Birch's theorem

In 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, ''k'', ''l'' and ''n'' be natural numbers, ''r''₁, ..., ''r''_''k'' be odd natural numbers, and ''f''₁, ..., ''f''_''k'' be homogeneous polynomials 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
:$n \ge \psi(r_1,\ldots,r_k,l,K)$
then there exists an ''l''- dimensional vector subspace ''V'' of ''K''^''n'' such that
:$f_1(x) = \cdots = f_k(x) = 0 \text x \in V.$

Remarks

The proof 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

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

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