Brauer's theorem on forms

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

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

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

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

Letting K be the field of p-adic numbers in the theorem, the equation (*) is satisfied, since ${\displaystyle \mathbb {Q} _{p}^{*}/\left(\mathbb {Q} _{p}^{*}\right)^{b}}$, 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.

• 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.