Bernstein–Sato Ideals

• Bernstein–Sato ideals are multivariable D-module invariants attached to tuples of functions or ideals, generalizing the classical Bernstein–Sato polynomial.
• They capture singularity-theoretic, birational, and topological information via zero loci comprised of rational affine hyperplanes linked to multiplier ideals and local system supports.
• Computational methods leveraging Gröbner bases and D-module techniques enable effective analysis of these invariants, connecting algebraic analysis with geometric and topological applications.

Bernstein–Sato ideals are multivariable $D$-module invariants attached to collections of functions or, more generally, to tuples of ideals. They extend the classical Bernstein–Sato polynomial $b_f(s)$, which is defined for a single function by a functional equation $b_f(s)\,f^s=P(s)\,f^{s+1}$, to settings in which several exponents vary simultaneously or in which a single ideal is encoded by auxiliary variables (Budur et al., 2019). In the one-variable case the Bernstein–Sato ideal is principal and generated by the classical $b$-function; in the multivariable case one obtains an ideal in a polynomial ring $\mathbb{C}[s_1,\dots,s_r]$, whose zero locus records singularity-theoretic, birational, and topological information, including multiplier-ideal jumping loci and monodromy support loci (Budur et al., 2019, Budur et al., 2021, Budur et al., 2024).

1. Classical origin and the passage from functions to ideals

For a nonzero function $f$ on a smooth complex algebraic variety or complex manifold, the Bernstein–Sato polynomial $b_f(s)\in\mathbb{C}[s]$ is the monic polynomial of minimal degree for which there exists a differential operator $P(s)\in D_X[s]$ satisfying

$b_f(s)\,f^s=P(s)\,f^{s+1}.$

Equivalently, if $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$, then $b_f(s)$0 is the monic generator of the annihilator in $b_f(s)$1 of the cyclic element $b_f(s)$2 (Montaner, 2021). In this classical case, roots of $b_f(s)$3 are negative rational numbers, and the negative of the largest root yields the log-canonical threshold or minimal exponent, depending on context (Mustata, 2019, Budur et al., 2024).

A first generalization replaces a principal ideal by an arbitrary ideal $b_f(s)$4. Budur–Mustaţă–Saito defined a Bernstein–Sato polynomial $b_f(s)$5 using a multi-parameter $b_f(s)$6-module construction, and Mustaţă later showed that this polynomial can be recovered from a single auxiliary hypersurface

$b_f(s)$7

on $b_f(s)$8, via the identity

$b_f(s)$9

where $b_f(s)\,f^s=P(s)\,f^{s+1}$0 is the classical Bernstein–Sato polynomial of $b_f(s)\,f^s=P(s)\,f^{s+1}$1 and $b_f(s)\,f^s=P(s)\,f^{s+1}$2 is its reduced Bernstein–Sato polynomial (Mustata, 2019). This reduction from an ideal to a single function is structurally important: it gives a new proof that $b_f(s)\,f^s=P(s)\,f^{s+1}$3 exists and depends only on the ideal, not on the chosen generators (Mustata, 2019).

The same construction underlies later developments for tuples of ideals. In particular, for a single ideal $b_f(s)\,f^s=P(s)\,f^{s+1}$4, Mustaţă’s hypersurface $b_f(s)\,f^s=P(s)\,f^{s+1}$5 is the model for defining reduced Bernstein–Sato invariants that are independent of generators and compatible with multiplier ideals (Montaner, 2021).

2. Multivariable Bernstein–Sato ideals for tuples of functions

For a tuple of functions

$b_f(s)\,f^s=P(s)\,f^{s+1}$6

the natural multivariable analogue is the Bernstein–Sato ideal $b_f(s)\,f^s=P(s)\,f^{s+1}$7. It is defined by requiring that $b_f(s)\,f^s=P(s)\,f^{s+1}$8 if there exists $b_f(s)\,f^s=P(s)\,f^{s+1}$9 such that

$b$0

When $b$1, this ideal is principal and generated by the classical Bernstein–Sato polynomial $b$2 (Budur et al., 2019, Wu, 11 Sep 2025). For $b$3, the ideal is typically not principal, and its zero locus may have components of codimension $b$4 (Budur et al., 2019).

A broader family of generalized ideals is indexed by $b$5. The ideal $b$6 consists of all polynomials $b$7 for which

$b$8

for some $b$9 (Wu, 11 Sep 2025). The case $\mathbb{C}[s_1,\dots,s_r]$0 recovers the usual $\mathbb{C}[s_1,\dots,s_r]$1 (Wu, 11 Sep 2025).

