  For a hypergraph $\mathcal H$, we consider the edge-induced and
vertex-induced subhypergraph polynomials and study their relation. We use this
relation to prove that both polynomials are reconstructible, and to prove a
theorem relating the Hilbert series of the Stanley-Reisner ring of the
independent complex of $\mathcal H$ and the edge-induced subhypergraph
polynomial. We also consider reconstruction of some algebraic invariants of
$\mathcal H$.
