$\displaystyle\partial_{i}\mathrm{Std}_{j}=\sum_{k=1}^{R}(P_{i})_{jk}\,\mathrm{Std}_{k}\hskip
5.69046pt+\hskip
5.69046pt\sum_{k=1}^{D}\Delta_{jk}\,d_{k}\quad\text{in}\quad\mathcal{R}\,,$
(5.3)

where each $\Delta_{jk}\in\mathcal{R}$. Our short-term goal is to aptly
rewrite the second sum above, which will culminate in (5.7). We warn that this
will require some index gymnastics, but the final result does simplify. To
ease the notation in intermediate expressions, we shall keep the index $i$
fixed (otherwise we would have to write $\Delta_{ijk}$ etc.). The index $i$
will be reinstated at the very end.

To start, we bring the operator $\Delta_{jk}$ into normally ordered form:

$\displaystyle\Delta_{jk}\hskip
5.69046pt=\sum_{q\,\in\,\mathrm{supp}\left(\mathrm{Der}\right)}\delta_{jkq}(z)\,\partial^{q}\,,\quad\partial^{q}\in\mathrm{Der}\,,$
(5.4)

where $\delta_{jkq}\in\mathbb{C}(z)^{R\times D\times|\mathrm{Der}|}$ is a
newly introduced rank-3 tensor of rational functions. The set $\mathrm{Der}$
contains all the derivative monomials appearing on the RHS.

If we substitute the operators $\Delta_{jk}$ from (5.4) into the second sum of
(5.3), then the monomials $\partial^{q}\in\mathrm{Der}$ will act on the
generators $d_{i}$ (which potentially contain rational functions in $z$ as
prefactors). In normally ordered form, this action is written as

$\displaystyle\partial^{q}\,d_{k}=\sum_{p\,\in\,\mathrm{supp}\left(\mathrm{Mons}\right)}M_{qkp}(z)\,\partial^{p}\,,\quad\partial^{p}\in\mathrm{Mons}\,.$
(5.5)

Here we introduced a rank-3 tensor of rational functions
$M_{qkp}\in\mathbb{C}(z)^{|\mathrm{Der}|\times D\times|\mathrm{Mons}|}$ and
defined a set $\mathrm{Mons}$ whose elements consists of all the monomials
$\partial^{p}$ appearing on the RHS. Upon concatenation of the first two
indices $q$ and $k$, the rank-3 tensor $M_{qkp}$ turns into the _Macaulay
matrix_ [87]

$\displaystyle M_{op}(z):=M_{(qk)p}(z)\in\mathbb{C}^{(|\mathrm{Der}|\cdot
D)\times|\mathrm{Mons}|}\,.$ (5.6)

The combined index $o=(qk)=(kq)$ runs over the Cartesian product of sets
$\mathrm{supp}\left(\mathrm{Der}\right)\times\\{1,\ldots,D\\}.$

Plugging equations (5.4), (5.5) and (5.6) into the second sum of (5.3), this
term now takes the form of a triple matrix product:

$\displaystyle\sum_{k=1}^{D}\Delta_{jk}\,d_{k}\quad=\quad\sum_{o,\,p}\delta_{jo}(z)\,M_{op}(z)\,\partial^{p}\quad=\quad\big{(}C\cdot
M\cdot\mathrm{Mons}\big{)}_{j}\,.$ (5.7)

$C\in\mathbb{C}(z)^{R\times(|\mathrm{Der}|\cdot D)}$ stands for the
coefficient matrix $C_{jo}=\delta_{j(kq)}(z),$ when $\mathrm{Der}$ is regarded
as a column vector in (5.4). The set $\mathrm{Mons}$ is here also regarded as
a column vector.

Some of the monomials in $\mathrm{Mons}$ may be standard, but all need not be.
The non-standard monomials in $\mathrm{Mons}$ shall henceforth be called
_exterior monomials_ :

$\displaystyle\mathrm{Ext}=\mathrm{Mons}\setminus\mathrm{Std}\,.$ (5.8)

The disjoint union

$\displaystyle\mathrm{Mons}=\mathrm{Ext}\ \sqcup\ \mathrm{Std}$ (5.9)

then induces a partition on the columns of the Macaulay matrix:

$\displaystyle M=\Big{[}\ M_{\mathrm{Ext}}$ $\displaystyle\ \Big{|}\
M_{\mathrm{Std}}\ \Big{]}\,,\quad\text{with}\quad$ (5.10) $\displaystyle
M_{\mathrm{Ext}}\in\mathbb{C}(z)^{(|\mathrm{Der}|\cdot
D)\times|\mathrm{Ext}|}\qquad$ $\displaystyle\text{and}\qquad
M_{\mathrm{Std}}\in\mathbb{C}(z)^{(|\mathrm{Der}|\cdot
D)\times|\mathrm{Std}|}$

