Transfer principles and the Kato-Kuzumaki conjecture

TL;DR

This paper proves a transfer principle for the $C_i$ property in tame valued fields and applies it to confirm the Kato-Kuzumaki conjecture for several key fields, also establishing $ extbf{Q}_p$ as strong $C_1^1$.

Contribution

It introduces a transfer principle for the $C_i$ property in tame valued fields and verifies the Kato-Kuzumaki conjecture for important arithmetical fields, including $ extbf{Q}_p$.

Findings

The $C_i$ property lifts from residue fields to valued fields under certain conditions.

The Kato-Kuzumaki conjecture is proved for fields like $ extbf{C}(x_1,...,x_m)((t_1))...((t_n))$ and their perfections.

$ extbf{Q}_p$ satisfies the strong $C_1^1$ property, answering Wittenberg's question.

Abstract

We show that for tame valued fields of equal characteristic with divisible value group, the $C_i$ property lifts from the residue field to the valued field under suitable hypotheses on the residue field. We apply this transfer principle to prove Kato-Kuzumaki's conjecture in full generality for several arithmetically significant fields, for instance the field $\mathbf{C}(x_1,\dots,x_m)(\!(t_1)\!)\dots(\!(t_n)\!)$, and the perfections of both $\overline{\mathbf{F}}_p(x_1,\dots,x_m)(\!(t_1)\!)\dots(\!(t_n)\!)$ and $\mathbf{F}_p(\!(t_1)\!)\dots(\!(t_n)\!)$. Finally, we prove that $\mathbf{Q}_p$ satisfies the strong $C_1^1$ property, thereby answering a question of Wittenberg.