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, r1, ..., rk be odd natural numbers, and f1, ..., fk be homogeneous polynomials with coefficients in K of degrees r1, ..., rk respectively in n variables. Then there exists a number ψ(r1, ..., rk, 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 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, ..., fk. 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 x1, ..., xn, 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.