NoetherianOperators -- algorithms for computing local dual spaces and sets of Noetherian operators

Description

The NoetherianOperators package includes algorithms for computing Noetherian operators and local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. In addition, the package provides symbolic methods for computing Noetherian operators and Noetherian multipliers of polynomial modules.

The problem of characterizing ideal membership with differential conditions was first addressed by Gröbner ("Uber eine neue idealtheoretische Grundlegung der algebraischen Geometrie", Math. Ann. 115 (1938), no. 1, 333–358). Despite this early algebraic interest by Gröbner, a complete description of primary ideals and modules in terms of differential operators was first obtained by analysts in the Fundamental Principle of Ehrenpreis and Palamodov. At the core of the Fundamental Principle, one has the notion of Noetherian operators to describe a primary module.

In case of an ideal supported at one point a set of Noetherian operators forms a Macaulay inverse system that spans the dual space of the ideal. These notions relate to the work of Macaulay ("The algebraic theory of modular systems", Cambridge Press, (1916)).

In this package, we implement several (exact symbolic and approximate numerical) algorithms for the computation of sets of Noetherian operators.

Methods and types for computing and manipulating Noetherian operators:

Methods for numerically computing and manipulating local dual spaces:

Auxiliary numerical linear algebra methods:

rationalInterpolation

For the task of computing Noetherian operators, here we implement the algorithms developed in the papers Noetherian Operators and Primary Decomposition, Primary ideals and their differential equations, and Primary decomposition of modules: a computational differential approach. These include both symbolic and numerical algorithms, and a hybrid algorithm, where numerical data is used to speed up the symbolic algorithm.

To compute the initial ideal and Hilbert regularity of positive dimensional ideals we use the algorithm of R. Krone ("Numerical algorithms for dual bases of positive-dimensional ideals." Journal of Algebra and Its Applications, 12(06):1350018, 2013.). These techniques are numerically stable, and can be used with floating point arithmetic over the complex numbers. They provide a viable alternative in this setting to purely symbolic methods such as standard bases.

Authors

Robert Krone <[email protected]>
Justin Chen <[email protected]>
Marc Harkonen <[email protected]>
Yairon Cid-Ruiz <[email protected]>
Anton Leykin <[email protected]>

Certification

Version 2.2.1 of this package was accepted for publication in volume 12 of The Journal of Software for Algebra and Geometry on 26 September 2022, in the article Noetherian operators in Macaulay2. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 2.2.1 of NoetherianOperators.

Source code

The source code from which this documentation is derived is in the file NoetherianOperators.m2.

