Stability of the $C^q_i$ properties under field extensions

TL;DR

This work investigates the stability of Kato-Kuzumaki $C_i^q$ properties under both transcendental and finite algebraic extensions, and develops transfer principles that move these norm-based conditions from a base field to its function fields and Laurent series. The core technical tool is a Lang-style construction and analysis of l-normic forms that control when a symbol in Milnor $K$-theory lies in a norm group, allowing finite-extension stability results and descent from extensions to the base field. The authors prove new transfer theorems for constant and non-constant Milnor symbols, derive a broad set of applications to function fields over finite fields, $p$-adic fields, and totally imaginary number fields, and obtain concrete verifications such as $F_p(x_1, rainspots x_n)$ satisfying $C_n^1$. The results advance understanding of how Diophantine-type properties encoded in $C_i^q$ behave under field operations, with potential implications for broader connections between cohomological dimension and Diophantine geometry.

Abstract

In this paper we study the stability of variations of Kato-Kuzumaki's $C_i^q$ property under transcendental and algebraic field extensions. As an application, we obtain the $C_n^1$ property for the field ${\bf F}_p(x_1,\dots,x_n)$.

Theorems & Definitions (84)

• Conjecture
• Theorem A: Corollary \ref{['cor Cn1 function finite']}
• Theorem B: Theorem \ref{['thm constant transfer']}
• Theorem C: Theorem \ref{['thm: transferability under finite alg extension']}
• Theorem D: Theorem \ref{['thm transfer applications']}
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof