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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. | |||
==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]] | |||
Revision as of 06:09, 28 April 2013
In mathematics, Birch's theorem,[1] 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, r1, . . . ,rk be odd natural numbers, and f1, . . . ,fk be homogeneous polynomials with coefficients in K of degrees r1, . . . ,rk respectively in n variables, then there exists a number ψ(r1, . . . ,rk,l,K) such that
implies that there exists an l-dimensional vector subspace V of Kn such that
Remarks
The proof of the theorem is by induction over the maximal degree of the forms f1, . . . ,fk. 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
has a solution in integers x1, . . . ,xn, not all of which are 0.
The restriction to odd r is necessary, since even-degree forms, such as positive definite quadratic forms, may take the value 0 only at the origin.
References
- ↑ B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika, 4, pages 102–105 (1957)