This is ConnectionMatrices.info, produced by Macaulay2, version 1.25.11

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 (see https://link.springer.com/book/10.1007/978-3-662-04112-3 ),
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 (see
https://arxiv.org/abs/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

  * Cosmological correlator for the 2-site chain
  * Massless one-loop triangle Feynman diagram
  * Gauss' hypergeometric function

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 (see
https://link.springer.com/book/10.1007/978-3-662-04112-3 )] M. Saito, B.
Sturmfels, and N. Takayama Gröbner Deformations of Hypergeometric Differential
Equations, volume 6 of Algorithms and Computation in Mathematics. Springer,
2000.

See also
========

  * Dmodules -- D-modules package collection
  * HolonomicSystems -- Examples of Holonomic
    D-modules

Authors
=======

  * Paul Goerlach (see  ) <paul.goerlach@ovgu.de>
  * Joris Koefler (see  ) <joris.koefler@mis.mpg.de>
  * Mahrud Sayrafi (see https://math.umn.edu/~mahrud/ )
    <mahrud@fields.utoronto.ca>
  * Anna-Laura Sattelberger (see  ) <anna-laura.sattelberger@mis.mpg.de>
  * Hendrik Schroeder (see  ) <h.schroeder@tu-berlin.de>
  * Nicolas Weiss (see  ) <nicolas.weiss@mis.mpg.de>
  * Francesca Zaffalon (see  ) <francesca.zaffalon@mis.mpg.de>

Version
=======

This documentation describes version 1.0 of ConnectionMatrices, released March
2025.

Citation
========

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

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

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 (Macaulay2Doc),
defined in ConnectionMatrices.m2, with auxiliary files
in ConnectionMatrices/.

-------------------------------------------------------------------------------

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

normalForm -- computes the normal form within the rational Weyl algebra
***********************************************************************

  * Usage:
        normalForm(P, Q)
        normalForm(P, G)
  * Inputs:
      * P, a ring element (Macaulay2Doc), in the Weyl
        algebra
      * Q, a ring element (Macaulay2Doc), in the Weyl
        algebra
      * G, a list (Macaulay2Doc), of elements in the Weyl algebra
        (typically a Gröbner basis of a $D_n$-ideal)
  * Outputs:
      * normalForm, a ring element (Macaulay2Doc),
        normalForm, computed in rational Weyl algebra with respect to a list of
        elements in the Weyl algebra

Description
===========

This method computes the normal form of an element $P$ in the Weyl algebra
$D_n$ with respect to another element in the Weyl algebra, or a whole list of
such elements. The reduction step is carried out over the rational Weyl algebra
$R_n$.

+--------------------------------+
|i1 : D = makeWA(QQ[x,y], {1,1});|
+--------------------------------+
|i2 : P = dx^2 ; Q = x*dx+1;     |
+--------------------------------+
|i4 : normalForm(P, Q)           |
|                                |
|      2                         |
|o4 = --                         |
|      2                         |
|     x                          |
|                                |
|o4 : frac(QQ[x..y])[dx, dy]     |
+--------------------------------+

References
==========

See [SST (see https://link.springer.com/book/10.1007/978-3-662-04112-3 ),
Theorem 1.1.7].

Caveat
======

Due to technical limitations, the output lives in the graded associative
algebra of the rational Weyl algebra, which is a commutative ring over the
base fraction field, of $D_n$ where the partial
differentials are adjoined as commuting variables.

Ways to use normalForm:
=======================

  * "normalForm(RingElement,List)"
  * "normalForm(RingElement,RingElement)"

For the programmer
==================

The object normalForm, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:96:0.

standardMonomials -- computes the standard monomials for a $D_n$-ideal
**********************************************************************

  * Usage:
        standardMonomials I
        standardMonomials G
  * Inputs:
      * I, an ideal (Macaulay2Doc), of the Weyl algebra
      * G, a list (Macaulay2Doc), of generators of the ideal
  * Outputs:
      * L, a list (Macaulay2Doc), of standard monomials

Description
===========

This method computes the standard monomials of a Gröbner basis of $R_nI$ with
respect to the order $\prec'$ on $R_n$ which is induced by the order $\prec$ of
the Weyl algebra $D_n$.

+-----------------------------------------------------+
|i1 : D = makeWeylAlgebra(QQ[x,y]);                   |
+-----------------------------------------------------+
|i2 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);|
|                                                     |
|o2 : Ideal of D                                      |
+-----------------------------------------------------+
|i3 : standardMonomials I                             |
|                                                     |
|o3 = {1, dy}                                         |
|                                                     |
|o3 : List                                            |
+-----------------------------------------------------+
|i4 : holonomicRank I                                 |
|                                                     |
|o4 = 2                                               |
+-----------------------------------------------------+

See also
========

  * holonomicRank (missing documentation)
  * gaugeMatrix -- computes the base change over the field
    of rational functions

Ways to use standardMonomials:
==============================

  * "standardMonomials(Ideal)"
  * "standardMonomials(List)"

For the programmer
==================

The object standardMonomials, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:292:0.

baseFractionField -- extracts the fraction field of the base polynomial ring of a Weyl algebra
**********************************************************************************************

  * Usage:
        baseFractionField D
  * Inputs:
      * D, a Weyl algebra,
  * Outputs:
      * a fraction field (Macaulay2Doc), the fraction
        field of the base polynomial ring of D

Description
===========

Several functions in this package return an object over the fraction field of
the base polynomial ring of a Weyl algebra. This function extracts that ring.

+----------------------------------------------------------+
|i1 : D = makeWA(frac(QQ[e])[a,b,c, DegreeRank => 0][x,y]);|
+----------------------------------------------------------+
|i2 : baseFractionField D                                  |
|                                                          |
|o2 = frac(QQ[x..y, a..c, e])                              |
|                                                          |
|o2 : FractionField                                        |
+----------------------------------------------------------+
|i3 : gens oo                                              |
|                                                          |
|o3 = {x, y, a, b, c, e}                                   |
|                                                          |
|o3 : List                                                 |
+----------------------------------------------------------+

Ways to use baseFractionField:
==============================

  * "baseFractionField(FractionField)"
  * "baseFractionField(PolynomialRing)"

For the programmer
==================

The object baseFractionField, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:120:0.

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

  * Usage:
        connectionMatrices I
        connectionMatrices(I, B)
  * Inputs:
      * I, an ideal (Macaulay2Doc), $D_n$-ideal
      * B, a list (Macaulay2Doc), a basis of $R_n/R_nI$
  * Outputs:
      * A, a list (Macaulay2Doc), the system of connection matrices
        of $I$ over the base fraction field, of $D_n$

Description
===========

Let $I$ be an ideal in the Weyl algebra $D_n$ and $B$ a basis for $R_n/R_nI$
over the base fraction field, of $D_n$. If no basis is
provided by the user, the basis is chosen to be the set of standard monomials
of a Gröbner basis on $R_nI$ with regards to the weighted Lex order
$(\partial_1 > \cdots > \partial_n > x_1 > \cdots > x_n)$ on the Weyl algebra.

