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, . . . ,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

n≥ψ(r1,…,rk,l,K){displaystyle ngeq psi (r_{1},ldots ,r_{k},l,K)}

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

f1(x)=⋯=fk(x)=0,∀x∈V.{displaystyle f_{1}(x)=cdots =f_{k}(x)=0,quad forall xin 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

c1x1r+⋯+cnxnr=0,ci∈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, . . . ,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

1. 
   ^ B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika, 4, pages 102–105 (1957)
   

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