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1. Bernstein–Sato functional equations, V-filtrations, and multiplier ideals of direct summands

   https://doi.org/10.1142/S0219199721500838  

   Àlvarez Montaner, Josep; Hernández, Daniel J.; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E. (December 2022, Communications in Contemporary Mathematics)

   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. 

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2. Bernstein’s Inequality and Holonomicity for Certain Singular Rings

   https://doi.org/10.1093/imrn/rnad121  

   Montaner, Josep Àlvarez; Hernández, Daniel J; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E (June 2023, International Mathematics Research Notices)

   Abstract In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic $$D$$-modules for rings of invariants of finite groups in characteristic zero, and for strongly $$F$$-regular finitely generated graded algebras with finite $$F$$-representation type in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic $$D$$-modules, in this context, have finite length, and we prove the existence of Bernstein–Sato polynomials in characteristic zero. We obtain these results using a more general version of Bernstein filtrations. 

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3. Bernstein-Sato theory modulo $p^m$

   https://doi.org/10.46298/epiga.2025.14739  

   Bitoun, Thomas; Quinlan-Gallego, Eamon (September 2025, Épijournal de Géométrie Algébrique)

   For fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $$\mathbb{Z} / p^m$$-coefficients, compatible with existing theory in the case $m = 1$. We show that the ``roots" of such polynomials are rational and we show that the negative roots agree with those of the mod-$$p$$ reduction. We give examples to show that, surprisingly, roots may be positive in this context. Moreover, our construction allows us to define a notion of ``strength" for roots by measuring $$p$$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$$p$$ reduction. Comments welcome. v3: final version. v2: fixed typos, small changes in notation, and additional example following suggestions from the referee 

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4. Lower bounds on Hilbert–Kunz multiplicities and maximal F -signatures

   https://doi.org/10.1017/S0305004122000238  

   JEFFRIES, JACK; NAKAJIMA, YUSUKE; SMIRNOV, ILYA; WATANABE, KEI–ICHI; YOSHIDA, KEN–ICHI (March 2023, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity. 

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5. Bernstein-Sato theory for arbitrary ideals in positive characteristic

   https://doi.org/10.1090/tran/8271  

   Quinlan-Gallego, Eamon (March 2021, Transactions of the American Mathematical Society)

   Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the F F -jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to F F -jumping numbers. 

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