time for it.<br><br>My webpage ... home std test kit, [http://btcsoc.com/index.php?do=/profile-4621/info/ go to website],