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This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct [Formula: see text]-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of [Formula: see text]-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings. 

We prove that if f f is a reduced homogeneous polynomial of degree d d , then its F F -pure threshold at the unique homogeneous maximal ideal is at least 1 d − 1 \frac {1}{d-1} . We show, furthermore, that its F F -pure threshold equals 1 d − 1 \frac {1}{d-1} if and only if f ∈ m [ q ] f\in \mathfrak m^{[q]} and d = q + 1 d=q+1 , where q q is a power of p p . Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain. 

This article describes the Macaulay2 package FrobeniusThresholds, designed to estimate and calculate F-pure thresholds, more general F-thresholds, and related numerical invariants arising in the study of singularities in prime characteristic commutative algebra. 

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”. 

COoking with gas Low concentrations of carbon monoxide (CO) have shown therapeutic benefit in preclinical models, but safe delivery of appropriate dose has been challenging to achieve. Here, inspired by molecular gastronomy, Byrne et al . designed gas-entrapping materials (GEMs) using components generally recognized as safe, including xanthan gum, methylcellulose, maltodextrin, and corn syrup. Solid, hydrogel, and foam GEMs containing CO could deliver different concentrations of the gas to healthy rodents and pigs through noninhaled routes. In rodent models of colitis, acetaminophen overdose, and radiation-induced proctitis, rectally administered foam GEMs reduced tissue injury and inflammation. Foam GEMs could help achieve safe therapeutic CO delivery. 

This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.