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arXiv · 2603.01815

Transfer principles and the Kato-Kuzumaki conjecture

Felipe Gambardella·Konstantinos Kartas
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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.

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Felipe Gambardella, Konstantinos Kartas. 2026-03-02. Transfer principles and the Kato-Kuzumaki conjecture. https://arxiv.org/abs/2603.01815

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