Thinking of the set $\mathrm{Std}$ as a column vector, we further decompose
the LHS of (5.3) as

$\displaystyle\partial_{i}\mathrm{Std}\quad=\quad\Big{[}\ C_{\mathrm{Ext}}\
\Big{|}\ C_{\mathrm{Std}}\ \Big{]}\cdot\mathrm{Mons}\quad=\quad
C_{\mathrm{Ext}}\cdot\mathrm{Ext}\hskip 5.69046pt+\hskip
5.69046ptC_{\mathrm{Std}}\cdot\mathrm{Std}\,,\quad\text{with}\quad$ (5.11)
$\displaystyle\hskip
62.59596ptC_{\mathrm{Ext}}\in\\{0,1\\}^{|\mathrm{Std}|\times|\mathrm{Ext}|}\qquad\text{and}\qquad
C_{\mathrm{Std}}\in\\{0,1\\}^{|\mathrm{Std}|\times|\mathrm{Std}|}\,.$

In other words, $C_{\mathrm{Ext}}$ and $C_{\mathrm{Std}}$ are two binary
matrices that encode whether the monomials in $\partial_{i}\mathrm{Std}$ are
exterior or standard.

Finally, inserting equations (5.7), (5.10) and (5.11) into the Pfaffian
equation (5.3), we arrive at

$\displaystyle(C_{\mathrm{Ext}}-C\cdot M_{\mathrm{Ext}})\hskip
5.69046pt+\hskip 5.69046pt(C_{\mathrm{Std}}-C\cdot
M_{\mathrm{Std}})\quad=\quad P_{i}\cdot\mathrm{Std}\,.$ (5.12)

But, because $\mathrm{Ext}$ and $\mathrm{Std}$ are linearly independent in
$\mathcal{R}$, this is equivalent to the system

$\displaystyle C_{\mathrm{Ext}}^{(i)}-C^{(i)}\cdot M_{\mathrm{Ext}}$
$\displaystyle=0$ (5.13) $\displaystyle C_{\mathrm{Std}}^{(i)}-C^{(i)}\cdot
M_{\mathrm{Std}}$ $\displaystyle=P_{i}\,.$ (5.14)

Here we also reinstated the dependence on the index $i$, as promised earlier
in this section. These two matrix equations are central to the algorithm for
computing Pfaffian matrices that we propose in the next section.

####  Section 5.2From Macaulay to Pfaffian matrix

Now we reverse the logic in the derivation leading to equations (5.13) and
(5.14). Although we started by assuming to know the Pfaffian system (5.2), we
now regard it as the goal. Supposing that $C_{\bullet}^{(i)}\ \text{and}\
M_{\bullet}$ in (5.13) and (5.14) are easily calculable
($\bullet=\mathrm{Ext},\,\mathrm{Std}$), then $C^{(i)}$ and $P_{i}$ are the
only unknowns. The purported strategy is this:

$\displaystyle\text{solve \eqref{MM_Ext_eq} for the rational matrix
$C^{(i)}$}\implies\text{insert $C^{(i)}$ into \eqref{MM_Std_eq} to obtain
$P_{i}$.}$

Let us formalize this strategy into a proper algorithm. To begin, define the
set of all derivative monomials bounded by a degree $Q\in\mathbb{Z}_{\geq 0}$:

$\displaystyle\mathrm{Der}_{Q}=\\{\partial^{q}\\}_{|q|\leq Q}\,.$ (5.15)

In analogy with $\mathrm{Mons}$ from (5.5), we let $\mathrm{Mons}_{Q}$ denote
the derivative monomials appearing in the normally order form of the set
$\big{\\{}\partial^{q}\,d_{k}\big{\\}}_{\partial^{q}\,\in\,\mathrm{Der}_{Q}}^{k=1,\ldots,D}.$
The Macaulay matrix $M_{Q}$ of degree $Q$ is defined as in (5.6), but with
$\mathrm{Der}$ and $\mathrm{Mons}$ replaced with $\mathrm{Der}_{Q}$ and
$\mathrm{Mons}_{Q}$. It thus holds that

$\displaystyle\Big{\\{}\partial^{q}\,d_{k}\Big{\\}}_{\partial^{q}\,\in\,\mathrm{Der}_{Q}}^{k=1,\ldots,D}\quad$
$\displaystyle=\quad\sum_{p\,\in\,\mathrm{supp}\left(\mathrm{Mons}_{Q}\right)}(M_{Q})_{(qk)p}\,\partial^{p}$
(5.16) $\displaystyle=\quad M_{Q}\cdot\mathrm{Mons}_{Q}\,.$ (5.17)

The set on the LHS of (5.16) is here viewed as a column vector of operators
with rational function coefficients, and $M_{Q}=M_{Q}(z)$ is the coefficient
matrix of the monomial derivatives $\mathrm{Mons}_{Q}$. The set
$\mathrm{Der}_{Q}$ can be thought of as "seeding" the Macaulay matrix.

