Birch's theorem

Short description: Statement about the representability of zero by odd degree forms

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, r₁, ..., r_k be odd natural numbers, and f₁, ..., f_k be homogeneous polynomials with coefficients in K of degrees r₁, ..., r_k respectively in n variables. Then there exists a number ψ(r₁, ..., r_k, l, K) such that if

$n\ge \psi (r_1,\dots ,r_k,l,K)$

then there exists an l-dimensional vector subspace V of Kⁿ such that

$f_1(x)=\cdots =f_k(x)=0\text{ for all }x\in V.$

Remarks

The proof of the theorem is by induction over the maximal degree of the forms f₁, ..., 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

$c_1{x}_1^r+\cdots +c_n{x}_n^r=0,c_i\in \mathbb{\mathbb{Z}},i=1,\dots ,n$

has a solution in integers x₁, ..., 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.