Exports

Types
- DiffOp -- differential operator
Functions and commands
- amult -- Computes the arithmetic multiplicity of a module
- coordinateChangeOps -- induced Noetherian operators under coordinate change
- differentialPrimaryDecomposition -- compute a differential primary decomposition
- diffOp -- create a differential operator
- diffOpRing -- create and cache the ring of differential operators
- eliminatingDual -- eliminating dual space of a polynomial ideal
- gCorners -- generators of the initial ideal of a polynomial ideal
- getIdealFromNoetherianOperators -- Computes a primary ideal corresponding to a list of Noetherian operators and a prime ideal
- getModuleFromNoetherianOperators -- Computes a primary submodule corresponding to a list of Noetherian operators and a prime ideal
- isPointEmbedded -- see isPointEmbedded(AbstractPoint,Ideal,List) -- numerically determine if the point is an embedded component of the scheme
- isPointEmbeddedInCurve -- see isPointEmbeddedInCurve(AbstractPoint,Ideal) -- numerically determine if the point is an embedded component of a 1-dimensional scheme
- joinIdeals -- Computes the join of two ideals
- localHilbertRegularity -- regularity of the local Hilbert function of a polynomial ideal
- mapToPunctualHilbertScheme -- maps an ideal into a point in a certain punctual Hilbert scheme
- noetherianOperators -- Noetherian operators
- noethOpsFromComponents -- merge Noetherian operators for non-primary ideals
- normalize -- rescale a differential operator
- numericalNoetherianOperators -- Noetherian operators via numerical interpolation
- orthogonalInSubspace -- Orthogonal of a space
- pairingMatrix -- Applies dual space functionals to polynomials
- rationalInterpolation -- numerically interpolate rational functions
- solvePDE -- see solvePDE(Module) -- solve linear systems of PDE with constant coefficients
- specializedNoetherianOperators -- Noetherian operators evaluated at a point
- truncatedDual -- truncated dual space of a polynomial ideal
- zeroDimensionalDual -- dual space of a zero-dimensional polynomial ideal
Methods
- amult(Ideal) -- see amult -- Computes the arithmetic multiplicity of a module
- amult(Module) -- see amult -- Computes the arithmetic multiplicity of a module
- coordinateChangeOps(Matrix,DiffOp) -- see coordinateChangeOps -- induced Noetherian operators under coordinate change
- coordinateChangeOps(Matrix,List) -- see coordinateChangeOps -- induced Noetherian operators under coordinate change
- coordinateChangeOps(RingMap,DiffOp) -- see coordinateChangeOps -- induced Noetherian operators under coordinate change
- coordinateChangeOps(RingMap,List) -- see coordinateChangeOps -- induced Noetherian operators under coordinate change
- differentialPrimaryDecomposition(Ideal) -- see differentialPrimaryDecomposition -- compute a differential primary decomposition
- differentialPrimaryDecomposition(Module) -- see differentialPrimaryDecomposition -- compute a differential primary decomposition
- DiffOp Matrix -- apply a differential operator
- DiffOp RingElement -- see DiffOp Matrix -- apply a differential operator
- diffOp(Matrix) -- create a differential operator
- diffOp(RingElement) -- see diffOp(Matrix) -- create a differential operator
- eliminatingDual(AbstractPoint,Ideal,List,ZZ) -- see eliminatingDual -- eliminating dual space of a polynomial ideal
- eliminatingDual(AbstractPoint,Matrix,List,ZZ) -- see eliminatingDual -- eliminating dual space of a polynomial ideal
- evaluate(DiffOp,AbstractPoint) -- evaluate coefficients of a differential operator
- evaluate(DiffOp,Matrix) -- see evaluate(DiffOp,AbstractPoint) -- evaluate coefficients of a differential operator
- gCorners(AbstractPoint,Ideal) -- see gCorners -- generators of the initial ideal of a polynomial ideal
- gCorners(AbstractPoint,Matrix) -- see gCorners -- generators of the initial ideal of a polynomial ideal
- getIdealFromNoetherianOperators(List,Ideal) -- see getIdealFromNoetherianOperators -- Computes a primary ideal corresponding to a list of Noetherian operators and a prime ideal
- getModuleFromNoetherianOperators(Ideal,List) -- see getModuleFromNoetherianOperators -- Computes a primary submodule corresponding to a list of Noetherian operators and a prime ideal
- hilbertFunction(DualSpace) -- see hilbertFunction(ZZ,DualSpace)
- hilbertFunction(List,DualSpace) -- see hilbertFunction(ZZ,DualSpace)
- hilbertFunction(ZZ,DualSpace)
- isPointEmbedded(AbstractPoint,Ideal,List) -- numerically determine if the point is an embedded component of the scheme
- isPointEmbeddedInCurve(AbstractPoint,Ideal) -- numerically determine if the point is an embedded component of a 1-dimensional scheme
- joinIdeals(Ideal,Ideal) -- see joinIdeals -- Computes the join of two ideals
- localHilbertRegularity(AbstractPoint,Ideal) -- see localHilbertRegularity -- regularity of the local Hilbert function of a polynomial ideal
- localHilbertRegularity(AbstractPoint,Matrix) -- see localHilbertRegularity -- regularity of the local Hilbert function of a polynomial ideal
