Transfer principles and the Kato-Kuzumaki conjecture

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 $\mathbf{C}(x_1,\dots ,x_m)(\text{\hspace{0.17em}⁣}(t_1)\text{\hspace{0.17em}⁣})\dots (\text{\hspace{0.17em}⁣}(t_n)\text{\hspace{0.17em}⁣})$, and the perfections of both ${\bar{\mathbf{F}}}_p(x_1,\dots ,x_m)(\text{\hspace{0.17em}⁣}(t_1)\text{\hspace{0.17em}⁣})\dots (\text{\hspace{0.17em}⁣}(t_n)\text{\hspace{0.17em}⁣})$ and ${\mathbf{F}}_p(\text{\hspace{0.17em}⁣}(t_1)\text{\hspace{0.17em}⁣})\dots (\text{\hspace{0.17em}⁣}(t_n)\text{\hspace{0.17em}⁣})$. Finally, we prove that ${\mathbf{Q}}_p$ satisfies the strong $C_1^1$ property, thereby answering a question of Wittenberg.