+-------------------------------------------------------+
|i1 : D = makeWeylAlgebra(QQ[x,y], {2, 1})              |
|                                                       |
|o1 = D                                                 |
|                                                       |
|o1 : PolynomialRing, 2 differential variable(s)        |
+-------------------------------------------------------+
|i2 : I = ideal (x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1)  |
|                                                       |
|                2             2                        |
|o2 = ideal (x*dx  + 2dx - y*dy  - 2dy, x*dx + y*dy + 1)|
|                                                       |
|o2 : Ideal of D                                        |
+-------------------------------------------------------+
|i3 : A = connectionMatrices I                          |
|                                                       |
|o3 = {| -1/x -y/x |, | 0 1    |}                       |
|      | 0    0    |  | 0 -2/y |                        |
|                                                       |
|o3 : List                                              |
+-------------------------------------------------------+

References
==========

For more details, see [SST (see
https://link.springer.com/book/10.1007/978-3-662-04112-3 ), pp. 37-40].

Ways to use connectionMatrices:
===============================

  * "connectionMatrices(Ideal)"
  * "connectionMatrices(Ideal,List)"

For the programmer
==================

The object connectionMatrices, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:192:0.

connectionMatrix -- computes the connection matrix
**************************************************

  * Usage:
        connectionMatrix I
        connectionMatrix A
  * Inputs:
      * I, an ideal (Macaulay2Doc), of the Weyl algebra
      * A, a list (Macaulay2Doc), a system of connection matrices
  * Outputs:
      * a net (Macaulay2Doc), the connection matrix

Description
===========

This method encodes the system of connection matrices in a single matrix. Its
entries are differential one-forms in the variables of the underlying Weyl
algebra.

+-----------------------------------------------------+
|i1 : D = makeWeylAlgebra(QQ[x,y]);                   |
+-----------------------------------------------------+
|i2 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);|
|                                                     |
|o2 : Ideal of D                                      |
+-----------------------------------------------------+
|i3 : connectionMatrix I                              |
|                                                     |
|o3 = | -1/xdx -y/xdx+dy |                            |
|     | 0      -2/ydy    |                            |
+-----------------------------------------------------+

Caveat
======

The output is purely for visualizing purposes.

See also
========

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

Ways to use connectionMatrix:
=============================

  * "connectionMatrix(Ideal)"
  * "connectionMatrix(List)"

For the programmer
==================

The object connectionMatrix, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:260:0.

gaugeMatrix -- computes the base change over the field of rational functions
****************************************************************************

  * Usage:
        gaugeMatrix(G, B)
  * Inputs:
      * I, an ideal (Macaulay2Doc), of the Weyl algebra
      * G, a list (Macaulay2Doc), of generators of a Gröbner basis
        for $I$
      * B, a list (Macaulay2Doc), of standard monomials for $I$,
        i.e. any basis of $R_n/R_nI$ over the field of rational functions
  * Outputs:
      * M, a matrix (Macaulay2Doc), encoding the change of basis
        with regard to to $B$

Description
===========

Let $I = D_n\langle G\rangle$ be a $D_n$-ideal, $B$ basis of $R_n/R_nI$. This
methods computes the matrix which encodes the change of basis from the set of
standard monomials of $R_nI$ to the basis $B$.

+-----------------------------------------------------+
|i1 : D1 = makeWeylAlgebra(QQ[x, y], w1 = {2, 1});    |
+-----------------------------------------------------+
|i2 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);|
|                                                     |
|o2 : Ideal of D1                                     |
+-----------------------------------------------------+
|i3 : SM1 = standardMonomials I                       |
|                                                     |
|o3 = {1, dy}                                         |
|                                                     |
|o3 : List                                            |
+-----------------------------------------------------+
|i4 : F = baseFractionField D1;                       |
+-----------------------------------------------------+
|i5 : D2 = makeWeylAlgebra(QQ[x, y], w2 = {1, 2});    |
+-----------------------------------------------------+
|i6 : SM2 = standardMonomials sub(I, D2)              |
|                                                     |
|o6 = {1, dx}                                         |
|                                                     |
|o6 : List                                            |
+-----------------------------------------------------+
|i7 : gaugeMatrix(I, SM2)                             |
|                                                     |
|o7 = | 1    0    |                                   |
|     | -1/x -y/x |                                   |
|                                                     |
|             2      2                                |
|o7 : Matrix F  <-- F                                 |
+-----------------------------------------------------+

See also
========

  * standardMonomials -- computes the standard
    monomials for a $D_n$-ideal
  * gaugeTransform -- computes the gauge transform of a
    system of connection matrices
  * Cosmological correlator for the 2-site chain

Ways to use gaugeMatrix:
========================

  * "gaugeMatrix(Ideal,List)"
  * "gaugeMatrix(List,List)"

For the programmer
==================

The object gaugeMatrix, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:159:0.

gaugeTransform -- computes the gauge transform of a system of connection matrices
*********************************************************************************

  * Usage:
        gaugeTransform(M, A, D)
        gaugeTransform(M, A)
  * Inputs:
      * M, a matrix (Macaulay2Doc), change of basis matrix
      * A, a list (Macaulay2Doc), a system of connection matrices
      * D, a ring (Macaulay2Doc), the Weyl algebra (if omitted, it
        will be inferred from first element of A)
  * Outputs:
      * L, a list (Macaulay2Doc), the system of connection matrices
        $\widetilde{A}_i = M A_i M^{-1} + (\partial_i \bullet M)M^{-1}$ after
        the gauge transformation with respect to $M$.

Description
===========

This method computes the gauge transform of a system of connection matrices for
a given invertible matrix that encodes a change of basis.

+-----------------------------------------------------+
|i1 : D = makeWeylAlgebra(QQ[x,y]);                   |
+-----------------------------------------------------+
|i2 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);|
|                                                     |
|o2 : Ideal of D                                      |
+-----------------------------------------------------+
|i3 : A = connectionMatrices I;                       |
+-----------------------------------------------------+
|i4 : M = matrix {{x,0}, {0,y}};                      |
|                                                     |
|             2      2                                |
|o4 : Matrix D  <-- D                                 |
+-----------------------------------------------------+
|i5 : gaugeTransform(M, A, D)                         |
|                                                     |
|o5 = {| 0 -1 |, | 0 x/y  |}                          |
|      | 0 0  |  | 0 -1/y |                           |
|                                                     |
|o5 : List                                            |
+-----------------------------------------------------+

It is also possible to compute the gauge transform of a system of connection
matrices containing parameters.

See also
========

  * gaugeMatrix -- computes the base change over the field
    of rational functions
  * Cosmological correlator for the 2-site chain

Ways to use gaugeTransform:
===========================

  * "gaugeTransform(Matrix,List)"
  * "gaugeTransform(Matrix,List,PolynomialRing)"

For the programmer
==================

The object gaugeTransform, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:228:0.

isEpsilonFactorized -- checks whether a system of connection matrices is in $\epsilon$-factorized form
******************************************************************************************************

  * Usage:
        isEpsilonFactorized(A,eps)
        isEpsilonFactorized(M,eps)
  * Inputs:
      * A, a list (Macaulay2Doc), a system of connection matrices
      * M, a matrix (Macaulay2Doc), a connection matrix
      * eps, a ring element (Macaulay2Doc), parameter for
        which the factorization property should be tested
  * Outputs:
      * b, a Boolean value (Macaulay2Doc), true or false

Description
===========

This method returns true if the system of connection matrices factors out (a
power of) eps, such that the resulting matrices are independent of eps.

