mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

, ''k'', ''l'' and ''n'' be natural numbers, ''r''₁, ..., ''r''_''k'' be odd natural numbers, and ''f''₁, ..., ''f''_''k'' be homogeneous polynomials with

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

s 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 if :

n \ge \psi(r_1,\ldots,r_k,l,K)

then there exists an ''l''- dimensional vector subspace ''V'' of ''K''^''n'' such that :

f_1(x) = \cdots = f_k(x) = 0 \text x \in V.

Remarks

The

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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+\cdots+c_nx_n^r=0,\quad c_i \in \mathbb{Z},\ i=1,\ldots,n

has 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

Diophantine equations Analytic number theory Theorems in number theory