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

TL;DR

This paper proves that every additive sextic form in seven variables over the fields () and () has a nontrivial zero, confirming a conjecture by Michael Knapp.

Contribution

The paper confirms Knapp's conjecture by establishing the minimal number of variables needed for additive sextic forms over these fields to have nontrivial zeros.

Findings

Confirmed Knapp's conjecture for () and ()

Established () and () as 7 for additive sextic forms

Proved universal solubility in seven variables over the specified fields

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^*(6, \mathbb{Q}_2(\sqrt{-1})) = \Gamma^*(6, \mathbb{Q}_2(\sqrt{-5})) = 7 $.