This repository contains Macaulay2 code accompanying the paper "Noetherian operators and primary decomposision" by J. Chen, M. Härkönen, R. Krone, and A. Leykin.
Algorithms are implemented in the package NoetherianOperators (distributed with Macaulay2 since version 1.17), and examples can be found in the examples directory. In particular, the file examples/paper_examples.m2 will run through every example in the aforementioned paper.
Open a Macaulay2 instance in this directory. Load the package by running needsPackage "NoetherianOperators".
The function noetherianOperators(I,P) computes a set of Noetherian operators for the (isolated) P-primary component of I. The function noetherianOperators I corresponds to noetherianOperators(I, radical I).
The function noetherianOperators(I,P, Strategy => "Hybrid") symbolically computes a set of Noetherian operators for the (isolated) P-primary component of I by using numerical data as a "template". We often see performance improvements compared to the purely symbolic noetherianOperators.
To compute specialized Noetherian operators at a point p numerically, use specializedNoetherianOperators(I,p). The output is a set of specialized Noetherian operators for the primary ideal corresponding to the irreducible component containing p.
The function numericalNoetherianOperators(I,pts,DependentSet => {...}) numerically computes Noetherian operators for the primary component of I containing the points in pts. At the moment, the option DependentSet is mandatory. To find a suitable dependent set numerically, one can use the package NumericalImplicitization. See the end of the file examples/carpet.m2 for details.
Rational function interpolation is implemented in rationalInterpolation(pts,vals, numBasis, denBasis), where pts is a list of points, vals is a list of values of the rational function at each point, and numBasis,denBasis are the monomial ansatzes for the numerators and denominator respectively.
The full documentation of the package can be viewed by running
needsPackage "NoetherianOperators"
viewHelp NoetherianOperators