Solubility of Additive Forms of Twice Odd Degree over $\mathbb{Q}_2(\sqrt{5})$

TL;DR

This paper establishes bounds on the number of variables needed for additive forms of certain degrees over the field _2() to have nontrivial zeros, with specific results depending on divisibility conditions.

Contribution

It provides new bounds for the solubility of additive forms of twice odd degree over _2(), including optimal bounds when 3 does not divide the degree.

Findings

Nontrivial zeros exist for s  4d+1 when m  3

Optimal bound s   3/2 d + 1 when 3  d

Examples of forms without zeros when 3  d in 3d variables

Abstract

We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there exists a nontrivial zero if $s \ge \frac{3}{2}d + 1$, this bound being optimal. We give examples of forms in $3d$ variables without a nontrivial zero in case that $3 \mid d$.