+---------------------------------------------+
|i1 : D = makeWeylAlgebra(frac(QQ[ϵ])[x]);    |
+---------------------------------------------+
|i2 : I = ideal(x*(1-x)*dx^2 - ϵ*(1-x)*dx);   |
|                                             |
|o2 : Ideal of D                              |
+---------------------------------------------+
|i3 : A = connectionMatrices(I, {1_D, 1/ϵ*dx})|
|                                             |
|o3 = {| 0 ϵ   |}                             |
|      | 0 ϵ/x |                              |
|                                             |
|o3 : List                                    |
+---------------------------------------------+
|i4 : isEpsilonFactorized(A, ϵ)               |
|                                             |
|o4 = true                                    |
+---------------------------------------------+

See also
========

  * isIntegrable -- checks whether a list of matrices
    fulfills the integrability conditions
  * gaugeTransform -- computes the gauge transform of a
    system of connection matrices

Ways to use isEpsilonFactorized:
================================

  * "isEpsilonFactorized(List,RingElement)"
  * "isEpsilonFactorized(Matrix,RingElement)"

For the programmer
==================

The object isEpsilonFactorized, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:327:0.

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

  * Usage:
        isIntegrable(D, A)
        isIntegrable A
  * Inputs:
      * D, a Weyl algebra,
      * A, a list (Macaulay2Doc), a system of matrices
  * Outputs:
      * p, a Boolean value (Macaulay2Doc), whether $[A_i,A_j] =
        \partial_i(A_j) - \partial_j(A_i)$ for all $i,j$

Description
===========

Checks whether a list of $n$ matrices $A_i$ in $\operatorname{Mat}_{m\times
m}(k(x_1..x_n))$ satisfy $[A_i,A_j] = \partial_i(A_j) - \partial_j(A_i)$ for
all $i,j$. This is the case, in particular, when they come from a $D_n$-module,
respectively from a $D_n$-ideal.

+-----------------------------------------------------+
|i1 : D = makeWeylAlgebra(QQ[x,y], {1, 2});           |
+-----------------------------------------------------+
|i2 : I = ideal(x*dx^2 - y*dy^2 + dx-dy, x*dx+y*dy+1);|
|                                                     |
|o2 : Ideal of D                                      |
+-----------------------------------------------------+
|i3 : A = connectionMatrices I;                       |
+-----------------------------------------------------+
|i4 : assert isIntegrable(D, A)                       |
+-----------------------------------------------------+
|i5 : assert isIntegrable A                           |
+-----------------------------------------------------+

Caveat
======

The matrices need to be defined over the base fraction field,
of $D_n$.

See also
========

  * connectionMatrices -- computes the connection
    matrices of a $D_n$-ideal $I$ for a chosen basis
  * isEpsilonFactorized -- checks whether a system
    of connection matrices is in $\epsilon$-factorized form

Ways to use isIntegrable:
=========================

  * "isIntegrable(List)"
  * "isIntegrable(PolynomialRing,List)"

For the programmer
==================

The object isIntegrable, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/docs.m2:363:0.

Cosmological correlator for the 2-site chain
********************************************

Differential equations for correlation functions in cosmology are studied in
[ABHJLP]. Therein, a basis of master integrals is constructed from the
underlying hyperplane arrangement via canonical forms in the setup of positive
geometries, which results in a matrix differential equation for the
cosmological correlator that is in $\varepsilon$-factorized form. For the
$2$-site chain (mathematically, the path graph on $2$ vertices), the underlying
$D$-ideal was investigated in [FPSW].

Here we revisit the $D$-ideal $I = \langle
\nabla_1+\nabla_3,\nabla_2+\nabla_3,H\rangle$ (see Equation (11) in [FPSW]),
and carry out the gauge transformation to write the connection matrices in
$\varepsilon$-factorized form. This form is especially useful, as it allows for
the construction of formal power series solutions in the variable $\varepsilon$
of such systems via the ``path-ordered exponential formalism.’’

+-----------------------------------------------------------------------------+
|i1 : w = {1,1,1};                                                            |
+-----------------------------------------------------------------------------+
|i2 : D = makeWeylAlgebra(frac(QQ[ϵ])[x,y,z], w);                             |
+-----------------------------------------------------------------------------+
|i3 : delta1 = (x^2-z^2)*dx^2+2*(1-ϵ)*x*dx-ϵ*(1-ϵ);                           |
+-----------------------------------------------------------------------------+
|i4 : delta2 = (y^2-z^2)*dy^2+2*(1-ϵ)*y*dy-ϵ*(1-ϵ);                           |
+-----------------------------------------------------------------------------+
|i5 : delta3 = (x+z)*(y+z)*dx*dy-ϵ*(x+z)*dx-ϵ*(y+z)*dy+ϵ^2;                   |
+-----------------------------------------------------------------------------+
|i6 : h = x*dx+y*dy+z*dz-2*ϵ;                                                 |
+-----------------------------------------------------------------------------+
|i7 : I = ideal(delta1+delta3, delta2+delta3, h)                              |
|                                                                             |
|             2  2    2  2                                        2           |
|o7 = ideal (x dx  - z dx  + x*y*dx*dy + x*z*dx*dy + y*z*dx*dy + z dx*dy + (- |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     3ϵ + 2)x*dx - ϵ*z*dx - ϵ*y*dy - ϵ*z*dy + 2ϵ  - ϵ, x*y*dx*dy + x*z*dx*dy |
|     ------------------------------------------------------------------------|
|                    2         2  2    2  2                                   |
|     + y*z*dx*dy + z dx*dy + y dy  - z dy  - ϵ*x*dx - ϵ*z*dx + (- 3ϵ + 2)y*dy|
|     ------------------------------------------------------------------------|
|                  2                                                          |
|     - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)                            |
|                                                                             |
|o7 : Ideal of D                                                              |
+-----------------------------------------------------------------------------+

First, we check that the system has finite holonomic rank using
holonomicRank (WeylAlgebras).

+---------------------------------+
|i8 : assert(holonomicRank I == 4)|
+---------------------------------+

Then, we compute the system in connection form and verify that it meets the
integrability conditions.

