Nero Budur, Mircea Mustata, Morihiko Saito
We describe the roots of the Bernstein-Sato polynomial of a monomial ideal using reduction mod p and invariants of singularities in positive chracteristic. We give in this setting a positive answer to a problem of Takagi, Watanabe and the second author, concerning the dependence on the characteristic for these invariants of singularities.
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We define the Bernstein-Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.
Following work of Mustaţă and Bitoun we recently developed a notion of Bernstein-Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein-Sato polynomial. Here we prove that for monomial ideals the roots of the Bernstein-Sato polynomial (over $\mathbb{C}$) agree with the Bernstein-Sato roots of the mod-$p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic-$p$ notion of Bernstein-Sato root is reasonable.
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.
We give a combinatorial description of the roots of the Bernstein-Sato polynomial of a monomial ideal using the Newton polyhedron and some semigroups associated to the ideal.
A combinatorial criterion for detecting roots of Bernstein-Sato polynomials
提出检测理想Bernstein-Sato多项式根的组合判据，并应用于证明行列式理想和Pfaffian理想的强单值猜想。
Estimates for roots of Bernstein-Sato polynomials via log resolutions by orbifolds
本研究利用orbifold对数分解，对Bernstein-Sato多项式的根进行了相关估计，属于数学领域的理论研究
Bernstein-Sato functional equations for ideals in positive characteristic
本文将 Bernstein–Sato 函数方程推广至正特征 F-有限环中的理想，刻画其与 Bernstein–Sato 根的关系，并给出加权齐次孤立奇点多项式的根及 Thom–Sebastiani 性质。
On the Multiplicative Thom-Sebastiani Property for Bernstein-Sato Polynomials of Ideals
该研究证明理想的Bernstein-Sato多项式满足乘性Thom-Sebastiani性质，正面解决了相关领域的一个公开问题
Analytic Criteria for Detecting Bernstein-Sato Roots
该研究针对理想的伯恩斯坦-佐藤多项式根，提出一般偏微分方程判据，齐次情形下转化为解析条件，还将其应用于一般行列式理想，证明了该类理想的强单值群猜想。
We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m x n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.
The Intersection Structure of Bernstein-Sato Ideals
本文利用对数 D 模与 Gröbner 基证明 Bernstein-Sato 理想具有对称交性质，回答 Budur 的问题，并据此给出多元多项式整数次幂的 Bernstein-Sato 多项式公式。
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 class…
We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V-filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier ideals, and also a criterion for rational singularities in terms of the maximal root of the polynomial in the case of a reduced complete intersection. These…
In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider the analogous D-module in positive characteristic, and use it to define a sequence of Bernstein-Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by the F-jumping exponents of the hypersurface, in the sense of Hara and Yoshida.
Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset \text{Spec}(\mathcal{C})$. We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results by H. Biosca. When $\mathcal{C}$ is the ring of an algebraic or analytic space, we deduce a stratification of the space of the parameters such that on each stratum, there is a non zero rational polynomial which is a Bernstein-Sato poly…
Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-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 $F$-jumping numbers.
In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein-Sato polynomials for the structure sheaf $\mathcal{O}_X$ and a hypersurface $(f=0)$ in $X$, where $X$ is a regular variety over an $F$-finite field of positive characteristic (see arxiv:0711.3794). He shows that the suitably interpreted zeros of his Bernstein-Sato polynomials correspond to the jumping numbers of the test ideal filtration $τ(X,f^t)$. In the present paper we generalize Mustaţă's construction replacing $\mathcal{O}_X$ by an arbitrary $F$-regular Cartier module $M$ on $X$ and show an analogous c…
The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After surveying algorithms for computing the global b-function, we develop a new method to compute the local b-function for a single polynomial. We then develop algorithms that compute generalized Bernstein-Sato polynomials of Budur-Mustata-Saito and Shibuta for an arbitrary polynomial ideal. These lead to computations of log canonical thresholds, jumping coefficients, and multiplier ideals. Our algorithm for multiplier i…
Bernstein--Sato ideals might not be linear
本文借助 AI 构造了两个反例，证明 Budur 关于 Bernstein--Sato 理想由线性多项式乘积生成的猜想不成立，并指出 [Wu26] 的结果提供了最优线性性质。
We introduce and study invariants of singularities in positive characteristic called F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals in characteristic zero. We discuss the connection between the invariants of an ideal in characteristic zero and the invariants of the different reduction mod p of this ideal. Our main point is that this relation depends on arithmetic properties of p. We also describe a new connection between invariants mod p and the roots of the Bernstein-Sato polynomial.
We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically equivalent.
We generalize the Bernstein-Sato polynomials of Budur, Mustata and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein-Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. In order to prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.
We have recently proved a precise relation between Bernstein-Sato ideals of collections of polynomials and monodromy of generalized nearby cycles. In this article we extend this result to other ideals of Bernstein-Sato type.
