Power structures on the Grothendieck--Witt ring and the motivic Euler characteristic

TL;DR

This paper develops a power structure on the Grothendieck--Witt ring of a field that could unify symmetric powers of varieties and the motivic Euler characteristic, with compatibility shown in dimension zero cases.

Contribution

It introduces a new power structure on the Grothendieck--Witt ring and demonstrates its compatibility with existing structures in specific cases.

Findings

Power structure on Grothendieck--Witt ring constructed

Compatibility with symmetric powers and motivic Euler characteristic proposed

Compatibility verified for zero-dimensional varieties

Abstract

For $k$ a field, we construct a power structure on the Grothendieck--Witt ring of $k$ which has the potential to be compatible with symmetric powers of varieties and the motivic Euler characteristic. We then show this power structure is compatible with the power structure when we restrict to varieties of dimension $0$.