+-----------------------------------------------------------------------------+
|i9 : elapsedTime A = connectionMatrices I;                                   |
| -- 3.38469s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i10 : elapsedTime assert isIntegrable A                                      |
| -- 6.08651s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i11 : netList A                                                              |
|                                                                             |
|      +----------------------------------------------------------------------|
|o11 = || 2ϵ/x                                                                |
|      || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4|
|      || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z|
|      || (-2x3yzϵ^3+2xy3zϵ^3-2x2yz2ϵ^3+8xy2z2ϵ^3+2y3z2ϵ^3+8xyz3ϵ^3+8y2z3ϵ^3+8|
|      +----------------------------------------------------------------------|
|      || 0                                                                   |
|      || (-4xyϵ^2-4z2ϵ^2+2x2ϵ+2xyϵ+2yzϵ+2z2ϵ)/(2x2y2+2xy3+x2yz-y3z-x2z2-2xyz2|
|      || (2xz2ϵ^2+2yz2ϵ^2+4z3ϵ^2-2x2yϵ-x2zϵ-xyzϵ-2yz2ϵ-2z3ϵ)/(2x2y2z+x2yz2+xy|
|      || (-2x2yzϵ^3+2y3zϵ^3-2x2z2ϵ^3-8xyz2ϵ^3+2y2z2ϵ^3-8xz3ϵ^3-8yz3ϵ^3-8z4ϵ^3|
|      +----------------------------------------------------------------------|
|      || 0                                                                   |
|      || (2xz2ϵ^2+2yz2ϵ^2+4z3ϵ^2-2x2yϵ-x2zϵ-xyzϵ-2yz2ϵ-2z3ϵ)/(2x2y2z+x2yz2+xy|
|      || 0                                                                   |
|      || (4x3y2zϵ^3-4xy4zϵ^3+8x2y2z2ϵ^3-8xy3z2ϵ^3+4x2yz3ϵ^3-4y3z3ϵ^3+8xyz4ϵ^3|
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)        |
|+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                          |
|yz4ϵ^3-6x3y2ϵ^2-6x2y3ϵ^2+x4zϵ^2-4x3yzϵ^2-21x2y2zϵ^2-4xy3zϵ^2+x3z2ϵ^2-8x2yz2ϵ^|
|-----------------------------------------------------------------------------|
|                                                                             |
|-y2z2)                                                                       |
|2z2-x2z3-y2z3-xz4-yz4)                                                       |
|+2x3yϵ^2+6x2y2ϵ^2+4xy3ϵ^2+2x3zϵ^2+5x2yzϵ^2+14xy2zϵ^2-y3zϵ^2+7x2z2ϵ^2+10xyz2ϵ^|
|-----------------------------------------------------------------------------|
|                                                                             |
|2z2-x2z3-y2z3-xz4-yz4)                                                       |
|                                                                             |
|-8y2z4ϵ^3-2x4y2ϵ^2+2x2y4ϵ^2-3x4yzϵ^2-5x3y2zϵ^2+7x2y3zϵ^2+9xy4zϵ^2+x4z2ϵ^2-10x|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|2-15xy2z2ϵ^2-6y3z2ϵ^2+4x2z3ϵ^2-8xyz3ϵ^2-16y2z3ϵ^2+4xz4ϵ^2-12yz4ϵ^2+2x4yϵ+10x2|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|2+3y2z2ϵ^2+16xz3ϵ^2+12yz3ϵ^2+16z4ϵ^2-6x3yϵ-6xy3ϵ-2x3zϵ-11x2yzϵ-6xy2zϵ-3y3zϵ-3|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|3yz2ϵ^2-18x2y2z2ϵ^2+22xy3z2ϵ^2+5y4z2ϵ^2+x3z3ϵ^2-21x2yz3ϵ^2-9xy2z3ϵ^2+13y3z3ϵ^|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|y3ϵ+x4zϵ+2x3yzϵ+21x2y2zϵ+2xy3zϵ+x3z2ϵ+8x2yz2ϵ+5xy2z2ϵ+4y3z2ϵ-2x2z3ϵ+8y2z3ϵ-2x|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x2z2ϵ-8xyz2ϵ-3y2z2ϵ-4xz3ϵ-6yz3ϵ-6z4ϵ)/(2x3y4+x3y3z+3x2y4z-3x3y2z2+x2y3z2-x3yz|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|2+4x2z4ϵ^2-12xyz4ϵ^2+8y2z4ϵ^2+4xz5ϵ^2+4yz5ϵ^2+4x4y2ϵ-4x2y4ϵ+3x4yzϵ+3x3y2zϵ-5x|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|z4ϵ+4yz4ϵ)/(2x5y3+3x5y2z+x4y3z+x4y2z2-3x3y3z2-x5z3-x4yz3-5x3y2z3-x2y3z3-x4z4-|
|-----------------------------------------------------------------------------|
|                                                                    1        |
|                                                                    (2x2yϵ+6x|
|                                                                    (-x2z2ϵ-2|
|3-5x2y2z3-xy3z3-y4z3+x3z4-x2yz4-xy2z4-y3z4+2x2z5+xyz5+y2z5+xz6+yz6) (x3yzϵ^2+|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|2y3zϵ-5xy4zϵ+x4z2ϵ+2x3yz2ϵ+10x2y2z2ϵ-6xy3z2ϵ-7y4z2ϵ+x3z3ϵ+11x2yz3ϵ+3xy2z3ϵ-7y|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                -y/x                         |
|                                                (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-|
|                                                (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+|
|x3yz4-x2y2z4+xy3z4+x3z5+x2yz5+2xy2z5+x2z6+xyz6) (x4yzϵ^2+x3y2zϵ^2-x2y3zϵ^2-xy|
|-----------------------------------------------------------------------------|
|                                                                             |
|y2ϵ+2xyzϵ-2y2zϵ-4x2y-4xy2)/(2x2y2+2xy3+x2yz-y3z-x2z2-2xyz2-y2z2)             |
|xyz2ϵ-y2z2ϵ-2xz3ϵ-2yz3ϵ+2x2y2+x2yz+xy2z+x2z2+y2z2+xz3+yz3)/(2x2y2z+x2yz2+xy2z|
|x2y2zϵ^2-xy3zϵ^2-y4zϵ^2+x3z2ϵ^2+5x2yz2ϵ^2+3xy2z2ϵ^2-y3z2ϵ^2+4x2z3ϵ^2+8xyz3ϵ^2|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|3z3ϵ-2x2z4ϵ+4xyz4ϵ-2y2z4ϵ-2xz5ϵ-2yz5ϵ)/(2x4y4z+x4y3z2+x3y4z2-3x4y2z3-3x2y4z3-|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3|
|2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3|
|4zϵ^2+x3yz2ϵ^2-3x2y2z2ϵ^2-5xy3z2ϵ^2-y4z2ϵ^2-4x2yz3ϵ^2-8xy2z3ϵ^2-4y3z3ϵ^2-4xyz|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|2-x2z3-y2z3-xz4-yz4)                                                         |
|+4y2z3ϵ^2+4xz4ϵ^2+4yz4ϵ^2-4x3y2ϵ-6x2y3ϵ-2xy4ϵ-3x3yzϵ-13x2y2zϵ-9xy3zϵ+y4zϵ-3x3|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x4yz4-2x3y2z4-2x2y3z4-xy4z4+x4z5+4x2y2z5+y4z5+x3z6+2x2yz6+2xy2z6+y3z6-x2z7-y2|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                          |
|-x2z4-xyz4)                                                                  |
|4ϵ^2-4y2z4ϵ^2+2x4y2ϵ+6x3y3ϵ+4x2y4ϵ-x4yzϵ+9x3y2zϵ+13x2y3zϵ+3xy4zϵ+x3yz2ϵ+9x2y2|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|z2ϵ-11x2yz2ϵ-9xy2z2ϵ-y3z2ϵ-8x2z3ϵ-12xyz3ϵ-4y2z3ϵ-6xz4ϵ-6yz4ϵ+8x3y2+4xy4+2x3yz|
|-----------------------------------------------------------------------------|
|            0                                                                |
|            (-x2z2ϵ-2xyz2ϵ-y2z2ϵ-2xz3ϵ-2yz3ϵ+2x2y2+x2yz+xy2z+x2z2+y2z2+xz3+yz|
|            0                                                                |
|z7-xz8-yz8) (-2x4y2zϵ^2-2x3y3zϵ^2+2x2y4zϵ^2+2xy5zϵ^2-4x3y2z2ϵ^2+4xy4z2ϵ^2-2x3|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|z2ϵ+11xy3z2ϵ+3y4z2ϵ+4x2yz3ϵ+12xy2z3ϵ+8y3z3ϵ+6xyz4ϵ+6y2z4ϵ-4x4y2-8x2y4-2x4yz-4|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|+16x2y2z+4xy3z+2y4z+2x3z2+4x2yz2+8xy2z2+2y3z2+4x2z3+2xyz3+2y2z3+2xz4+2yz4)/(2|
|-----------------------------------------------------------------------------|
|                                                                             |
|3)/(2x2y2z+x2yz2+xy2z2-x2z3-y2z3-xz4-yz4)                                    |
|                                                                             |
|yz3ϵ^2-2x2y2z3ϵ^2+2xy3z3ϵ^2+2y4z3ϵ^2-4x2yz4ϵ^2+4y3z4ϵ^2+2x4y3ϵ-2x2y5ϵ+6x4y2zϵ|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x3y2z-16x2y3z-2xy4z-2x3yz2-8x2y2z2-4xy3z2-2y4z2-2x2yz3-2xy2z3-4y3z3-2xyz4-2y2|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x3y4+x3y3z+3x2y4z-3x3y2z2+x2y3z2-x3yz3-5x2y2z3-xy3z3-y4z3+x3z4-x2yz4-xy2z4-y3|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|+6x3y3zϵ-6x2y4zϵ-6xy5zϵ+2x4yz2ϵ+16x3y2z2ϵ-16xy4z2ϵ-2y5z2ϵ+6x3yz3ϵ+6x2y2z3ϵ-6x|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|z4)/(2x5y3+3x5y2z+x4y3z+x4y2z2-3x3y3z2-x5z3-x4yz3-5x3y2z3-x2y3z3-x4z4-x3yz4-x|
|-----------------------------------------------------------------------------|
|                            0                                                |
|                            (-2x2zϵ+2yz2ϵ+4z3ϵ-2yz2-2z3)/(2x2y2+2xy3+x2yz-y3z|
