- 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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