Local solvability of diagonal equations (again)

Diagonal equations over unramified extensions of Qp

“…Moreover, there is no known counterexample to Artin's conjecture for a diagonal form for any such and . Upper bounds for have been given in .…”

Simultaneous diagonal equations over p-adic fields

“…Theorem 2 is a natural generalisation of Knapp [7,Theorem 1] and improves Dodson [6,Theorem 1] and Knapp [7,Theorem 3]. See also Skinner [11] for other references.…”

“…Theorem 1 has the virtue of being independent of K and can be compared with Skinner [11] where the bound N > k 6τ +4 is given for a single diagonal equation. Theorem 2 is a natural generalisation of Knapp [7,Theorem 1] and improves Dodson [6,Theorem 1] and Knapp [7,Theorem 3].…”

“…This is shown using a well-known "contraction" argument (see examples in [4] and [11]). The idea is to construct a primitive solution modulo p a+b in N = (Φ(R, k, a) + 1) · (Φ(R, k, b) + 1) variables as follows: First divide the left hand side of ( * ) into Φ(R, k, a) + 1 subsystems of diagonal forms, each in Φ(R, k, b) + 1 variables, and solve each system primitively modulo p b .…”

“…The next two lemmas and the final proof of Theorem 1 are much inspired by the ideas presented in Skinner [11].…”