Solubility of Additive Forms of Twice Odd Degree over Ramified Quadratic Extensions of ${\mathbb{Q}}_2$

Abstract

We determine the minimal number of variables ${\Gamma }^{\ast }(d,K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m\ge 3$ over the six ramified quadratic extensions of ${\mathbb{Q}}_2$. We prove that if $K$ is one of $\{{\mathbb{Q}}_2(\sqrt{2}),{\mathbb{Q}}_2(\sqrt{10}),{\mathbb{Q}}_2(\sqrt{-2}),{\mathbb{Q}}_2(\sqrt{-10})\}$, ${\Gamma }^{\ast }(d,K)=\frac{3}{2}d$, and if $K$ is one of $\{{\mathbb{Q}}_2(\sqrt{-1}),{\mathbb{Q}}_2(\sqrt{-5})\}$, ${\Gamma }^{\ast }(d,K)=d+1$. The case $d=6$ was previously known.

arXiv admin note: text overlap with arXiv:2005.09770