More Like this

1. 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. 

   more » « less
2. Bernstein-Sato theory for singular rings in positive characteristic

   https://doi.org/10.1090/tran/8917  

   Jeffries, Jack; Núñez-Betancourt, Luis; Quinlan-Gallego, Eamon (July 2023, Transactions of the American Mathematical Society)

   The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case. 

   more » « less
3. A generalization of the b -function lemma

   https://doi.org/10.1112/S0010437X21007491  

   Raskin, Sam (October 2021, Compositio Mathematica)

   We establish some cohomological bounds in $$D$$ -module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the $$b$$ -function lemma for non-holonomic $$D$$ -modules. 

   more » « less
4. Limit F -signature functions of two-variable binomial hypersurfaces

   https://doi.org/10.1017/nmj.2025.10094  

   Brosowsky, Anna; Coskun, Izzet; Pande, Suchitra; Tucker, Kevin (January 2026, Nagoya Mathematical Journal)

   Abstract TheF-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strongF-regularity, an important class of singularities related to KLT singularities in characteristic zero. In this article, we compute the limitingF-signature function of binomial and other related hypersurfaces in two variables as the characteristic$$p \to \infty $$. In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume. 

   more » « less
5. Higher resonance schemes and Koszul modules of simplicial complexes

   https://doi.org/10.1007/s10801-024-01313-2  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Sammartano, Alessio; Suciu, Alexander I (March 2024, Journal of Algebraic Combinatorics)

   Abstract Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$ $k$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$ $A=k⟨\Delta ⟩$ is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$ $\Delta$ , we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. 

   more » « less