Solubility of additive forms of twice odd degree over ramified quadratic extensions of $\mathbb {Q}_2$
Volume 201 / 2021
Acta Arithmetica 201 (2021), 149-164
MSC: 11D72, 11D88, 11E76.
DOI: 10.4064/aa201116-30-4
Published online: 28 October 2021
Abstract
We determine the minimal number $\Gamma ^*(d, K)$ of variables 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 $\{\mathbb {Q}_2(\sqrt {2}), \mathbb {Q}_2(\sqrt {10}), \mathbb {Q}_2(\sqrt {-2}), \mathbb {Q}_2(\sqrt {-10})\}$, then $\Gamma ^*(d,K) = \frac {3}{2}d$, and if $K$ is one of $\{\mathbb {Q}_2(\sqrt {-1}), \mathbb {Q}_2(\sqrt {-5})\}$, $\Gamma ^*(d,K) = d+1$. The case $d=6$ was previously known.