The idea is now to adjust the degree $Q$ until (5.13) admits a solution for
$C^{(i)}$:

Input: Standard monomials $\mathrm{Std}$, generators $\\{d_{1},\ldots
d_{D}\\}$ of a holonomic ideal $\mathcal{I}$, direction $i$.
---
Output: Pfaffian matrix $P_{i}$.

1:$Q=0$

2:$M=M_{0}=\big{[}\ M_{\mathrm{Ext}}\ \big{|}\ M_{\mathrm{Std}}\ \big{]}$

3:$\mathrm{Mons}=\mathrm{Mons}_{0}$

4:while $C_{\mathrm{Ext}}^{(i)}=C^{(i)}\cdot M_{\mathrm{Ext}}$ is not solvable
w.r.t. $C^{(i)}$ do

5: $Q++$

6: $M\leftarrow M_{Q}$

7: $\mathrm{Mons}\leftarrow\mathrm{Mons}_{Q}$

8:end while

9:Solve $C_{\mathrm{Ext}}^{(i)}=C^{(i)}\cdot M_{\mathrm{Ext}}$ for $C^{(i)}$

10:return $P_{i}=C_{\mathrm{Std}}^{(i)}-C^{(i)}\cdot M_{\mathrm{Std}}$

Algorithm 1 : Pfaffian matrix by the Macaulay matrix method

There are a couple of immediate optimizations to this algorithm. First, notice
that (5.13) has a solution when the rows of $C_{\mathrm{Ext}}^{(i)}$ lie in
the row space of $M_{\mathrm{Ext}}$. The condition in the while statement in
line 4 of Algorithm (1) then translates into the requirement

$\displaystyle\mathrm{rank}\left[\begin{array}[]{c}M_{\mathrm{Ext}}\\\
\hline\cr
C_{\mathrm{Ext}}^{(i)}\end{array}\right]=\mathrm{rank}\big{[}M_{\mathrm{Ext}}\big{]}\,.$
(5.20)

It is quick to check this equality by setting every parameter in the problem
to a number in $\mathbb{Q}$ or a finite field
$\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ where $p$ is prime.

Second, to solve the system (5.13), it often suffices to pick only a subset of
the rows in $M_{\mathrm{Ext}}$. The reason is that some rows may be linearly
dependent. Such a subset can be identified by replacing parameters with
numbers and row reducing.

Lastly, since every matrix in the problem is defined over the rational
function field $\mathbb{C}(z)$, it is vastly more efficient to employ
_rational reconstruction_ algorithms rather than Gaussian elimination. These
algorithms reconstruct analytic expressions for rational functions by repeated
evaluations over $\mathbb{F}_{p}$. One need not even reconstruct the
intermediate "dummy matrix" $C^{(i)}$, as we are ultimately only interested in
the Pfaffian matrix $P_{i}$. I.e., rational reconstruction can be postponed
until all the matrix operations have been carried out in (5.14). This thesis
employs the rational reconstruction software FiniteFlow [88], because it
allows one to easily chain together several operations (such as matrix
multiplication and addition of matrices), with reconstruction postponed until
the final step. See [89, 90, 91, 92] for alternative implementations.

Example 5.1. Consider the holonomic ideal

$\displaystyle\mathcal{I}=\langle z\partial-\beta\rangle\,,$ (5.21)

where $z_{1}=z,\,\partial_{1}=\partial$ and $\beta\in\mathbb{C}$ is a
parameter. Although this example is too simple for an interesting application
of the Macaulay matrix method, we include it to showcase the many definitions
in the above text.

To build the Macaulay matrix, we must first choose the seeds
$\mathrm{Der}_{Q}$ defined in (5.16). It turns out that $Q=0$ is a sufficient
degree, but for the sake of illustration let us pick $Q=1$. Then
$\mathrm{Der}_{Q}=\\{1,\partial\\}.$ Following (5.16), we apply the monomials
in $\mathrm{Der}_{Q}$ to the generator of $\mathcal{I}$:

$\displaystyle 1(z\partial-\beta)=z\partial-\beta$ (5.22)
$\displaystyle\partial(z\partial-\beta)=\partial
z\partial-\beta\partial=(1+z\partial)\partial-\beta\partial=z\partial^{2}+(1-\beta)\partial\,.$
(5.23)

The set $\mathrm{Mons}_{Q}$ is formed from all the derivative monomials
appearing above:

