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 ≥ ψ ( r 1 , … , r k , l , K ) {\displaystyle n\geq \psi (r_{1},\ldots ,r_{k},l,K)}
then there exists an l-dimensional vector subspace V of Kn such that
f 1 ( x ) = ⋯ = f k ( x ) = 0 for all x ∈ V . {\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, ..., 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 1 x 1 r + ⋯ + c n x n r = 0 , c i ∈ Z , i = 1 , … , n {\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, ..., 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.
References