Transfer principles and the Kato-Kuzumaki conjecture
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912987304951808 |
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| author | Gambardella, Felipe Kartas, Konstantinos |
| author_facet | Gambardella, Felipe Kartas, Konstantinos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01815 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transfer principles and the Kato-Kuzumaki conjecture Gambardella, Felipe Kartas, Konstantinos Number Theory K-Theory and Homology Logic 12L12, 14J70, 12G10, 19D45, 03C60 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. |
| title | Transfer principles and the Kato-Kuzumaki conjecture |
| topic | Number Theory K-Theory and Homology Logic 12L12, 14J70, 12G10, 19D45, 03C60 |
| url | https://arxiv.org/abs/2603.01815 |