Pairs of additive sextic forms

On Artin's conjecture: Pairs of additive forms

“…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 missing primes are p = 2 in the case k = 3 • 2 τ and p if k = p τ (p − 1). Here, they gave the bounds s 8 3 k 2 for p = 2 and k = 3 • 2 τ , s 8k 2 for p = 2 and k = 2 τ , and s 4k 2 for p 3 and k = p τ (p − 1). All in all, the bound s 8k 2 was established for all p and all k.…”

On Artin's Conjecture: Pairs of Additive Forms

“…For p ≥ 3 and k = p τ (p − 1) on the other hand, the bound was further sharpened by Godinho and de Souza Neto [6,7] who proved that s ≥ 2 p p−1 k 2 − 2k suffices for p ∈ {3, 5} and if τ ≥ p−1 2 for p ≥ 7 as well. 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 diagonal forms of equal degree k in more than 2k 2 variables have a non-trivial 3-adic solution.…”

“…The missing primes are p = 2 in the case k = 3⋅2 τ and p if k = p τ (p − 1). Here, they gave the bounds s ≥ 8 3 k 2 for p = 2 and k = 3 ⋅ 2 τ , s ≥ 8k 2 for p = 2 and k = 2 τ , and s ≥ 4k 2 for p ≥ 3 and k = p τ (p − 1). All in all, the bound s ≥ 8k 2 holds for all p and all k.…”

Two Cases of Artin's Conjecture

“…For k = 6 = 3 ⋅ 2, the bound s > 2k 2 was reached by Godinho, Knapp and Rodrigues [22] while later Godinho and Ventura [23] showed that this bound suffices for k = 3 τ ⋅ 2 with τ ≥ 2 as well. Therefore, all pairs of diagonal forms of equal degree k in more than 2k 2 variables have a non-trivial 3-adic solution.…”