ConnectionMatrices -- writing $D$-ideals in connection form

Description

Systems of homogeneous, linear partial differential equations (PDEs) with polynomial coefficients are encoded by left ideals in the Weyl algebra, denoted $$ D_n=\CC[x_1,\ldots,x_n]\langle \partial_1,\ldots,\partial_n \rangle. $$ Such systems can be systematically written as a first-order matrix system known as a "Pfaffian system" or "connection form," by utilizing Gröbner bases in the Weyl algebra [SST, p. 37]. The systematic computation of connection matrices requires Gröbner basis computations in the rational Weyl algebra $$ R_n=\CC(x_1,\ldots,x_n)\langle \partial_1,\ldots,\partial_n \rangle. $$

The theoretical foundations of our algorithms is described the companion paper to this package, available at arXiv:2504.01362.

Working with the rational Weyl algebra

normalForm -- computes the normal form within the rational Weyl algebra
standardMonomials -- computes the standard monomials for a $D_n$-ideal
baseFractionField -- extracts the fraction field of the base polynomial ring of a Weyl algebra

Computing and displaying $D$-ideals in connection form

connectionMatrices -- computes the connection matrices of a $D_n$-ideal $I$ for a chosen basis
connectionMatrix -- computes the connection matrix

Changing basis of a system of connection matrices

gaugeMatrix -- computes the base change over the field of rational functions
gaugeTransform -- computes the gauge transform of a system of connection matrices
isEpsilonFactorized -- checks whether a system of connection matrices is in $\epsilon$-factorized form

Testing integrability of a list of matrices

isIntegrable -- checks whether a list of matrices fulfills the integrability conditions

Examples

Acknowledgement

Work on this package began at the workshop Macaulay2 in the Sciences held at MPI-MiS in Leipzig in November 2024. Devlin Mallory and Carlos Gustavo Rodriguez Fernandez contributed to the development of this package during the workshop.

References

The main reference for our algorithms is the book [SST] M. Saito, B. Sturmfels, and N. Takayama Gröbner Deformations of Hypergeometric Differential Equations, volume 6 of Algorithms and Computation in Mathematics. Springer, 2000.

Authors

Version

This documentation describes version 1.0 of ConnectionMatrices.

Citation

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

@misc{ConnectionMatricesSource,
  title = {{ConnectionMatrices: connection matrices and integrable systems from D-ideals. Version~1.0}},
  author = {Paul Goerlach and Joris Koefler and Mahrud Sayrafi and Anna-Laura Sattelberger and Hendrik Schroeder and Nicolas Weiss and Francesca Zaffalon},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- baseFractionField -- extracts the fraction field of the base polynomial ring of a Weyl algebra
- connectionMatrices -- computes the connection matrices of a $D_n$-ideal $I$ for a chosen basis
- connectionMatrix -- computes the connection matrix
- gaugeMatrix -- computes the base change over the field of rational functions
- gaugeTransform -- computes the gauge transform of a system of connection matrices
- isEpsilonFactorized -- checks whether a system of connection matrices is in $\epsilon$-factorized form
- isIntegrable -- checks whether a list of matrices fulfills the integrability conditions
- normalForm -- computes the normal form within the rational Weyl algebra
- standardMonomials -- computes the standard monomials for a $D_n$-ideal
Methods
- baseFractionField(FractionField) -- see baseFractionField -- extracts the fraction field of the base polynomial ring of a Weyl algebra
- baseFractionField(PolynomialRing) -- see baseFractionField -- extracts the fraction field of the base polynomial ring of a Weyl algebra
- connectionMatrices(Ideal) -- see connectionMatrices -- computes the connection matrices of a $D_n$-ideal $I$ for a chosen basis
- connectionMatrices(Ideal,List) -- see connectionMatrices -- computes the connection matrices of a $D_n$-ideal $I$ for a chosen basis
- connectionMatrix(Ideal) -- see connectionMatrix -- computes the connection matrix
- connectionMatrix(List) -- see connectionMatrix -- computes the connection matrix
- gaugeMatrix(Ideal,List) -- see gaugeMatrix -- computes the base change over the field of rational functions
- gaugeMatrix(List,List) -- see gaugeMatrix -- computes the base change over the field of rational functions
- gaugeTransform(Matrix,List) -- see gaugeTransform -- computes the gauge transform of a system of connection matrices
- gaugeTransform(Matrix,List,PolynomialRing) -- see gaugeTransform -- computes the gauge transform of a system of connection matrices
- isEpsilonFactorized(List,RingElement) -- see isEpsilonFactorized -- checks whether a system of connection matrices is in $\epsilon$-factorized form
- isEpsilonFactorized(Matrix,RingElement) -- see isEpsilonFactorized -- checks whether a system of connection matrices is in $\epsilon$-factorized form
- isIntegrable(List) -- see isIntegrable -- checks whether a list of matrices fulfills the integrability conditions
- isIntegrable(PolynomialRing,List) -- see isIntegrable -- checks whether a list of matrices fulfills the integrability conditions
- normalForm(RingElement,List) -- see normalForm -- computes the normal form within the rational Weyl algebra
- normalForm(RingElement,RingElement) -- see normalForm -- computes the normal form within the rational Weyl algebra
- standardMonomials(Ideal) -- see standardMonomials -- computes the standard monomials for a $D_n$-ideal
- standardMonomials(List) -- see standardMonomials -- computes the standard monomials for a $D_n$-ideal

For the programmer

The object ConnectionMatrices is a package, defined in ConnectionMatrices.m2, with auxiliary files in ConnectionMatrices/.

The source of this document is in ConnectionMatrices/docs.m2:60:0.