Determining the Betti numbers of $R/(x^{p^e},y^{p^e},z^{p^e})$ for most even degree hypersurfaces in odd characteristic

Abstract: Let $k$ be a field of odd characteristic $p$. Fix an even number $d<p+1$ and a power $q\geq d+3$ of $p$. For most choices of degree $d$ standard graded hypersurfaces $R=k[x,y,z]/(f)$ with homogeneous maximal ideal $\mathfrak{m}$, we can determine the graded Betti numbers of $R/\mathfrak{m}^{[q]}$. In fact, given two fixed powers $q_0,q_1\geq d+3$, for most choices of $R$ the graded Betti numbers in high homological degree of $R/\mathfrak{m}^{[q_0]}$ and $R/\mathfrak{m}^{[q_1]}$ are the same up to a constant shift. This thesis shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-$q$-compressed polynomials in Betti numbers of the frobenius powers of the maximal ideal over certain hypersurfaces. We show that link-$q$-compressed polynomials are indeed fairly common in many polynomial rings.