The zero locus

$\mathbb{C}[s_1,\dots,s_r]$2

is a fundamental geometric object. In the one-variable case, it is simply the set of roots of $\mathbb{C}[s_1,\dots,s_r]$3. In the multivariable case, codimension-one components are hyperplanes, and components of codimension $\mathbb{C}[s_1,\dots,s_r]$4 can occur (Budur et al., 2019). The multivariable theory is therefore genuinely richer than the principal case and is designed to capture interactions among several hypersurfaces rather than a single divisor.

A different but compatible multivariable formalism appears in the generalized Bernstein–Sato ideals used for arrangements and restricted functional equations. For a factorization $\mathbb{C}[s_1,\dots,s_r]$5, a divisor $\mathbb{C}[s_1,\dots,s_r]$6, and another divisor $\mathbb{C}[s_1,\dots,s_r]$7, one considers the ideal $\mathbb{C}[s_1,\dots,s_r]$8 of polynomials $\mathbb{C}[s_1,\dots,s_r]$9 satisfying

$f$0

The standard Bernstein–Sato ideal in the sense of Budur is recovered by taking $f$1 and $f$2 (Bath, 2019). This generality is particularly effective for hyperplane arrangements, where explicit combinatorial descriptions of zero loci become possible (Bath, 2019).

3. Bernstein–Sato ideals for tuples of ideals

A further extension, introduced in "A note on Bernstein-Sato ideals" (Montaner, 2021), associates a Bernstein–Sato ideal to a tuple of ideals

$f$3

in $f$4 or $f$5. For each $f$6, choosing generators

$f$7

one introduces new variables $f$8 and defines

$f$9

This produces a tuple of hypersurfaces $b_f(s)\in\mathbb{C}[s]$0 in a larger ring $b_f(s)\in\mathbb{C}[s]$1, and one defines the Bernstein–Sato ideal $b_f(s)\in\mathbb{C}[s]$2 by the condition that

$b_f(s)\in\mathbb{C}[s]$3

for some $b_f(s)\in\mathbb{C}[s]$4 (Montaner, 2021).

Because the $b_f(s)\in\mathbb{C}[s]$5 are pairwise without common factors, one has

$b_f(s)\in\mathbb{C}[s]$6

and the reduced Bernstein–Sato ideal $b_f(s)\in\mathbb{C}[s]$7 is defined by dividing out this universal factor (Montaner, 2021). The Bernstein–Sato ideal of the tuple of ideals is then

$b_f(s)\in\mathbb{C}[s]$8

A key theorem states that $b_f(s)\in\mathbb{C}[s]$9 is independent of the chosen generators of the $P(s)\in D_X[s]$0, so $P(s)\in D_X[s]$1 is a well-defined invariant of the tuple (Montaner, 2021).

This construction specializes correctly in two basic directions. If $P(s)\in D_X[s]$2, then $P(s)\in D_X[s]$3 is generated by the reduced Bernstein–Sato polynomial of Mustaţă’s universal linear combination $P(s)\in D_X[s]$4, hence recovers the one-variable Bernstein–Sato polynomial of an ideal (Montaner, 2021). If each $P(s)\in D_X[s]$5 is principal, $P(s)\in D_X[s]$6, then $P(s)\in D_X[s]$7 coincides with the reduced Bernstein–Sato ideal of the tuple of functions $P(s)\in D_X[s]$8 in Sabbah’s sense (Montaner, 2021).

The note also records basic properties inherited from $P(s)\in D_X[s]$9: nontriviality, localization as an intersection of local Bernstein–Sato ideals, and divisibility by $b_f(s)\,f^s=P(s)\,f^{s+1}.$0 before reduction (Montaner, 2021). These properties place tuple-of-ideals Bernstein–Sato theory on the same formal footing as the already established theories for one function, one ideal, and tuples of functions.

4. Zero loci, monodromy, multiplier ideals, and jumping loci

The most developed geometric interpretation of Bernstein–Sato ideals concerns their zero loci. For a tuple of functions $b_f(s)\,f^s=P(s)\,f^{s+1}.$1, the zero locus $b_f(s)\,f^s=P(s)\,f^{s+1}.$2 is related to the support $b_f(s)\,f^s=P(s)\,f^{s+1}.$3 of Sabbah’s specialization complex and, equivalently, to cohomology support loci of rank-one local systems on local complements (Budur et al., 2019). The decisive statement is

$b_f(s)\,f^s=P(s)\,f^{s+1}.$4

which generalizes the Malgrange–Kashiwara theorem from one function to several (Budur et al., 2019). In particular, $b_f(s)\,f^s=P(s)\,f^{s+1}.$5 is a finite union of torsion-translated complex affine subtori of codimension $b_f(s)\,f^s=P(s)\,f^{s+1}.$6, and every codimension-one irreducible component of $b_f(s)\,f^s=P(s)\,f^{s+1}.$7 is a hyperplane

$b_f(s)\,f^s=P(s)\,f^{s+1}.$8

with $b_f(s)\,f^s=P(s)\,f^{s+1}.$9 and $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$0 (Budur et al., 2019).

