Generalized F -depth and graded nilpotent singularities

In this article, we provide explicit and effective bounds for Frobenius test exponents, integers crucial for computing Frobenius closures of parameter ideals, in new settings and articulate their explicit computation along common graded constructions. We do this via the study of variants of F-nilpotent singularities. In particular, we explore how (generalized) weakly F-nilpotent singularities behave under Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. To accomplish these tasks, we introduce the generalized F-depth in analogy to Lyubeznik's F-depth. These depth-like invariants track (generalized) weakly F-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.