# Solubility of Additive Forms over Local Fields

7-2-2021

2021

## Degree Name

Doctor of Philosophy (PhD)

## Document Type

Doctoral Dissertation

## College

Arts and Sciences

## Department/School/Program

Mathematics

Dr. David B. Leep

## Abstract

Michael Knapp, in a previous work, conjectured that every additive sextic form over $\mathbb{Q}_2(\sqrt{-1})$ and $\mathbb{Q}_2(\sqrt{-5})$ in seven variables has a nontrivial zero. In this dissertation, I show that this conjecture is true, establishing that $$\Gamma^*(6, \mathbb{Q}_2(\sqrt{-1})) = \Gamma^*(6, \mathbb{Q}_2(\sqrt{-5})) = 7.$$ I then 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 $$K \in \{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\},$$ then $$\Gamma^*(d,K) = \frac{3}{2}d,$$ and if $$K \in \{\mathbb{Q}_2(\sqrt{-1}), \mathbb{Q}_2(\sqrt{-5})\},$$ then $$\Gamma^*(d,K) = d+1$$. Finally, I show that for degree $d=4$, if $$K \in \{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\},$$ then $$\Gamma^*(4, K) = 11.$$

## Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2021.277

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