In
mathematics, Birch's theorem,
named for
Bryan John Birch
Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture.
Biography
Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith 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 number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s, ''r''
1, ..., ''r''
''k'' be odd natural numbers, and ''f''
1, ..., ''f''
''k'' be
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s with
coefficients in ''K'' of
degrees ''r''
1, ..., ''r''
''k'' respectively in ''n'' variables. Then there exists a number ''ψ''(''r''
1, ..., ''r''
''k'', ''l'', ''K'') such that if
:
then there exists an ''l''-
dimensional vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
''V'' of ''K''
''n'' such that
:
Remarks
The
proof of the theorem is by
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
over the maximal degree of the forms ''f''
1, ..., ''f''
''k''. Essential to the proof is a special case, which can be proved by an application of the
Hardy–Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History
The initial idea is usually at ...
, of the theorem which states that if ''n'' is sufficiently large and ''r'' is odd, then the equation
:
has a solution in
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''x''
1, ..., ''x''
''n'', not all of which are 0.
The restriction to odd ''r'' is necessary, since even degree forms, such as
positive definite quadratic form
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a pos ...
s, may take the value 0 only at the origin.
References
Diophantine equations
Analytic number theory
Theorems in number theory