Transfer principles and the Kato-Kuzumaki conjecture

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.

arXiv comment31 pages. Comments are welcome :)