Stable birational equivalence and geometric Chevalley-Warning (Q2855885)

The classical Chevalley-Warning theorem in number theory states that a certain set of polynomial equations has a solution over a finite field as long as the total degree of the polynomials is smaller than the number of variables. The paper under review proposes a conjecture, which can be viewed as a geometric version of the Chevalley-Warning theorem. Given an algebraically closed base field \(k\), let \(K_0(var)\) be the Grothendieck ring of varieties and let \(\mathbb{L}\) be the class of \(\mathbb{A}_k^1\). The conjecture predicts that in \(K_0(var)\) the class of a variety, which is defined by a set of homogeneous polynomials of total degree less than the number of variables, is congruent to 1 modulo \(\mathbb{L}\). By a result of \textit{M. Larsen} and \textit{V. A. Lunts} [Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)], the property of being ``congruent to 1 modulo \(\mathbb{L}\)'' is essentially the same as being stably rational. Hence the geometric Chevalley-Warning becomes interesting in the study of rationality problems. The author gives some examples of low degree hypersurfaces being conguent to 1 modulo \(\mathbb{L}\). These includes quadratic hypersurfaces and singular cubic hypersurfaces.