We give estimates for the zero loci of Bernstein-Sato ideals. An upper bound is proved as a multivariate generalisation of the upper bound by Lichtin for the roots of Bernstein-Sato polynomials. The lower bounds generalise the fact that log-canonical thresholds, small jumping numbers of multiplier ideals, and their real versions provide roots of Bernstein-Sato polynomials.
We study the structure of $D$-modules over a ring $R$ which is a direct summand of a polynomial or a power series ring $S$ with coefficients over a field. We relate properties of $D$-modules over $R$ to $D$-modules over $S$. We show that the localization $R_f$ and the local cohomology module $H^i_I(R)$ have finite length as $D$-modules over $R$. Furthermore, we show the existence of the Bernstein-Sato polynomial for elements in $R$. In positive characteristic, we use this relation between $D$-modules over $R$ and $S$ to show that the set of $F$-jumping numbers of an ideal $I\subseteq R$ is con…
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 famil…
For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standar…
A Polynomial PDE Criterion for the $-n/d$ Root of the Bernstein-Sato polynomial of Homogeneous Ideals
针对由次数为d的齐次多项式生成的理想，给出判定其Bernstein-Sato多项式$-n/d$根的多项式PDE准则，应用于极大子式理想得特殊根$-n$，且该准则具稳定性。
Given $p$ polynomials of $n$ variables over a field $k$ of characteristic 0 and a point $a \in k^n$, we propose an algorithm computing the local Bernstein-Sato ideal at $a$. Moreover with the same algorithm we compute a constructible stratification of $k^n$ such that the local Bernstein-Sato ideal is constant along each stratum. Finally, we present non-trivial examples computed with our algorithm.
Zero loci of Bernstein-Sato ideals are not necessarily linear
本文构造了一个多项式映射，其Bernstein-Sato理想的零点轨迹非线性，从而否证了N. Budur关于该轨迹线性的猜想。
Bernstein-Sato polynomial and related invariants for meromorphic functions
本文建立亚纯函数的 Bernstein-Sato 多项式理论，用于分析阿基米德局部 zeta 函数的解析延拓，并提出乘子理想的解析与代数理论，揭示其跳跃数与多项式根的关系。
Bernstein-Sato ideals
本文综述了与超曲面奇点相关的Bernstein-Sato理想零点轨迹的若干结果，并介绍了相关历史、动机及进一步发展。
This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. We also make an extensive study of an example for which we give an expression of a generic (and under some conditions, a relative) Bernstein-Sato polynomial.
We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the product of X with an r-dimensional affine space. By combining this with results from [BMS], we relate invariants and properties of I to those of g. We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals.
We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice. We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the mu…
Given a smooth complex algebraic variety $X$ and a nonzero regular function $f$ on $X$, we give an effective estimate for the difference between the jumping numbers of $f$ and the $F$-jumping numbers of a reduction $f_p$ of $f$ to characteristic $p\gg 0$, in terms of the roots of the Bernstein-Sato polynomial $b_f$ of $f$. As an application, we show that if $b_f$ has no roots of the form $-{\rm lct}(f)-n$, with $n$ a positive integer, then the $F$-pure threshold of $f_p$ is equal to the log canonical threshold of $f$ for $p\gg 0$ with $(p-1){\rm lct}(f)\in {\mathbf Z}$.
We prove a conjecture of the first author relating the Bernstein-Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology.
We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at the origin are the same and moreover their graded Milnor algebras have the same Hilbert series. This shows that the roots of the Bernstein-Sato polynomials strictly supported at the origin cannot be determined uniquely by the Hilbert series of the Milnor algebras. This is contrary to certain hyperplane arrangement cases. It also implies that a nonzero torsio…
In this note we determine the Bernstein-Sato polynomial $b_Q(s)$ of a generic central arrangement $Q=\prod_{i=1}^kH_i$ of hyperplanes. We establish a connection between the roots of $b_Q(s)$ and the degrees of the generators for the top cohomology of the corresponding Milnor fiber. This connection holds for all homogeneous polynomials. We also introduce certain subschemes of the arrangement determined by the roots of $b_Q(s)$.
From Differential Values to Roots of the Bernstein-Sato Polynomial
本文证明了尖点的微分值半模如何决定其Bernstein-Sato多项式根的子集，并在重数$n\leq 4$时给出了更精确的结果。
We show that the Bernstein-Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange's formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than -2 and the multiplicity of the root -1 is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a metho…
In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein--Sato theory. We investigate the critical points of very affine varieties and study their asymptotic behavior. We relate these asymptotics to particular rays in the tropical variety as well as to Bernstein--Sato ideals and give a connection to Maximum Likelihood Estimation in Statistics.
On the Tensor Property of Bernstein-Sato Polynomial
本文证明了Bernstein-Sato多项式的乘法Thom-Sebastiani规则，推广至有效除子张量，并拓展到理想情形，回答了长期未解问题。
We compute the Bernstein-Sato polynomial of $f$, a function which given a pair $(M,v)$ in $X = M_n(\mathbf{C}) \times \mathbf{C}^n$ tests whether $v$ is a cyclic vector for $M$. The proof includes a description of shift operators corresponding to the Calogero-Moser operator $L_k$ in the rational case.
