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