Solubility of Additive Forms of Twice Odd Degree over Q₂(^√5)

We prove that an additive form of degree d = 2m, m odd, m ≥ 3, over the unramified quadratic extension ℚ₂(^√5) has a nontrivial zero if the number of variables s satisifies s ≥ 4d+1. If 3 ∤ d, then there exists a nontrivial zero if s ≥3/2d + 1, this bound being optimal. We give examples of forms in 3d variables without a nontrivial zero in case that 3 | d.