# 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.

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 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 ideal.

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 a set of Noetherian operators describing a primary ideal.

Methods and types for computing and manipulating Noetherian operators:

Methods for 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 and Primary ideals and their differential equations. 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
• Justin Chen
• Marc Harkonen
• Yairon Cid-Ruiz
• Anton Leykin

## Version

This documentation describes version 1.1 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 .