|                            (2x2yϵ+x2zϵ+xyzϵ-xz2ϵ-3yz2ϵ-4z3ϵ+2yz2+2z3)/(2x2y2|
|z4+2x2z5+xyz5+y2z5+xz6+yz6) (-2x3yzϵ^2+2x2y2zϵ^2-2x3z2ϵ^2-2x2yz2ϵ^2+2xy2z2ϵ^2|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|y3z3ϵ-6y4z3ϵ+2x2yz4ϵ-2y3z4ϵ-4x4y3+4x2y5-4x4y2z-4x3y3z+4x2y4z+4xy5z-4x4yz2-4x3|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                         -z/x                                |
|                                         (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+|
|                                         (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z|
|2y2z4+xy3z4+x3z5+x2yz5+2xy2z5+x2z6+xyz6) (2x3y2zϵ^2-2x2y3zϵ^2-x4z2ϵ^2+3x3yz2ϵ|
|-----------------------------------------------------------------------------|
|                                                                             |
|-x2z2-2xyz2-y2z2)                                                            |
|+x2yz+xy2z-x2z2-y2z2-xz3-yz3)                                                |
|-2y3z2ϵ^2-4x2z3ϵ^2+6xyz3ϵ^2-2y2z3ϵ^2+4xz4ϵ^2+8yz4ϵ^2+8z5ϵ^2+6x3yzϵ-4x2y2zϵ-4x|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|y2z2+4xy4z2+4y5z2-4x3yz3-4x2y2z3+4xy3z3+4y4z3)/(2x4y4z+x4y3z2+x3y4z2-3x4y2z3-|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z|
|2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z|
|^2-3x2y2z2ϵ^2-3xy3z2ϵ^2-x3z3ϵ^2+3x2yz3ϵ^2-9xy2z3ϵ^2-y3z3ϵ^2+4x2z4ϵ^2-4xyz4ϵ^2|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|y3zϵ+2x3z2ϵ+10x2yz2ϵ-10xy2z2ϵ+2x2z3ϵ-6xyz3ϵ-2y2z3ϵ-12xz4ϵ-14yz4ϵ-16z5ϵ+4xy3z+|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|3x2y4z3-x4yz4-2x3y2z4-2x2y3z4-xy4z4+x4z5+4x2y2z5+y4z5+x3z6+2x2yz6+2xy2z6+y3z6|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|3-2x2yz3-xy2z3)                                                              |
|2-x2z3-xyz3)                                                                 |
|-4y2z4ϵ^2+4xz5ϵ^2-4yz5ϵ^2-2x4yzϵ+2x3y2zϵ+10x2y3zϵ-x4z2ϵ-x3yz2ϵ+23x2y2z2ϵ+5xy3|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|4xy2z2+2y3z2+4xyz3+2y2z3+4xz4+6yz4+6z5)/(2x3y4+x3y3z+3x2y4z-3x3y2z2+x2y3z2-x3|
|-----------------------------------------------------------------------------|
|                    1                                                        |
|                    (2x2yϵ+x2zϵ+xyzϵ-xz2ϵ-3yz2ϵ-4z3ϵ+2yz2+2z3)/(2x2y2+x2yz+xy|
|                    0                                                        |
|-x2z7-y2z7-xz8-yz8) (2x4y2ϵ^2-4x3y3ϵ^2+2x2y4ϵ^2+3x4yzϵ^2-x3y2zϵ^2+x2y3zϵ^2+5x|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|z2ϵ-x3z3ϵ+3x2yz3ϵ+13xy2z3ϵ+3y3z3ϵ-10x2z4ϵ+8y2z4ϵ-10xz5ϵ+6yz5ϵ-8x2y3z-16x2y2z2|
|-----------------------------------------------------------------------------|
|                                                                      0      |
|                                                                      (x2z-z3|
|                                                                      (-xz+z2|
|yz3-5x2y2z3-xy3z3-y4z3+x3z4-x2yz4-xy2z4-y3z4+2x2z5+xyz5+y2z5+xz6+yz6) (2x2y2ϵ|
|-----------------------------------------------------------------------------|
|                                                                             |
|2z-x2z2-y2z2-xz3-yz3)                                                        |
|                                                                             |
|y4zϵ^2-x4z2ϵ^2+6x3yz2ϵ^2-14x2y2z2ϵ^2+10xy3z2ϵ^2-y4z2ϵ^2-x3z3ϵ^2-9x2yz3ϵ^2-7xy|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|-2xy3z2-4x2yz3-4xy2z3-2y3z3+4x2z4+2xyz4-4y2z4+4xz5-2yz5)/(2x5y3+3x5y2z+x4y3z+|
|-----------------------------------------------------------------------------|
|                                                                             |
|)/(2x2y+2xy2-x2z-2xyz-y2z)                                                   |
|)/(2xy-xz-yz)                                                                |
|-2x2yzϵ+2x2z2ϵ-2xyz2ϵ+2xz3ϵ-2z4ϵ-2x2y2+xy2z-2x2z2+y2z2-xz3+3z4)/(2x2y3-x2y2z+|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|2z3ϵ^2+y3z3ϵ^2+4x2z4ϵ^2-12xyz4ϵ^2+8y2z4ϵ^2+4xz5ϵ^2+4yz5ϵ^2-4x4y2ϵ+2x3y3ϵ-6x2y|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x4y2z2-3x3y3z2-x5z3-x4yz3-5x3y2z3-x2y3z3-x4z4-x3yz4-x2y2z4+xy3z4+x3z5+x2yz5+2|
|-----------------------------------------------------------------------------|
|                                          |                                  |
|                                          |                                  |
|                                          |                                  |
|xy3z-2x2yz2-xy2z2-y3z2+x2z3-xyz3+xz4+yz4) |                                  |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|4ϵ-3x4yzϵ-x3y2zϵ-7x2y3zϵ-9xy4zϵ-x4z2ϵ+36x2y2z2ϵ-16xy3z2ϵ-3y4z2ϵ-x3z3ϵ+27x2yz3|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                 0                                                           |
|                 (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                  |
|                 (-yz+z2)/(2xy-xz-yz)                                        |
|xy2z5+x2z6+xyz6) (2x2y2ϵ-2xy2zϵ-2xyz2ϵ+2y2z2ϵ+2yz3ϵ-2z4ϵ-2x2y2+x2yz+x2z2-2y2z|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|ϵ+21xy2z3ϵ-7y3z3ϵ-10x2z4ϵ+14xyz4ϵ-12y2z4ϵ-10xz5ϵ-10yz5ϵ+4x2y4+4x2y3z+4xy4z-16|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                  |          |
|                                                                  |          |
|                                                                  |          |
|2-yz3+3z4)/(2x3y2+x3yz-x2y2z-x3z2-x2yz2-2xy2z2-xyz3+y2z3+xz4+yz4) |          |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x2y2z2+4xy3z2+4y4z2-12x2yz3-8xy2z3+4y3z3+4x2z4-4xyz4+4y2z4+4xz5+4yz5)/(2x4y4+|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|x4y3z+x3y4z-3x4y2z2-3x2y4z2-x4yz3-2x3y2z3-2x2y3z3-xy4z3+x4z4+4x2y2z4+y4z4+x3z|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                        0                                    |
|                                        (-xz+z2)/(2xy-xz-yz)                 |
|                                        1                                    |
|5+2x2yz5+2xy2z5+y3z5-x2z6-y2z6-xz7-yz7) (2x3y2ϵ+2x2y3ϵ-6x3yzϵ-6xy3zϵ+2x3z2ϵ+2|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|y3z2ϵ+12xyz3ϵ-4xz4ϵ-4yz4ϵ-x3y2-x2y3+6x3yz+6xy3z-x3z2-3x2yz2-3xy2z2-y3z2-12xyz|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|3+5xz4+5yz4)/(2x3y3-x3y2z-x2y3z-2x3yz2-2xy3z2+x3z3+x2yz3+xy2z3+y3z3+2xyz4-xz5|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-------+                                                                     |
|       |                                                                     |
|       |                                                                     |
|       |                                                                     |
|       |                                                                     |
|-------+                                                                     |
|       |                                                                     |
|       |                                                                     |
|       |                                                                     |
|       |                                                                     |
|-------+                                                                     |
|      ||                                                                     |
|      ||                                                                     |
|      ||                                                                     |
|-yz5) ||                                                                     |
|-------+                                                                     |
+-----------------------------------------------------------------------------+

