Bernstein-Sato polynomials in commutative algebra
This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.
math.AC↗
arXiv subjects
Jack Jeffries
Publications and source records attributed to Jack Jeffries.
Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summands
This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-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 $V$-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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Faithfulness of Top Local Cohomology Modules in Domains
We study the conditions under which the highest nonvanishing local cohomology module of a domain $R$ with support in an ideal $I$ is faithful over $R$, i.e., which guarantee that $H^c_I(R)$ is faithful, where $c$ is the cohomological dimension of $I$. In particular, we prove that this is true for the case of positive prime characteristic when $c$ is the number of generators of $I$.
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Extensions of primes, flatness, and intersection flatness
We study when $R \to S$ has the property that prime ideals of $R$ extend to prime ideals or the unit ideal of $S$, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if $R$ is reduced, every maximal ideal of $R$ contains only finitely many minimal primes of $R$, and prime ideals of $R[X_1,\dots,X_n]$ extend to prime ideals of $S[X_1,\dots,X_n]$ for all $n$, then $S$ is flat over $R$. We give a counterexample to flatness over a reduced quasilocal ring $R$ with infinitely many minimal primes by constructing a non-flat $R$-module $M$ such that $M = PM$ for every minimal prime $P$ of $R$. We study the notion of intersection flatness and use it to prove that in certain graded cases it suffices to examine just one closed fiber to prove the stable prime extension property.
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A transformation rule for natural multiplicities
For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension étale in codimension one. Our result can be applied to bound the size of the local étale fundamental group of a singularity in terms of F-signature, recovering a recent result of Carvajal-Rojas, Schwede, and Tucker, and differential signature, providing the first characteristic-free effective bound.
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Quantifying singularities with differential operators
The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to compute. Motivated by different aspects of the $F$-signature, we define a numerical invariant for rings of characteristic zero or $p>0$ that exhibits many of the useful properties of the $F$-signature. We also compute many examples of this invariant, including cases where the $F$-signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials.
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Derived Functors of Differential Operators
In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive characteristic. In this note, we provide formulas for the ring of differential operators as well as these derived functors of differential operators. We apply these descriptions to show that differential operators behave well under reduction to positive characteristic under certain hypotheses. We show that these functors also detect a number of interesting properties of singularities.
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A Zariski-Nagata theorem for smooth $\mathbb{Z}$-algebras
In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the $n$-th symbolic power of a given prime ideal consists of the elements that vanish up to order $n$ on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use $p$-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of $p$-derivations to commutative algebra.
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Algebraic signatures of convex and non-convex codes
A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.
q-bio.NC↗
Mapping toric varieties into low dimensional spaces
A smooth $d$-dimensional projective variety $X$ can always be embedded into $2d+1$-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any $d$-dimensional projective variety can be mapped injectively to $2d+1$-dimensional projective space. A natural question then arises: what is the minimal $m$ such that a projective variety can be mapped injectively to $m$-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties.
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Polarization of Neural Rings
The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently, commutative algebra are some the mathematical areas that are involved in analyzing these codes. Neural rings and ideals are algebraic objects that create a bridge between mathematical neuroscience and commutative algebra. A neural ideal is an ideal in a polynomial ring that encodes the combinatorial firing data of a neural code. Using some algebraic techniques one hopes to understand more about the structure of a neural code via neural rings and ideals. In this paper, we introduce an operation, called "polarization," that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra.
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Local Okounkov bodies and limits in prime characteristic
This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call $p$-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature). We associate to each $p$-family of ideals an object in Euclidean space that is analogous to the Newton-Okounkov body of a graded family of ideals, which we call a $p$-body. Generalizing the methods used to establish volume formulas for the Hilbert-Kunz multiplicity and $F$-signature of semigroup rings, we relate the volume of a $p$-body to a certain asymptotic invariant determined by the corresponding $p$-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn-Minkowski theorem for Hilbert-Kunz multiplicity, and a uniformity result concerning the positivity of a $p$-family.
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What makes a neural code convex?
Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.
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On the behavior of singularities at the $F$-pure threshold
We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of \mustata-Takagi-Watanabe). We then study the $F$-signature function in the case where either the $F$-pure threshold and log canonical threshold coincide or where $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves similarly in those two cases. Finally, we include an appendix which shows that the test ideal can still behave in surprising ways even when the $F$-pure threshold and log canonical threshold coincide.
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Multiplicities of Classical Varieties
The $j$-multiplicity plays an important role in the intersection theory of Stückrad-Vogel cycles, while recent developments confirm the connections between the $ε$-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the $j$-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$- and $ε$-multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.
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Separating Invariants and Local Cohomology
The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G-action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples.
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The $j$-Multiplicity of Monomial Ideals
We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissier's volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the epsilon-multiplicity of a monomial ideal in terms of the volume of a region.
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Non-simplicial decompositions of Betti diagrams of complete intersections
We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij--Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from (Eisenbud-Schreyer, 2009). In higher codimension, obstructions arise that inspire our work on an alternative algorithm.
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