This picture extends to generalized ideals $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$1. If $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$2 and $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$3 is not invertible, there is a corresponding restricted monodromy support $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$4, and one has

$$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$5

Codimension-one components of $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$6 are again hyperplanes of the same rational type, with the additional positivity condition that some coefficient attached to an index $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$7 with $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$8 is nonzero (Budur et al., 2020, Budur et al., 2024).

A parallel geometric interpretation arises from multiplier ideals. In the one-variable case, jumping numbers of multiplier ideals in $$M_f=D_X[s]\cdot f^s\subset \mathcal{O}_X[f^{-1}][s]\cdot f^s$$9 give roots of the Bernstein–Sato polynomial for a function or an ideal (Montaner, 2021). For tuples of ideals $b_f(s)$00, mixed multiplier ideals are defined by

$b_f(s)$01

where $b_f(s)$02 is a common log resolution and $b_f(s)$03 (Montaner, 2021). Their constancy regions and jumping walls are unions of rational hyperplanes determined by the numerical data of the resolution (Montaner, 2021).

The main theorem of (Montaner, 2021) states that if $b_f(s)$04 is a jumping point of $b_f(s)$05 with $b_f(s)$06, then

$b_f(s)$07

Thus every mixed jumping point near the origin yields a point on the zero locus of the Bernstein–Sato ideal (Montaner, 2021). The statement is an inclusion, not an equality, and the note explicitly does not claim the converse (Montaner, 2021).

Further estimates for zero loci were obtained in (Budur et al., 2021). For $b_f(s)$08 and a multi-index $b_f(s)$09, every codimension-one irreducible component of $b_f(s)$10 has the form

$b_f(s)$11

for some irreducible component $b_f(s)$12 of a strong log resolution and some integer $b_f(s)$13 (Budur et al., 2021). Conversely, facets of jumping walls and of the $b_f(s)$14-polytope that intersect the region $b_f(s)$15 give actual codimension-one components of $b_f(s)$16 (Budur et al., 2021). This multivariate estimate generalizes Lichtin-type upper bounds and the classical principle that small jumping numbers give roots of the $b_f(s)$17-function (Budur et al., 2021).

5. Structural properties, symmetry, and functorial operations

Bernstein–Sato ideals exhibit several nontrivial structural operations. One concerns the whole family $b_f(s)$18. Budur had proved an inclusion expressing $b_f(s)$19 as contained in an intersection of shifted elementary ideals $b_f(s)$20; "The Intersection Structure of Bernstein-Sato Ideals" (Wu, 11 Sep 2025) proves this inclusion is always an equality. For every permutation $b_f(s)$21 of $b_f(s)$22,

$b_f(s)$23

where $b_f(s)$24 acts by the shift $b_f(s)$25 (Wu, 11 Sep 2025). This symmetric intersection property is proved by interpreting $b_f(s)$26 as annihilators of logarithmic $b_f(s)$27-modules and decomposing monomial quotients via Gröbner bases (Wu, 11 Sep 2025). In the one-variable case, it implies that $b_f(s)$28 is generated by the least common multiple of $b_f(s)$29 and $b_f(s)$30 (Wu, 11 Sep 2025).

A direct application is a formula for powers of a function: $b_f(s)$31 derived from the identity $b_f(s)$32 and the intersection theorem (Wu, 11 Sep 2025).

Another operation is the tensor or multiplicative Thom–Sebastiani rule. For functions $b_f(s)$33 and $b_f(s)$34 on smooth varieties, one has

$b_f(s)$35

and for effective divisors $b_f(s)$36 on nonsingular varieties,

$b_f(s)$37

(Shi et al., 2024). For ideals, full multiplicativity fails in general: Example 3.8 in (Shi et al., 2024) gives monomial ideals $b_f(s)$38 for which $b_f(s)$39. However, if one factor is principal then

$b_f(s)$40

and for monomial ideals one has root-level inclusions and congruence modulo $b_f(s)$41: $b_f(s)$42 (Shi et al., 2024).