Microlocal Bernstein--Sato polynomials on singular ambient varieties
研究奇异环境簇上函数的微局部伯恩斯坦 - 佐藤多项式，扩展斋藤理论。定义最小指数研究相关奇点与霍奇滤过，得出包括与有理奇点关联等多个主要结果，还给出完全交情形下的有效算法及实现。
In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the $D[s]$-module $D[s] h^s$ admits a Spencer logarithmic resolution satisfies the symmetry property $b(-s-2) = \pm b(s)$. This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection $E$ and of its dual $E^*$ with respect to a free divisor of linear Jacobian type are related by the equality $b_{E}(s)=\…
This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in \textsc{Singular}. Among other, we use this algorithm in computational $D$-module theory for computing e. g. the Bernstein-Sato…
Let n and d be positive integers, let k be a field and let P(n,d;k) be the space of the polynomials in n variables of degree at most d with coefficients in k. Let B(n,d) be the set of the Bernstein-Sato polynomials of all polynomials in P(n,d;k) as k varies over all fields of characteristic 0. G. Lyubeznik proved that B(n,d) is a finite set and asked if, for a fixed k, the set of the polynomials corresponding to each element of B(n,d) is a constructible subset of P(n,d;k). In this paper we give an affirmative answer to Lyubeznik's question by showing that the set in question is indeed construc…
For a homogeneous polynomial of $n$ variables, we present a new method to compute the roots of Bernstein-Sato polynomial supported at the origin, assuming that general hyperplane sections of the associated projective hypersurface have at most weighted homogeneous isolated singularities. Calculating the dimensions of certain $E_r$-terms of the pole order spectral sequence for a given integer $r\in[2,n]$, we can detect its degeneration at $E_r$ for certain degrees. In the case of strongly free, locally positively weighted homogeneous divisors on ${\mathbb P}^3$, we can prove its degeneration alm…
Bernstein--Sato Theory for D-modules in Positive Characteristic
本文发展了复数域上微分D-模块伯恩斯坦-萨托理论的正特征版本，研究了有限生成单位Cartier模的根的有限性和有理性。
Topological roots of the Bernstein-Sato polynomial of plane curves
研究任意平面曲线奇点局部Bernstein-Sato多项式的一组拓扑根，用除子赋值和对数极小分解数值数据刻画，证明对n=2的强单值性猜想的重数部分。
The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of…
Multiplicative Thom-Sebastiani for Bernstein-Sato polynomials
该研究在光滑复代数簇上的正则函数乘积场景，证明了乘积函数的Bernstein-Sato多项式等于各因子多项式的乘积，解决了Budur与Popa提出的相关问题。
On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities
本文给出了 Thom-Sebastiani 奇点约化 Bernstein-Sato 多项式关系的纯代数证明，适用于任意可定义该多项式的设定。
An algorithm for annihilator and Bernstein-Sato polynomial of a rational function
本文提出一个算法用于计算有理函数的Bernstein-Sato多项式，通过非交换PBW代数中的Gröbner基实现，支持现有猜想并展示非平凡例子。
Given a family of analytic functions near 0 \in C^n parametrized by a smooth space, we study the Bernstein polynomial of the fiber on an irreducible variety V of the space of parameters and we show that it is generically constant. We prove that this polynomial b(s) satisfies a functional equation on V from which we derive a contructible stratification of the space of parameters by the Bernstein polynomial of the fiber. When the hypersurface admits generically a unique singularity at 0 \in C^n, we prove that b(s) is the generic Bernstein polynomial in the sense of Briançon-Geandier-Maisonobe. T…
On a Bernstein-Sato polynomial of a meromorphic function
该研究定义了亚纯函数的Bernstein-Sato多项式，证明了相关Kashiwara-Malgrange型定理，发现亚纯情形下多个b函数对应相同Milnor单值群特征值，还引入亚纯函数乘子理想层并建立其与b函数的关联。
For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically only a few roots are obtained this way. Following ideas of Veys, we study the following question. Is it possible to find a collection G of polynomials g in C[x_1,...,x_n], such that, for all g in G, every pole of the topological zeta function associated to I and the volume form gdx on the affine n-space, is a root of the Bernstein-Sato polynomial of I, and su…
For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying the functional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in \text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdots f_{r}^{s_{r} + 1}.$ Generalizing techniques due to Maisonobe, we compute the zero locus of the standard Bernstein--Sato ideal in the sense of Budur (…
The Bernstein-Sato Polynomial of the Hankel Determinant
本文确定通用对称Hankel矩阵行列式的整体Bernstein--Sato多项式，通过近旁循环、阿基米德zeta积分及Gauss--Radau参数化等方法，证明了极点与根的关系及单值类的唯一性。
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