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,...,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

\[ n\ge\psi(r_1,\ldots,r_k,l,K) \]

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

\[ f_1(x)=\ldots f_k(x)=0,\quad\forall x\in V. \]

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

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

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

^ B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika, 4, pages 102-105 (1957)