Birch's theorem

In mathematics, Birch's theorem,[1] 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

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

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

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

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

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

has a solution in integers 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.

References

1. Birch, B. J. (1957). "Homogeneous forms of odd degree in a large number of variables". Mathematika. 4 (2): 102–105. doi:10.1112/S0025579300001145.