Unramified cohomology, $\A^1$-connectedness, and Chevalley-Warning problem in Grothendieck ring

TL;DR

This paper investigates the Chevalley-Warning problem within the Grothendieck ring framework, utilizing $ ext{A}^1$-homotopy theory to identify invariants and provide counterexamples to longstanding conjectures.

Contribution

It introduces $ ext{A}^1$-homotopy invariants in the Grothendieck ring and constructs a concrete counterexample to the Chevalley-Warning conjecture over a $C_1$-field.

Findings

The Brauer group is an invariant in the Grothendieck ring modulo $old{L}$.

A counterexample to the Chevalley-Warning conjecture over a $C_1$-field is provided.

The results give a negative answer to a question posed by Bilgin (2011).

Abstract

We study the Chevalley-Warning problem in the Grothendieck ring $K_0(Var/k)$. We show that the $\A^1$-homotopy theory yields well defined invariants on $K_0(Var/k)/\bold{L}$, in particular the Brauer group is such an invariant. We use this to give a concrete counter-example to the Chevalley-Warning conjecture over a $C_1$-field (Brown and Schnetz, 2011). This also gives a negative answer to the question in Bilgin (2011).