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Birch's theorem

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Statement about the representability of zero by odd degree forms

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 K 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

  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.
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