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, ..., 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

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

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

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