Solubility of Additive Forms of Twice Odd Degree over ${\mathbb{Q}}_2(\sqrt{5})$

Abstract

We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge 3$, over the unramified quadratic extension ${\mathbb{Q}}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s\ge 4d+1$. If $3\nmid d$, then there exists a nontrivial zero if $s\ge \frac{3}{2}d+1$, this bound being optimal. We give examples of forms in $3d$ variables without a nontrivial zero in case that $3\mid d$.