Solubility of Additive Quartic Forms over Ramified Quadratic Extensions of $\mathbb{Q}_2$

TL;DR

This paper determines the minimal number of variables needed for additive quartic forms over four specific ramified quadratic extensions of 2 to always have a nontrivial solution, extending understanding of solubility in local fields.

Contribution

It computes the exact minimal variable count * for additive quartic forms over these ramified quadratic extensions, a novel result for such extensions where degree is a power of the prime.

Findings

* = 11 for all four fields

First such computation for a proper extension of 2 with degree > 2

Extends solubility criteria to ramified quadratic extensions of 2

Abstract

We determine the minimal number of variables $\Gamma^*(d, K)$ 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 $\Gamma^*(4,K) = 11$. 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 $p$.