- mapToPunctualHilbertScheme(Ideal) -- see mapToPunctualHilbertScheme -- maps an ideal into a point in a certain punctual Hilbert scheme
- noetherianOperators(Ideal) -- Noetherian operators of a primary ideal
- noetherianOperators(Ideal,Ideal) -- Noetherian operators of a primary component
- noetherianOperators(Module) -- Noetherian operators of a primary submodule
- noetherianOperators(Module,Ideal) -- Noetherian operators of a primary component
- noethOpsFromComponents(List) -- see noethOpsFromComponents -- merge Noetherian operators for non-primary ideals
- normalize(DiffOp) -- see normalize -- rescale a differential operator
- numericalNoetherianOperators(Ideal) -- see numericalNoetherianOperators -- Noetherian operators via numerical interpolation
- orthogonalInSubspace(DualSpace,PolySpace,Number) -- see orthogonalInSubspace -- Orthogonal of a space
- orthogonalInSubspace(PolySpace,PolySpace,Number) -- see orthogonalInSubspace -- Orthogonal of a space
- pairingMatrix(PolySpace,DualSpace) -- see pairingMatrix -- Applies dual space functionals to polynomials
- pairingMatrix(PolySpace,PolySpace) -- see pairingMatrix -- Applies dual space functionals to polynomials
- pairingMatrix(PolySpace,RingElement) -- see pairingMatrix -- Applies dual space functionals to polynomials
- pairingMatrix(RingElement,DualSpace) -- see pairingMatrix -- Applies dual space functionals to polynomials
- pairingMatrix(RingElement,RingElement) -- see pairingMatrix -- Applies dual space functionals to polynomials
- rationalInterpolation(List,List,Matrix) -- see rationalInterpolation -- numerically interpolate rational functions
- rationalInterpolation(List,List,Matrix,Matrix) -- see rationalInterpolation -- numerically interpolate rational functions
- rationalInterpolation(List,List,Ring) -- numerically interpolate rational functions
- solvePDE(Ideal) -- see solvePDE(Module) -- solve linear systems of PDE with constant coefficients
- solvePDE(Matrix) -- see solvePDE(Module) -- solve linear systems of PDE with constant coefficients
- solvePDE(Module) -- solve linear systems of PDE with constant coefficients
- specializedNoetherianOperators(Ideal,AbstractPoint) -- see specializedNoetherianOperators -- Noetherian operators evaluated at a point
- specializedNoetherianOperators(Ideal,Matrix) -- see specializedNoetherianOperators -- Noetherian operators evaluated at a point
- truncate(DualSpace,List,ZZ) -- truncate a polynomial space or dual space
- truncate(PolySpace,List,ZZ) -- see truncate(DualSpace,List,ZZ) -- truncate a polynomial space or dual space
- truncate(DualSpace,ZZ) -- truncate a polynomial space or dual space
- truncate(PolySpace,ZZ) -- see truncate(DualSpace,ZZ) -- truncate a polynomial space or dual space
- truncatedDual(AbstractPoint,Ideal,ZZ) -- see truncatedDual -- truncated dual space of a polynomial ideal
- truncatedDual(AbstractPoint,Matrix,ZZ) -- see truncatedDual -- truncated dual space of a polynomial ideal
- truncatedDual(Matrix,Ideal,ZZ) -- see truncatedDual -- truncated dual space of a polynomial ideal
- truncatedDual(Matrix,Matrix,ZZ) -- see truncatedDual -- truncated dual space of a polynomial ideal
- zeroDimensionalDual(AbstractPoint,Ideal) -- see zeroDimensionalDual -- dual space of a zero-dimensional polynomial ideal
- zeroDimensionalDual(AbstractPoint,Matrix) -- see zeroDimensionalDual -- dual space of a zero-dimensional polynomial ideal
- zeroDimensionalDual(Matrix,Ideal) -- see zeroDimensionalDual -- dual space of a zero-dimensional polynomial ideal
- zeroDimensionalDual(Matrix,Matrix) -- see zeroDimensionalDual -- dual space of a zero-dimensional polynomial ideal
Symbols
- DependentSet -- option for computing Noetherian operators
- StandardBasis -- see gCorners -- generators of the initial ideal of a polynomial ideal
- AllVisible -- see isPointEmbedded(AbstractPoint,Ideal,List) -- numerically determine if the point is an embedded component of the scheme
- Regularity -- see isPointEmbeddedInCurve(AbstractPoint,Ideal) -- numerically determine if the point is an embedded component of a 1-dimensional scheme
- Rational -- see noetherianOperators(Ideal,Ideal) -- Noetherian operators of a primary component
- InterpolationDegreeLimit -- see numericalNoetherianOperators -- Noetherian operators via numerical interpolation
- InterpolationTolerance -- see numericalNoetherianOperators -- Noetherian operators via numerical interpolation
- NoetherianDegreeLimit -- see numericalNoetherianOperators -- Noetherian operators via numerical interpolation
- TrustedPoint -- see numericalNoetherianOperators -- Noetherian operators via numerical interpolation
- Sampler -- optional sampler function
- IntegralStrategy -- see Strategy => "MacaulayMatrix" -- strategy for computing Noetherian operators
- KernelStrategy -- see Strategy => "MacaulayMatrix" -- strategy for computing Noetherian operators

For the programmer

The object NoetherianOperators is a package.