Solubility of Additive Sextic Forms over ${\mathbb{Q}}_2(\sqrt{-1})$ and ${\mathbb{Q}}_2(\sqrt{-5})$

Abstract

Michael Knapp, in a previous work, conjectured that every additive sextic form over ${\mathbb{Q}}_2(\sqrt{-1})$ and ${\mathbb{Q}}_2(\sqrt{-5})$ in seven variables has a nontrivial zero. In this paper, we show that this conjecture is true, establishing that ${\Gamma }^{\ast }(6,{\mathbb{Q}}_2(\sqrt{-1}))={\Gamma }^{\ast }(6,{\mathbb{Q}}_2(\sqrt{-5}))=7$.