  The main goal of this paper is to prove, in positive characteristic $p$,
stability behavior for the graded Betti numbers in the periodic tails of the
minimal resolutions of Frobenius powers of the homogeneous maximal ideals for
very general choices of hypersurface in three variables whose degree has the
opposite parity to that of $p$. We also find some of the structure of the
matrix factorization giving the resolution. We achieve this by developing a
method for obtaining the degrees of the generators of the defining ideal of an
$\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from its socle
degree, where $\mathfrak{c}$ is a Frobenius power of the homogeneous maximal
ideal. As an application, we also obtain the Hilbert-Kunz function of the
hypersurface ring, as well as the Castelnuovo-Mumford regularity of the
quotients by Frobenius powers of the homogeneous maximal ideal.