Next, we use gauge matrix, for changing base to a base given
by suitable set of standard monomials, and compute the gauge transform,
with respect to this gauge matrix.

+-----------------------------------------------------------------------------+
|i12 : F = baseFractionField D;                                               |
+-----------------------------------------------------------------------------+
|i13 : B = {1_D,dx,dy,dx*dy};                                                 |
+-----------------------------------------------------------------------------+
|i14 : elapsedTime g = gaugeMatrix(I, B);                                     |
| -- .650641s elapsed                                                         |
|                                                                             |
|              4      4                                                       |
|o14 : Matrix F  <-- F                                                        |
+-----------------------------------------------------------------------------+
|i15 : elapsedTime A1 = gaugeTransform(g, A);                                 |
| -- 1.60881s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i16 : elapsedTime assert isIntegrable A1                                     |
| -- .956932s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i17 : netList A1                                                             |
|                                                                             |
|      +----------------------------------------------------------------------|
|o17 = || 0                            1                      0               |
|      || (-2ϵ^2+ϵ)/(x2-z2)            (3xϵ+zϵ-2x)/(x2-z2)    (yϵ+zϵ)/(x2-z2) |
|      || 0                            0                      0               |
|      || (-2ϵ^3+ϵ^2)/(x3+x2y-xz2-yz2) (2ϵ^2-ϵ)/(x2+xy-xz-yz) (-xϵ^2+yϵ^2+2zϵ^|
|      +----------------------------------------------------------------------|
|      || 0                            0                                      |
|      || 0                            0                                      |
|      || (-2ϵ^2+ϵ)/(y2-z2)            (xϵ+zϵ)/(y2-z2)                        |
|      || (-2ϵ^3+ϵ^2)/(xy2+y3-xz2-yz2) (xϵ^2-yϵ^2+2zϵ^2+yϵ-zϵ)/(xy2+y3-xz2-yz2|
|      +----------------------------------------------------------------------|
|      || 2ϵ/z                                             -x/z               |
|      || (2xϵ^2-xϵ)/(x2z-z3)                              (-x2ϵ-xzϵ-2z2ϵ+x2+z|
|      || (2yϵ^2-yϵ)/(y2z-z3)                              (-xyϵ-yzϵ)/(y2z-z3)|
|      || (2xyϵ^3-2z2ϵ^3-xyϵ^2+z2ϵ^2)/(x2y2z-x2z3-y2z3+z5) (-xyϵ^2-yzϵ^2+2z2ϵ^|
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                          0                                                  |
|                          (-y-z)/(x-z)                                       |
|                          1                                                  |
|2+xϵ-zϵ)/(x3+x2y-xz2-yz2) (2x2ϵ-2xzϵ-2yzϵ-2z2ϵ-2x2-xy+xz+yz+z2)/(x3+x2y-xz2-y|
|-----------------------------------------------------------------------------|
|  1                      0                                                   |
|  0                      1                                                   |
|  (3yϵ+zϵ-2y)/(y2-z2)    (-x-z)/(y-z)                                        |
|) (2ϵ^2-ϵ)/(xy+y2-xz-yz) (2y2ϵ-2xzϵ-2yzϵ-2z2ϵ-xy-2y2+xz+yz+z2)/(xy2+y3-xz2-yz|
|-----------------------------------------------------------------------------|
|                              -y/z                                           |
|2)/(x2z-z3)                   (-xyϵ-xzϵ)/(x2z-z3)                            |
|                              (-y2ϵ-yzϵ-2z2ϵ+y2+z2)/(y2z-z3)                 |
|2+yzϵ-z2ϵ)/(xy2z-y2z2-xz3+z4) (-xyϵ^2-xzϵ^2+2z2ϵ^2+xzϵ-z2ϵ)/(x2yz-x2z2-yz3+z4|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|    |                                                                        |
|    |                                                                        |
|    |                                                                        |
|z2) |                                                                        |
|-----------------------------------------------------------------------------|
|   |                                                                         |
|   |                                                                         |
|   |                                                                         |
|2) |                                                                         |
|-----------------------------------------------------------------------------|
|  0                                                                          |
|  (x+y)/(x-z)                                                                |
|  (x+y)/(y-z)                                                                |
|) (2x2yzϵ+2xy2zϵ-2xz3ϵ-2yz3ϵ+x2y2-x2yz-xy2z+xz3+yz3-z4)/(x2y2z-x2z3-y2z3+z5) |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-+                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
|-+                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
| |                                                                           |
|-+                                                                           |
|||                                                                           |
|||                                                                           |
|||                                                                           |
|||                                                                           |
|-+                                                                           |
+-----------------------------------------------------------------------------+

