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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.
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| ==Statement of Birch's theorem==
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| 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
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| :<math>n\ge\psi(r_1,\ldots,r_k,l,K)</math>
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| implies that there exists an ''l''-dimensional vector subspace ''V'' of ''K''<sup>''n''</sup> such that
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| :<math>f_1(x)=\cdots = f_k(x)=0,\quad\forall x\in V.</math>
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| ==Remarks==
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| 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
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| :<math>c_1x_1^r+\cdots+c_nx_n^r=0,\quad c_i\in\mathbb{Z}, i=1,\ldots,n</math>
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| has a solution in integers ''x''<sub>1</sub>, . . . ,''x''<sub>''n''</sub>, not all of which are 0.
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| 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.
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| ==References==
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| <references/>
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| [[Category:Diophantine equations]]
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| [[Category:Analytic number theory]]
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| [[Category:Theorems in number theory]]
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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 (click through the following website page) past we moved to Puerto Rico and my family enjoys it. Bookkeeping is her day job now.