Solubility of Additive Quartic Forms 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 $d=4$ over the four ramified quadratic extensions ${\mathbb{Q}}_2(\sqrt{2}),{\mathbb{Q}}_2(\sqrt{10}),{\mathbb{Q}}_2(\sqrt{-2}),{\mathbb{Q}}_2(\sqrt{-10})$ of ${\mathbb{Q}}_2$. In all four fields, we prove that ${\Gamma }^{\ast }(4,K)=11$. This is the first example of such a computation for a proper extension of ${\mathbb{Q}}_p$ where the degree is a power of $p$ greater than $p$.

arXiv commentarXiv admin note: text overlap with arXiv:2010.06833