Fedder-type criterion for quasi-$F^e$-splitting and quasi-$F$-regularity

We study quasi-$F^e$-split and quasi-$F$-regular singularities, which generalize Yobuko’s quasi-$F$-splitting. We establish Fedder type criteria that characterize these properties for hypersurfaces. These criteria offer explicit tools for computation and verification. As an application, we construct a counterexample to the inversion of adjunction for quasi-$F$-regularity and compute the quasi-$F$-split threshold of the cone over the ordinary cusp.