Bernstein–Sato polynomial
E934438
The Bernstein–Sato polynomial is a fundamental object in algebraic analysis and singularity theory that encodes deep information about the behavior of functions and their singularities via differential equations.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
invariant of singularities
ⓘ
mathematical object ⓘ polynomial ⓘ |
| alsoKnownAs |
Bernstein polynomial of a function
ⓘ
b-function ⓘ |
| appearsIn |
Hodge theory of singularities
ⓘ
theory of perverse sheaves NERFINISHED ⓘ |
| associatedWith |
each nonzero holomorphic function germ
ⓘ
each nonzero polynomial function ⓘ |
| centralConceptIn |
algebraic analysis of differential equations
ⓘ
study of singularities via D-modules ⓘ |
| definedOver |
complex analytic functions
ⓘ
polynomials over C ⓘ |
| dependsOn |
holomorphic function
ⓘ
polynomial function ⓘ |
| encodes |
behavior of a function under differential operators
ⓘ
information about singularities of a function ⓘ monodromy information of singularities ⓘ spectrum of singularities ⓘ |
| field |
D-module theory
ⓘ
algebraic analysis ⓘ algebraic geometry ⓘ microlocal analysis ⓘ singularity theory ⓘ |
| generalizationOf | Bernstein polynomial for one-variable polynomials ⓘ |
| historicalDevelopment | introduced in the 1970s ⓘ |
| namedAfter |
Ilya N. Bernstein
NERFINISHED
ⓘ
Mikio Sato NERFINISHED ⓘ |
| property |
degree depends on complexity of singularity
ⓘ
has rational roots ⓘ roots are negative rational numbers for many classes of functions ⓘ |
| relatedTo |
D-modules
NERFINISHED
ⓘ
Igusa zeta function NERFINISHED ⓘ Malgrange’s formula NERFINISHED ⓘ V-filtration ⓘ holonomic D-modules ⓘ local zeta function ⓘ log canonical threshold ⓘ microlocal b-function ⓘ multiplier ideals ⓘ |
| satisfies | functional equation with differential operator P(s) ⓘ |
| usedFor |
analyzing asymptotic behavior of integrals
ⓘ
computing invariants of singular points ⓘ studying monodromy of Milnor fibers ⓘ studying singularities of hypersurfaces ⓘ |
| variable | s ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.