MultigradedImplicitization -- Package for levaraging multigradings to solve implicitization problems

Description

The MultigradedImplicitization package provides methods for computing the maximal $\mathbb{Z}^k$ grading in which the kernel of a polynomial map $F$ is homogeneous and exploiting it to find generators of $\ker(F)$. This package is particularly useful for problems from algebraic statistics which often involve highly structured maps F which are often naturally homogeneous in a larger multigrading than the standard $\mathbb{Z}$-multigrading given by total degree. For more information on this approach see the following:

References:

[1] Cummings, J., & Hollering , B. (2023). Computing Implicitizations of Multi-Graded Polynomial Maps. arXiv preprint arXiv:2311.07678.

[2] Cummings, J., & Hauenstein, J. (2023). Multi-graded Macaulay Dual Spaces. arXiv preprint arXiv:2310.11587.

[3] Cummings, J., Hollering, B., & Manon, C. (2024). Invariants for level-1 phylogenetic networks under the cavendar-farris-neyman model. Advances in Applied Mathematics, 153, 102633.

Authors

Version

This documentation describes version 1.0 of MultigradedImplicitization.

Citation

If you have used this package in your research, please cite it as follows:

@misc{MultigradedImplicitizationSource,
  title = {{MultigradedImplicitization: solving implicitization problems using multigradings. Version~1.0}},
  author = {Joseph Cummings and Benjamin Hollering},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- componentOfKernel -- Finds all minimal generators of a given degree in the kernel of a ring map
- componentsOfKernel -- Finds all minimal generators up to a given total degree in the kernel of a ring map
- maxGrading -- computes the maximal $\mathbb{Z}^k$ grading such that $\ker(F)$ is homogeneous
- trimBasisInDegree -- Finds a basis for the homogeneous component of a graded ring but removes basis elements which correspond to previously computed generators.
Methods
- componentOfKernel(List,Ring,RingMap) -- see componentOfKernel -- Finds all minimal generators of a given degree in the kernel of a ring map
- componentOfKernel(List,Ring,RingMap,Matrix) -- see componentOfKernel -- Finds all minimal generators of a given degree in the kernel of a ring map
- componentOfKernel(List,Ring,RingMap,MutableHashTable) -- see componentOfKernel -- Finds all minimal generators of a given degree in the kernel of a ring map
- componentsOfKernel(Number,RingMap) -- see componentsOfKernel -- Finds all minimal generators up to a given total degree in the kernel of a ring map
- maxGrading(RingMap) -- see maxGrading -- computes the maximal $\mathbb{Z}^k$ grading such that $\ker(F)$ is homogeneous
- trimBasisInDegree(List,Ring,List,MutableHashTable) -- see trimBasisInDegree -- Finds a basis for the homogeneous component of a graded ring but removes basis elements which correspond to previously computed generators.
- trimBasisInDegree(List,Ring,MutableHashTable) -- see trimBasisInDegree -- Finds a basis for the homogeneous component of a graded ring but removes basis elements which correspond to previously computed generators.
Symbols
- Grading -- optional argument
- PreviousGens -- optional argument
- ReturnTargetGrading -- optional argument
- UseMatroidSpeedup -- optional argument

For the programmer

The object MultigradedImplicitization is a package, defined in MultigradedImplicitization.m2.

The source of this document is in MultigradedImplicitization.m2:304:0.