On Artin's Conjecture, Ii: Pairs of Additive Forms

Abstract

It is shown that the system of two additive equations a1x1k+…+asxsk=b1x1k+…+bsxsk=0 where k ⩾ 2 and aj, bj are any given integers, has non‐trivial solutions in all p‐adic fields provided only that s > 8k2. The constant 8 can be reduced when k is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound 8k2 can be replaced by 2k2. 2000 Mathematical Subject Classification: 11D72.

Pairs of homogeneous additive equations

“…In both parts of the proof, we use the notion of colored variables developed by Brüdern and Godinho in [4]. Consider the vectors…”

On Artin's conjecture: Pairs of additive forms

“…Davenport and Lewis [4] proved that this statement is true for odd k, whereas for even k the bound s 7k 3 is sufficient to ensure solubility. It was then proved by Brüdern and Godinho [2] that the expected bound s > 2k 2 holds for even k which are not of the shape…”

“…For k = 6 = 3 • 2, the bound s > 2k 2 was reached by Godinho, Knapp and Rodrigues [8] while later Godinho and Ventura [9] showed that this bound suffices for k = 3 τ • 2 with τ 2 as well. Therefore, all pairs of additive forms of equal degree k in more than 2k 2 variables have a non-trivial 3-adic solution. The aim of this paper is to prove the following theorem, which shows that this statement does not only hold for p = 3 but for all p 3, by taking care of the degrees k = p τ (p − 1) for p 5 and τ 1.…”

On Artin’s conjecture: Linear slices of diagonal hypersurfaces

“…Our main tool in this section is a contraction method. The basic ideas go back to Davenport and Lewis [8,11], as developed by Brüdern and Godinho [5]. We require a highly refined version of the methods in [5], but only in a 2-adic context.…”