Now we are ready to perform the gauge transform from this basis to the
$\epsilon$-factorized form.

+-----------------------------------------------------------------------------+
|i18 : changeEps = transpose((1/(2*z*ϵ^2)) * matrix {                         |
|              {2*z*ϵ^2, -ϵ^2*(x-z), -ϵ^2*(y-z), -ϵ^2*(x+y)},                 |
|              {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},                            |
|              {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},                            |
|              {0, 0, 0, -(x+y)*(x+z)*(y+z)}});                               |
|                                                                             |
|              4      4                                                       |
|o18 : Matrix F  <-- F                                                        |
+-----------------------------------------------------------------------------+
|i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);                        |
| -- .462339s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i20 : elapsedTime assert isIntegrable A2                                     |
| -- .886792s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i21 : netList A2                                                             |
|                                                                             |
|      +----------------------------------------------------------------------|
|o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                |
|      || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                |
|      || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                |
|      || 0       0           0       2ϵ/(x+y)               |                |
|      +----------------------------------------------------------------------|
|      || ϵ/(y+z) 0       2zϵ/(y2-z2) 0                      |                |
|      || 0       ϵ/(y+z) 0           (-xϵ+zϵ)/(xy+y2+xz+yz) |                |
|      || 0       0       ϵ/(y-z)     ϵ/(x+y)                |                |
|      || 0       0       0           2ϵ/(x+y)               |                |
|      +----------------------------------------------------------------------|
|      || (xϵ+yϵ+2zϵ)/(xy+xz+yz+z2) -2xϵ/(x2-z2)              -2yϵ/(y2-z2)    |
|      || 0                         (xϵ-yϵ-2zϵ)/(xy+xz-yz-z2) 0               |
|      || 0                         0                         (-xϵ+yϵ-2zϵ)/(xy|
|      || 0                         0                         0               |
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|---------------------+                                                       |
|                     |                                                       |
|                     |                                                       |
|                     |                                                       |
|                     |                                                       |
|---------------------+                                                       |
|                     |                                                       |
|                     |                                                       |
|                     |                                                       |
|                     |                                                       |
|---------------------+                                                       |
|           0        ||                                                       |
|           -ϵ/(y+z) ||                                                       |
|-xz+yz-z2) -ϵ/(x+z) ||                                                       |
|           0        ||                                                       |
|---------------------+                                                       |
+-----------------------------------------------------------------------------+

Finally, we verify that only the last system is in the $\epsilon$-factorized
form using isEpsilonFactorized.

+-------------------------------------------+
|i22 : assert isEpsilonFactorized(A2, ϵ)    |
+-------------------------------------------+
|i23 : assert not isEpsilonFactorized(A1, ϵ)|
+-------------------------------------------+
|i24 : assert not isEpsilonFactorized(A, ϵ) |
+-------------------------------------------+

References
==========

[FPSW (see https://arxiv.org/pdf/2410.14757 )]  C. Fevola, G. L. Pimentel,
A.-L. Sattelberger, and T. Westerdijk. Algebraic Approaches to Cosmological
Integrals. Preprint arXiv:2410.14757. To appear in Le Matematiche.

[ABHJLP (see https://arxiv.org/pdf/2312.05303 )]   N. Arkani-Hamed, D. Baumann,
A. Hillman, A. Joyce, H. Lee, and G. L. Pimentel. Differential Equations for
Cosmological Correlators. Preprint arXiv:2312.05303.

See also
========

  * Massless one-loop triangle Feynman diagram
  * Gauss' hypergeometric function
-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/examples.m2:71:0.

Massless one-loop triangle Feynman diagram
******************************************

Consider the annihilating $D$-ideal of a massless one-loop triangle Feynman
integral, appearing in Equation (5.14) in [HPSZ]. In what follows, we write
this system of PDEs in connection form.

+----------------------------------------------------------------+
|i1 : D = makeWeylAlgebra(QQ[x,y],{1,1});                        |
+----------------------------------------------------------------+
|i2 : Q1 = x^2*dx^2+2*x*y*dx*dy+(y-1)*y*dy^2+3*x*dx+(3*y-1)*dy+1;|
+----------------------------------------------------------------+
|i3 : Q2 = x*dx^2-y*dy^2+dx-dy;                                  |
+----------------------------------------------------------------+
|i4 : I = ideal(Q1,Q2);                                          |
|                                                                |
|o4 : Ideal of D                                                 |
+----------------------------------------------------------------+

We can check that the ideal has holonomic rank $4$ and compute the standard
monomials of the Gröbner basis of $RI$.

+-----------------------------------+
|i5 : assert( 4 == holonomicRank I )|
+-----------------------------------+
|i6 : standardMonomials I           |
|                                   |
|                   2               |
|o6 = {1, dx, dy, dy }              |
|                                   |
|o6 : List                          |
+-----------------------------------+

Finally, we can compute the connection matrices.

