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

Drew Duncan; David B. Leep

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arXiv · 2010.06833

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

Drew Duncan·David B. Leep

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Abstract

We determine the minimal number of variables $\Gamma^*(d, K)$ 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 $\{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\}$, $\Gamma^*(d,K) = \frac{3}{2}d$, 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.

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BibTeX RIS

Drew Duncan, David B. Leep. 2020-10-12. Solubility of Additive Forms of Twice Odd Degree over Ramified Quadratic Extensions of $\mathbb{Q}_2$. https://arxiv.org/abs/2010.06833

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