Solubility of Additive Forms of Twice Odd Degree over ℚ 2 (5)

We prove that an additive form of degree d=2m, m odd, m≥3, over the unramified quadratic extension ℚ 2 (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.