Exact values of Γ∗(10,p).

For k ∈ ℕ and p a prime number, define Γ ∗ (k , p) to be the smallest positive integer n ∈ ℕ such that any diagonal form f (x 1 , ... , x s) = a 1 x 1 k + ⋯ + a s x s k , with integer coefficients, has nontrivial zero over ℚ p whenever s ≥ n. A special case of a conjecture attributed to Artin states that Γ ∗ (k , p) ≤ k 2 + 1. It is well known that the equality occurs when p = k + 1. In this paper, we obtain the exact values of Γ ∗ (1 0 , p) for all primes p and, except for p = 1 1 , these values are much lower than those established in the conjecture, as might be expected. [ABSTRACT FROM AUTHOR]