On the equation of degree 6

In this paper we study the Schwarz genus for the covering of the space of polynomials with distinct roots by its roots. We show that, for the first unknown case (degree 6), the genus is strictly less than the one predicted by dimension arguments, contrary to what happens in all other reflection groups

Vector partition functions and index of transversally elliptic operators

Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah-Singer which is in a sense universal for index computations

Vector partition function and generalized Dahmen-Micchelli spaces

21 pages.This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in a subsequent paper