We study lower bound estimates for the number of solutions systems equations over finite fields. Heath-Brown improved bounds given by classical Chevalley-Warning Theorems excluding whose form an affine space. improve each Heath-Brown's results and demonstrate sharpness in several cases.

In this paper, we construct a new sequence of generators of the BRST complex and reformulate the BRST differential so that it acts on elements of the complex much like the Maurer-Cartan differential acts on left-invariant forms. Thus our BRST differential is formally analogous to the differential defined on the BRST formulation of the Chevalley-Eilenberg cochain complex of a Lie algebra. Moreov...

where H;(_) denotes the Lie algebra homology, HC*(_) cyclic homology, and A is an algebra with unit over a field of characteristic zero. In this paper we give an alternative proof of this theorem which does not involve Weyl's invariant theory for GL(C). We use this approach to compute Chevalley-Eilenberg homology in some interesting new cases. In Section 1, we begin by proving (in characteristi...

Let G be a connected reductive algebraic group with Lie algebra g. The ground field k is algebraically closed and of characteristic zero. Fundamental results in invariant theory of the adjoint representation ofG are primarily associated with C. Chevalley and B. Kostant. Especially, one should distinguish the ”Chevalley restriction theorem” and seminal article of Kostant [5]. Later, Kostant and ...

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P. The formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. The construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxte...