  In this paper, we consider multivariate hyperedge elimination polynomials and
multivariate chromatic polynomials for hypergraphs. The first set of
polynomials is defined in terms of a deletion-contraction-extraction
recurrence, previously investigated for graphs by Averbouch, Godlin, and
Makowsky. The multivariate chromatic polynomial is an equivalent polynomial
defined in terms of colorings, and generalizes the coboundary polynomial of
Crapo, and the bivariate chromatic polynomial of Dohmen, P\"onitz and Tittman.
We show that specializations of these new polynomials recover polynomials which
enumerate hyperedge coverings, matchings, transversals, and section
hypergraphs. We also prove that the polynomials can be defined in terms of
M\"obius inversion on the bond lattice of a hypergraph, as well as compute
these polynomials for various classes of hypergraphs.
