Title: A generalization of the Hasse-Witt matrix of a hypersurface,
Abstract: The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite
field of characteristic $p$ gives essentially complete mod $p$ information
about the zeta function of the hypersurface. But if the degree $d$ of the
hypersurface is $\leq n$, the zeta function is trivial mod $p$ and the
Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the
Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the
zeta function for all $d$. We also describe the differential equations
satisfied by this matrix and prove that it is generically invertible.