On Artin’s conjecture, I: Systems of diagonal forms

Jörg Brüdern and Hemar Godinho

As a special case of a well-known conjecture of Artin, it is expected that a system of R additive forms of degree k, say

∑_{i=1}^{N} a_{ij}x_{i}^{k} = 0 (1 ≤ j ≤ R)  (1)

with integer coefficients a_{ij}, has a non-trivial solution in ℚ_{p} for all primes p whenever

N > Rk^{2}.  (2)

In this paper we show that one can reach the expected order of magnitude k^{2}.

Theorem 1. Let k ≥ 3 and R ≥ 3. Then the system of equations (1) has a non-trivial solution in ℚ_{p} for all primes p provided that

N ≥ R^{3}k^{2}

unless R = 3 and k is a power of 2 in which case the condition on N has to be replaced by N ≥ 36k^{2}. For small values of R or k our analysis can be considerably refined. We shall discuss in greater detail the case of pairs of equations R = 2. In the light of the aforementioned result of Davenport and Lewis, only even k deserve attention. Davenport and Lewis [5] showed that for even k, the pair of equations

∑_{i=1}^{N} a_{i}x_{i}^{k} = ∑_{i=1}^{N} b_{i}x_{i}^{k} = 0  (3)

with a_{i}, b_{i} ∈ ℤ has a non-trivial p-adic solution for all primes p when N ≥ 7k^{3}, and this remained unimproved until very recently when Godinho[9] obtained bounds on N which are dependent on the prime factorisation of the degree. However, it does not follow from the work of [9] that a condition like N ≥ Ck^{2} with some constant C suffices to guarantee solubility of (3) in all ℚ_{p}. Our second theorem provides such a bound with C = 16.

Theorem 2. If k is of the form

k = 2 · 5^{τ} or k = (p − 1)p^{τ} for some prime p > 2  (4)

then the pair (3) has a non-trivial solution in all p-adic fields whenever

N ≥ 6k(k − 1).

If k is not of the form (4) but

k = k_{0}2^{τ} with k_{0} = 1 or 3 or 5 or 7  (5)

then the same conclusion holds if N ≥ 16k^{2}k_{0}^{−1} − 4k. If k is neither of the form (4) or (5) but takes the shape

k = 2p^{τ} (p − 1),  (6)

then for N ≥ 3k(k − 2) the equations (3) have a non-trivial solution in all p-adic fields. When k is neither of the forms (4), (5) or (6) then the the condition N ≥ 2k^{2} + 1 suffices.