Brauer's theorem on forms

Last updated September 01, 2023

There also is Brauer's theorem on induced characters.

In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables.^[1]

Statement of Brauer's theorem

Let K be a field such that for every integer r > 0 there exists an integer ψ(r) such that for n ≥ ψ(r) every equation

(*)\qquad a_{1}x_{1}^{r}+\cdots +a_{n}x_{n}^{r}=0,\quad a_{i}\in K,\quad i=1,\ldots ,n

has a non-trivial (i.e. not all x_i are equal to 0) solution in K. Then, given homogeneous polynomials f₁,...,f_k of degrees r₁,...,r_k respectively with coefficients in K, for every set of positive integers r₁,...,r_k and every non-negative integer l, there exists a number ω(r₁,...,r_k,l) such that for n ≥ ω(r₁,...,r_k,l) there exists an l-dimensional affine subspace M of Kⁿ (regarded as a vector space over K) satisfying

f_{1}(x_{1},\ldots ,x_{n})=\cdots =f_{k}(x_{1},\ldots ,x_{n})=0,\quad \forall (x_{1},\ldots ,x_{n})\in M.

An application to the field of p-adic numbers

Letting K be the field of p-adic numbers in the theorem, the equation (*) is satisfied, since $\mathbb {Q} _{p}^{*}/\left(\mathbb {Q} _{p}^{*}\right)^{b}$ , b a natural number, is finite. Choosing k = 1, one obtains the following corollary:

A homogeneous equation f(x₁,...,x_n) = 0 of degree r in the field of p-adic numbers has a non-trivial solution if n is sufficiently large.

One can show that if n is sufficiently large according to the above corollary, then n is greater than r². Indeed, Emil Artin conjectured^[2] that every homogeneous polynomial of degree r over Q_p in more than r² variables represents 0. This is obviously true for r = 1, and it is well known that the conjecture is true for r = 2 (see, for example, J.-P. Serre, A Course in Arithmetic, Chapter IV, Theorem 6). See quasi-algebraic closure for further context.

In 1950 Demyanov^[3] verified the conjecture for r = 3 and p ≠ 3, and in 1952 D. J. Lewis ^[4] independently proved the case r = 3 for all primes p. But in 1966 Guy Terjanian constructed a homogeneous polynomial of degree 4 over Q₂ in 18 variables that has no non-trivial zero.^[5] On the other hand, the Ax–Kochen theorem shows that for any fixed degree Artin's conjecture is true for all but finitely many Q_p.

Notes

Davenport, Harold (2005). Analytic methods for Diophantine equations and Diophantine inequalities. Cambridge Mathematical Library. Edited and prepared by T. D. Browning. With a preface by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman (2nd ed.). Cambridge University Press. ISBN 0-521-60583-0. Zbl 1125.11018.

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References

↑ R. Brauer, A note on systems of homogeneous algebraic equations, Bulletin of the American Mathematical Society, 51, pages 749-755 (1945)
↑ Collected papers of Emil Artin, page x, Addison–Wesley, Reading, Mass., 1965
↑ Demyanov, V. B. (1950). "На кубических форм дискретных линейных нормированных полей" [On cubic forms over discrete normed fields]. Doklady Akademii Nauk SSSR . 74: 889–891.
↑ D. J. Lewis, Cubic homogeneous polynomials over p-adic number fields, Annals of Mathematics, 56, pages 473–478, (1952)
↑ Guy Terjanian, Un contre-exemple à une conjecture d'Artin, C. R. Acad. Sci. Paris Sér. A–B, 262, A612, (1966)

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[1] R. Brauer, A note on systems of homogeneous algebraic equations, Bulletin of the American Mathematical Society, 51, pages 749-755 (1945)

[2] Collected papers of Emil Artin, page x, Addison–Wesley, Reading, Mass., 1965

[3] Demyanov, V. B. (1950). "На кубических форм дискретных линейных нормированных полей" [On cubic forms over discrete normed fields]. Doklady Akademii Nauk SSSR . 74: 889–891.

[4] D. J. Lewis, Cubic homogeneous polynomials over p-adic number fields, Annals of Mathematics, 56, pages 473–478, (1952)

[5] Guy Terjanian, Un contre-exemple à une conjecture d'Artin, C. R. Acad. Sci. Paris Sér. A–B, 262, A612, (1966)

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