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


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:

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.


Certification a gold star

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 (DOI: 10.2140/jsag.2022.12.33). That version can be obtained from the journal or from the Macaulay2 source code repository.


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.


For the programmer

The object NoetherianOperators is a package.