In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Birch's theorem,
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
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
, ''k'', ''l'' and ''n'' be
natural numbers, ''r''
1, ..., ''r''
''k'' be odd natural numbers, and ''f''
1, ..., ''f''
''k'' be
homogeneous polynomials with
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s 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 ''V'' of ''K''
''n'' such that
:
Remarks
The
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
of the theorem is by
induction 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, of the theorem which states that if ''n'' is sufficiently large and ''r'' is odd, then the equation
:
has a solution in
integers ''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 forms, may take the value 0 only at the origin.
References
Diophantine equations
Analytic number theory
Theorems in number theory