  An integer vector $b \in \mathbb{Z}^d$ is a degree sequence if there exists a
hypergraph with vertices $\{1,\dots,d\}$ such that each $b_i$ is the number of
hyperedges containing $i$. The degree-sequence polytope $\mathscr{Z}^d$ is the
convex hull of all degree sequences. We show that all but a $2^{-\Omega(d)}$
fraction of integer vectors in the degree sequence polytope are degree
sequences. Furthermore, the corresponding hypergraph of these points can be
computed in time $2^{O(d)}$ via linear programming techniques. This is
substantially faster than the $2^{O(d^2)}$ running time of the current-best
algorithm for the degree-sequence problem. We also show that for $d\geq 98$,
the degree-sequence polytope $\mathscr{Z}^d$ contains integer points that are
not degree sequences. Furthermore, we prove that the linear optimization
problem over $\mathscr{Z}^d$ is $\mathrm{NP}$-hard. This complements a recent
result of Deza et al. (2018) who provide an algorithm that is polynomial in $d$
and the number of hyperedges.
