Macaulay_Inverse_Systems

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title: Macaulay Inverse Systems permalink: wiki/Macaulay_Inverse_Systems/ layout: wiki

The goal of the project is to implement Macaulay inverse systems extending the existing commands fromDual and toDual to more general setting, like non-graded (local) algebras, modules over polynomial rings and other rings, as quotients of polynomial rings.

We have created new versions of fromDual and toDual that work also for inverse systems of submodules of a free module. These commands are called newFromDual and newToDual. As for the old versions, they don't work over quotient rings.

However, we have also another version of fromDual, called modFromDual, that works also for modules over a quotient ring R = S/I where S is a polynomial ring. The downside of this version is that the running time is much longer than the other.

In addition we have a version of newToDual that takes the intersection of the inverse system with the inverse system of a power of the maximal ideal rather than the box used in toDual.

The last method is intersectInverseSystems which computes the intersection of a list or sequence of finitely generated inverse systems.

The package is posted in the gitHub repository for the Berkeley workshop. Work remains on the documentation of the package, including the tests.

We noticed that there is a bug in the fromDual since it uses the degrees of the generators of the inverse system in producing the monomial g = (x_1x_2...x_n)^d. The degree could very well be negative, causing an error. Our modified version of fromDual uses the least common multiple of all the monomials in the inverse system.

There is a list of things we would like to add to the package:

1. a modToDual command, analogous to modFromDual, that would work for modules over quotient rings;

2. adding tests to all the commands in the package that verified if the input the user introduces is of the right type;

3. find an algorithm to make a change of basis in the inverse system of a ring that corresponds to a given change of basis on the ring.