+-----------------------------------------------------------------------------+
|i7 : elapsedTime A = connectionMatrices I;                                   |
| -- .260473s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i8 : elapsedTime assert isIntegrable A                                       |
| -- .212052s elapsed                                                         |
+-----------------------------------------------------------------------------+
|i9 : netList A                                                               |
|                                                                             |
|     +-----------------------------------------------------------------------|
|o9 = || 0                                                       1            |
|     || 0                                                       -1/x         |
|     || -1/2xy                                                  -1/y         |
|     || (x2-4xy+3y2-2x-4y+1)/(2x3y2-4x2y3+2xy4-4x2y2-4xy3+2xy2) (x2-5xy+2y2-2|
|     +-----------------------------------------------------------------------|
|     || 0                                  0                                 |
|     || -1/2xy                             -1/y                              |
|     || 0                                  0                                 |
|     || (x-y-1)/(x2y2-2xy3+y4-2xy2-2y3+y2) (2x2-2x)/(x2y2-2xy3+y4-2xy2-2y3+y2|
|     +-----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                   0                                         |
|                                   1/x                                       |
|                                   (-x-3y+1)/2xy                             |
|x-3y+1)/(x2y2-2xy3+y4-2xy2-2y3+y2) (x3-7x2y+xy2+9y3-3x2-15y2+3x+7y-1)/(2x3y2-|
|-----------------------------------------------------------------------------|
|  1                                               0                          |
|  (-x-3y+1)/2xy                                   (-x-y+1)/2x                |
|  0                                               1                          |
|) (x2+4xy-7y2-2x+2y+1)/(x2y2-2xy3+y4-2xy2-2y3+y2) (-x2+7xy-6y2+2x+7y-1)/(x2y-|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                            0                                                |
|                            y/x                                              |
|                            (-x-y+1)/2x                                      |
|4x2y3+2xy4-4x2y2-4xy3+2xy2) (x3-9x2y+5xy2+3y3-3x2+4xy-7y2+3x+5y-1)/(2x3y-4x2y|
|-----------------------------------------------------------------------------|
|                   |                                                         |
|                   |                                                         |
|                   |                                                         |
|2xy2+y3-2xy-2y2+y) |                                                         |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------+                                                     |
|                      ||                                                     |
|                      ||                                                     |
|                      ||                                                     |
|2+2xy3-4x2y-4xy2+2xy) ||                                                     |
|-----------------------+                                                     |
|                       |                                                     |
|                       |                                                     |
|                       |                                                     |
|                       |                                                     |
|-----------------------+                                                     |
+-----------------------------------------------------------------------------+

References
==========

[HPSZ (see https://link.springer.com/article/10.1007/s11005-024-01835-7 )] J.
Henn, E. Pratt, A.-L. Sattelberger, and S. Zoia. $D$-Module Techniques for
Solving Differential Equations behind Feynman Integrals. Letters in
Mathematical Physics, 114(28), 2024.

See also
========

  * Cosmological correlator for the 2-site chain
  * Gauss' hypergeometric function
-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/examples.m2:100:0.

Gauss' hypergeometric function
******************************

Consider the GKZ system representing the Gauss hypergeometric function as in
[SST (see https://link.springer.com/book/10.1007/978-3-662-04112-3 ), Example
1.2.9].

+-----------------------------------------------------------------------------+
|i1 : D = makeWeylAlgebra(frac(QQ[a,b,c])[x_1..x_4]);                         |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(dx_2*dx_3 - dx_1*dx_4,x_1*dx_1 - x_4*dx_4 + 1 - c,x_2*dx_2 +  |
|                                                                             |
|o2 : Ideal of D                                                              |
|-----------------------------------------------------------------------------|
|x_4*dx_4 + a,x_3*dx_3 + x_4*dx_4 + b);                                       |
+-----------------------------------------------------------------------------+
|i3 : assert(2 == holonomicRank I)                                            |
+-----------------------------------------------------------------------------+
|i4 : standardMonomials I                                                     |
|                                                                             |
|o4 = {1, dx }                                                                |
|           4                                                                 |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

[SST (see https://link.springer.com/book/10.1007/978-3-662-04112-3 ), Example
1.4.23] computes the connection matrices for this system with constants
$a=1/2,b=1/2,c=1$. Using the connectionMatrices,
function, we can find the system for arbitrary constants.

+-----------------------------------------------------------------------------+
|i5 : A = connectionMatrices I;                                               |
+-----------------------------------------------------------------------------+
|i6 : isIntegrable A                                                          |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : netList(Boxes => false, VerticalSpace => 1, apply(4, i -> i+1 => A#i))  |
|                                                                             |
|o7 = 1 => | (c-1)/x_1          x_4/x_1                              |        |
|          | ab/(x_2x_3-x_1x_4) (x_4a+x_4b-x_4c+x_4)/(x_2x_3-x_1x_4) |        |
|                                                                             |
|     2 => | -a/x_2                      -x_4/x_2                             |
|          | -x_1ab/(x_2^2x_3-x_1x_2x_4) (-x_2x_3a-x_1x_4b+x_2x_3c-x_2x_3)/(x_|
|                                                                             |
|     3 => | -b/x_3                      -x_4/x_3                             |
|          | -x_1ab/(x_2x_3^2-x_1x_3x_4) (-x_1x_4a-x_2x_3b+x_2x_3c-x_2x_3)/(x_|
|                                                                             |
|     4 => | 0                          1                                     |
|          | x_1ab/(x_2x_3x_4-x_1x_4^2) (x_1x_4a+x_1x_4b-x_2x_3c+x_1x_4)/(x_2x|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                  |                                                          |
|2^2x_3-x_1x_2x_4) |                                                          |
|                                                                             |
|                  |                                                          |
|2x_3^2-x_1x_3x_4) |                                                          |
|                                                                             |
|                |                                                            |
|_3x_4-x_1x_4^2) |                                                            |
+-----------------------------------------------------------------------------+

Substituting the constants, we note that example in [SST] contains a small
misprint.

  * $A_1=\left(\!\begin{array}{cc} 0&\frac{x_{4}}{x_{1}}\\
    \frac{1}{4\,x_{2}x_{3}-4\,x_{1}x_{4}}&\frac{x_{4}}{x_{2}x_{3}-x_{1}x_{4}}
    \end{array}\!\right)$
  * $A_2=\left(\!\begin{array}{cc} -\frac{1}{2\,x_{2}}&-\frac{x_{4}}{x_{2}}\\
    -\frac{x_{1}}{4\,x_{2}^{2}x_{3}-4\,x_{1}x_{2}x_{4}}&\frac{-x_{2}x_{3}-x_{1}
    x_{4}}{2\,x_{2}^{2}x_{3}-2\,x_{1}x_{2}x_{4}} \end{array}\!\right)$
  * $A_3=\left(\!\begin{array}{cc} -\frac{1}{2\,x_{3}}&-\frac{x_{4}}{x_{3}}\\
    -\frac{x_{1}}{4\,x_{2}x_{3}^{2}-4\,x_{1}x_{3}x_{4}}&\frac{-x_{2}x_{3}-x_{1}
    x_{4}}{2\,x_{2}x_{3}^{2}-2\,x_{1}x_{3}x_{4}} \end{array}\!\right)$
  * $A_4=\left(\!\begin{array}{cc} 0&1\\
    \frac{x_{1}}{4\,x_{2}x_{3}x_{4}-4\,x_{1}x_{4}^{2}}&\frac{-x_{2}x_{3}+2\,x_{
    1}x_{4}}{x_{2}x_{3}x_{4}-x_{1}x_{4}^{2}} \end{array}\!\right)$

References
==========

[SST (see https://link.springer.com/book/10.1007/978-3-662-04112-3 )] M. Saito,
B. Sturmfels, and N. Takayama. Gröbner Deformations of Hypergeometric
Differential Equations. Volume 6 of Algorithms and Computation in Mathematics.
Springer, 2000.

See also
========

  * Cosmological correlator for the 2-site chain
  * Massless one-loop triangle Feynman diagram
-------------------------------------------------------------------------------

The source of this document is in ConnectionMatrices/examples.m2:141:0.