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Birch's theorem

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In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.

Contents

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 ≥ ψ ( r 1 , … , r k , l , K )

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

f 1 ( x ) = ⋯ = f k ( x ) = 0 , ∀ x ∈ 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

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

Birch's theorem Wikipedia
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