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

TL;DR

This paper establishes conditions under which additive forms of twice odd degree over totally ramified extensions of 2 have nontrivial zeros, extending understanding of solubility in local fields.

Contribution

It proves a new bound on the number of variables needed for additive forms of degree 2m, with m odd, over totally ramified extensions of 2 to have nontrivial solutions.

Findings

Nontrivial zeros exist when variables s ge 8d^2 + 3d + 1

Extends solubility results to totally ramified extensions of 2

Provides explicit bounds for additive forms of twice odd degree

Abstract

We prove that an additive form of degree $d=2m$, $m$ odd over any totally ramified extension of $\mathbb{Q}_2$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge \frac{d^2}{4} + 3d + 1$.