Stable $\A^1$-connectedness

TL;DR

This paper introduces the concept of stable $ ext{A}^1$-connectedness, a stabilized form of $ ext{A}^1$-connectedness, linking it to zero cycles of degree one rather than rational points, and proves a related conjecture.

Contribution

It defines stable $ ext{A}^1$-connectedness and proves a stabilized version of a conjecture on $ ext{A}^1$-connectedness.

Findings

Introduction of stable $ ext{A}^1$-connectedness concept

Proof of a stabilized conjecture on $ ext{A}^1$-connectedness

Connection to existence of zero cycles of degree one

Abstract

We prove in this note a stabilized version of a conjecture on $\A^1$-connectedness. For the stabilized version of this conjecture, we introduce the notion of stable $\A^1$-connectedness, which is can be seen as the stabilization of $\A^1$-connectedness. The notion of stable $\A^1$-connectedness is in connection to the existence of zero cycles of degree one rather than rational points.