Hyperplane arrangements provide another source of structure. For strongly Euler-homogeneous, Saito-holonomic, and tame divisors, the modules underlying multivariate Bernstein–Sato ideals are $b_f(s)$43-Cohen–Macaulay, which forces the corresponding ideals to be principal and their zero loci to be purely codimension one (Bath, 2020). In that setting, different factorizations are related by a diagonal property: if $b_f(s)$44 is a coarser factorization of the same divisor $b_f(s)$45, then

$b_f(s)$46

where $b_f(s)$47 is the diagonal embedding induced by merging factors (Bath, 2020). For hyperplane arrangements factored into linear forms, the corresponding Bernstein–Sato ideals are even reduced (Bath, 2020).

6. Computation, examples, and variants

The computational theory of Bernstein–Sato polynomials and generalized Bernstein–Sato polynomials is based on Gröbner bases in Weyl algebras, graph embeddings, $b_f(s)$48-filtrations, and elimination (Berkesch et al., 2010). For a hypersurface $b_f(s)$49, the global $b_f(s)$50-function is obtained from the initial ideal $b_f(s)$51 and a minimal polynomial computation for $b_f(s)$52 (Berkesch et al., 2010). The same framework yields local Bernstein–Sato polynomials $b_f(s)$53, generalized Bernstein–Sato polynomials $b_f(s)$54 for an ideal $b_f(s)$55, and Shibuta’s $b_f(s)$56-generalized Bernstein–Sato polynomials $b_f(s)$57, which characterize multiplier-ideal membership (Berkesch et al., 2010).

These algorithms were implemented in the D-modules package of Macaulay2 (Berkesch et al., 2010). They compute global and local $b_f(s)$58-functions, generalized Bernstein–Sato polynomials of arbitrary polynomial ideals, log canonical thresholds, jumping coefficients, and multiplier ideals (Berkesch et al., 2010). In the examples of (Berkesch et al., 2010), the roots of $b_f(s)$59 and the resulting multiplier ideals show that not every root in $b_f(s)$60 is a jumping coefficient, so Bernstein–Sato data is finer than the jumping-locus data alone (Berkesch et al., 2010).

For arrangements, explicit formulas are available in several cases. For central generic arrangements in $b_f(s)$61 of degree $b_f(s)$62, if $b_f(s)$63 is the factorization into linear forms, then

$b_f(s)$64

(Bath, 2020). For other factorizations, the zero locus is

$b_f(s)$65

(Bath, 2020). These formulas refine earlier estimates of Maisonobe and illustrate how zero loci can often be described as explicit hyperplane arrangements.

Monomial ideals admit a different form of explicit analysis. In characteristic $b_f(s)$66, Budur–Mustaţă–Saito described the roots of $b_f(s)$67 in terms of Newton polyhedra; in characteristic $b_f(s)$68, (Quinlan-Gallego, 2019) proves that for monomial ideals the roots of $b_f(s)$69 coincide with the Bernstein–Sato roots of the mod-$b_f(s)$70 reductions $b_f(s)$71 for $b_f(s)$72 large enough (Quinlan-Gallego, 2019). This result concerns roots rather than multivariable ideals, but it supports the view that Bernstein–Sato theory has a meaningful Frobenius-theoretic analogue in positive characteristic (Quinlan-Gallego, 2019, Blickle et al., 2014).

A further extension replaces holomorphic functions by meromorphic functions $b_f(s)$73. The paper (Montaner et al., 2021) develops Bernstein–Sato polynomials $b_f(s)$74 and $b_f(s)$75 from Sabbah’s two-variable Bernstein–Sato ideal for the pair $b_f(s)$76, by specializing along lines $b_f(s)$77, $b_f(s)$78 or $b_f(s)$79 (Montaner et al., 2021). The resulting roots are again negative rational numbers and are bounded by the log-resolution data of the meromorphic germ (Montaner et al., 2021). This suggests a broader principle: multivariable Bernstein–Sato ideals provide the natural ambient structure from which one-variable invariants of more specialized objects can often be extracted by linear specialization.

Across these variants, a common pattern emerges. Bernstein–Sato ideals are annihilator ideals in parameter rings attached to $b_f(s)$80-modules generated by symbols such as $b_f(s)$81, $b_f(s)$82, or their ideal-theoretic analogues; their zero loci are unions of rational affine hyperplanes or their translated intersections; and these loci interface directly with multiplier ideals, mixed jumping walls, local-system support loci, nearby cycles, and monodromy (Budur et al., 2019, Montaner, 2021, Budur et al., 2021, Budur et al., 2024). This conjunction of $b_f(s)$83-module theory, birational geometry, and topology is the defining feature of Bernstein–Sato ideals as a research topic.