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

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).