Birch's theorem

Birch's theorem

In mathematics, Birch's theorem, [B. J. Birch, "Homogeneous forms of odd degree in a large number of variables", Mathematika, 4, pages 102-105 (1957)] named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.

tatement of Birch's theorem

Let "K" be an algebraic number field, "k", "l" and "n" be natural numbers, "r"1,...,"r""k" be odd natural numbers, and "f"1,...,"f""k" be homogeneous polynomials with coefficients in "K" of degrees "r"1,...,"r""k" respectively in "n" variables, then there exists a number ψ("r"1,...,"r""k","l","K") such that:ngepsi(r_1,ldots,r_k,l,K)implies that there exists an "l"-dimensional vector subspace "V" of "K""n" such that:f_1(x)=ldots f_k(x)=0,quadforall xin V.

Remarks

The proof of the theorem is by induction over the maximal degree of the forms "f"1,...,"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:c_1x_1^r+ldots+c_nx_n^r=0,quad c_iinmathbb{Z}, i=1,ldots,nhas a solution in integers "x"1,...,"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


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