Determining the Betti numbers of R/(xpe,ype,zpe) for most even degree hypersurfaces in odd characteristic

Let k be a field of odd characteristic p. Fix an even number $d<p+1$ and a power $q\ge d+3$ of p. For most choices of degree d standard graded hypersurfaces $R=k[x,y,z]/\left(f\right)$ with homogeneous maximal ideal $m$, we can determine the graded Betti numbers of $R/m^{\left[q\right]}$. In fact, given two fixed powers $q_0,q_1\ge d+3$, for most choices of R the graded Betti numbers in high homological degree of $R/m^{\left[q_0\right]}$ and $R/m^{\left[q_1\right]}$ are the same up to a constant shift. This paper shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-q-compressed polynomials in [13]. We show that link-q-compressed polynomials are indeed fairly common in many polynomial rings.