Birch's theorem
# 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

    $n\geq \psi (r_{1},\ldots ,r_{k},l,K)$

then there exists an l-dimensional vector subspace V of $K^n$ such that

    $f_{1}(x)=\cdots =f_{k}(x)=0{\text{ for all }}x\in V.$

## 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

    $c_{1}x_{1}^{r}+\cdots +c_{n}x_{n}^{r}=0,\quad c_{i}\in \mathbb {Z} ,\ i=1,\ldots ,n$

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.

