Stability of the Kato-Kuzumaki's properties under field extensions

TL;DR

This paper investigates how certain Diophantine properties of fields, known as $C_i^q$ properties, behave under field extensions, and applies these results to specific fields like $ extbf{F}_p(x_1, imes,x_n)$.

Contribution

It demonstrates the stability of variants of $C_i^q$ properties under field extensions and establishes the $C_n^1$ property for rational function fields over finite fields.

Findings

Proves stability of $C_i^q$ properties under algebraic and transcendental extensions.

Establishes the $C_n^1$ property for $ extbf{F}_p(x_1, imes,x_n)$.

Provides insights into the cohomological dimension characterization of fields.

Abstract

In 1986, Kato and Kuzumaki introduced some Diophantine properties of fields, called the $C_i^q$ properties, and they hoped they would provide a good characterization of the cohomological dimension of fields. In this paper, we study the stability of some variants of the $C_i^q$ properties under transcendental and algebraic extensions. As an application, we obtain the $C_n^1$ property for the field $\mathbf{F}_p(x_1,\cdots,x_n)$.