On Artin’s conjecture: Linear slices of diagonal hypersurfaces

Abstract

Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.

Two Cases of Artin's Conjecture

“…The proof of Theorem 2 follows a pattern by Brüdern and Robert [9]. The difficulty of finding a non-trivial p-adic solution for all systems of equations (1.0.4) depends on the residue class of p modulo 3.…”

“…However, there are cases in which it does hold. For example, the case (k 1 , k 2 ) = (3, 2) was proved by Wooley [41] and (k 1 , k 2 ) = (k, 1) for general k by Brüdern and Robert [9].…”

“…The cases R ∈ N with k 1 = k and k 2 = ⋅ ⋅ ⋅ = k R = 1 of the generalisation of Artin's conjecture for diagonal forms merit particular attention. As Brüdern and Robert [9] pointed out, they could be used as a means to prove Artin's conjecture for some k. The following lemma [9, Section 2], an immediate conclusion of a theorem by Ellison [19], which works over Q p as well, describes why this is the case, and which values of R are important for that.…”

“…For r = 0, this is the case of one diagonal form which was proved by Davenport and Lewis [12] as mentioned before. Brüdern and Robert [9] took care of the case r = 1 by proving that s > k 2 + 1 variables suffice to ensure the existence of a non-trivial p-adic solution for all primes. Nonetheless, the condition s > k 2 + r cannot be sufficient for all r ∈ N, because this would prove Artin's conjecture for all k, but it would be of interest to know up to which r this is true.…”

“…This indicates a necessity to distinguish between primes congruent to 1 modulo 3 and the prime 3. An equivalence relation on the set of systems (1.0.4) which preserves the solubility in Q s p {0}, introduced by Brüdern and Robert [9,Section 6], is used in Section 3.3 to pick representatives with useful properties to fulfil the requirements of the version of Hensel's lemma. Most cases for p congruent to 1 modulo 3 can be worked on with a simple combinatorial approach in Section 3.4, where one finds a solution using only the variables whose coefficients a i and b i are not both divisible by p. This leaves three cases which require more attention.…”