  We propose a 'geometric Chevalley-Warning' conjecture, that is a motivic
extension of the Chevalley-Warning theorem in number theory. It is equivalent
to a particular case of a recent conjecture of F. Brown and O.Schnetz. In this
paper, we show the conjecture is true for linear hyperplane arrangements,
quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces
in $\Pbb^3$. The last section is devoted to verifying the conjecture for
certain special kinds of hypersurfaces of any dimension. As a by-product, we
obtain information on the Grothendieck classes of the affine 'Potts model'
hypersurfaces considered in \cite{aluffimarcolli1}.

Algebraic Geometry