$\displaystyle\mathrm{Mons}_{Q}=\\{\partial^{2},\partial,1\\}=\mathrm{Ext}\
\sqcup\ \mathrm{Std}=\\{\partial^{2},\partial\\}\ \sqcup\ \\{1\\}\,.$ (5.24)

The Macaulay matrix can now be constructed as

$\displaystyle\left[\begin{array}[]{c}1(z\partial-\beta)\\\
\partial(z\partial-\beta)\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c|c}M_{\mathrm{Ext}}&M_{\mathrm{Std}}\end{array}\right]\cdot\mathrm{Mons}_{Q}$
(5.28) $\displaystyle=\left[\begin{array}[]{cc|c}0&z&-\beta\\\
z&1-\beta&0\end{array}\right]\cdot\left[\begin{array}[]{c}\partial^{2}\\\
\partial\\\ 1\end{array}\right]\,.$ (5.34)

Next we build the two matrices $C_{\mathrm{Ext}}$ and $C_{\mathrm{Std}}$ as
per (5.11):

$\displaystyle\partial\mathrm{Std}$
$\displaystyle=\partial\big{[}1\big{]}=\big{[}\partial\big{]}$ (5.35)
$\displaystyle=\left[\begin{array}[]{c|c}C_{\mathrm{Ext}}&C_{\mathrm{Std}}\end{array}\right]\cdot\mathrm{Mons}_{Q}$
(5.37) $\displaystyle=C_{\mathrm{Ext}}\cdot\mathrm{Ext}\hskip 5.69046pt+\hskip
5.69046ptC_{\mathrm{Std}}\cdot\mathrm{Std}$ (5.38)
$\displaystyle=\left[\begin{array}[]{cc}0&1\end{array}\right]\cdot\left[\begin{array}[]{c}\partial^{2}\\\
\partial\end{array}\right]\hskip 5.69046pt+\hskip
5.69046pt\big{[}0\big{]}\cdot\big{[}1\big{]}\,.$ (5.42)

At this point we have all ingredients present in (5.13) and (5.14). The first
equation is

$\displaystyle C_{\mathrm{Ext}}-C\cdot
M_{\mathrm{Ext}}=\left[\begin{array}[]{cc}0&1\end{array}\right]-\left[\begin{array}[]{cc}C_{11}&C_{12}\end{array}\right]\cdot\left[\begin{array}[]{cc}0&z\\\
z&1-\beta\end{array}\right]=\left[\begin{array}[]{cc}0&0\end{array}\right]\,.$
(5.48)

Its solution is
$C=\left[\begin{array}[]{cc}\frac{1}{z}&0\end{array}\right]\,.$ Inserting this
into (5.14), we get

$\displaystyle C_{\mathrm{Std}}-C\cdot
M_{\mathrm{Std}}=\big{[}0\big{]}-\left[\begin{array}[]{cc}\frac{1}{z}&0\end{array}\right]\cdot\left[\begin{array}[]{c}-\beta\\\
0\end{array}\right]=\left[\frac{\beta}{z}\right]=P\,,$ (5.52)

where $P$ is the Pfaffian matrix (albeit just a scalar in this case). In other
words, we have derived that

$\displaystyle\partial\mathrm{Std}=P\cdot\mathrm{Std}\implies\big{[}\partial\big{]}=\left[\frac{\beta}{z}\right]\,,$
(5.53)

which is exactly the relation dictated by the ideal (5.21)! So we have
obtained a trivial result, but we can at least be satisfied that everything is
self-consistent. $\blacksquare$

Example 5.2. Let $\mathcal{I}=\langle d_{1},d_{2},d_{3},d_{4}\rangle$ be the
ideal generated by

$\displaystyle d_{1}$
$\displaystyle=z_{1}\partial_{1}+z_{2}\partial_{2}+z_{3}\partial_{3}+z_{4}\partial_{4}-\beta_{1}$
$\displaystyle d_{2}$
$\displaystyle=z_{2}\partial_{2}+z_{4}\partial_{4}-\beta_{2}$ (5.54)
$\displaystyle d_{3}$
$\displaystyle=z_{3}\partial_{3}+z_{4}\partial_{4}-\beta_{3}$ $\displaystyle
d_{4}$ $\displaystyle=\partial_{1}\partial_{4}-\partial_{2}\partial_{3}\,.$

In Section (6.5), we show that this is the annihilating ideal for Gauss’
${}_{2}F_{1}$ hypergeometric function. The $\mathcal{D}$-module
$\mathcal{D}/\mathcal{I}$ turns out to have a basis of standard monomials
given by $\mathrm{Std}=\\{\partial_{4},1\\}$ (later on, in Section (6.4), we
explain how to easily find such a basis). The goal of this example is to find
the Pfaffian system in direction $z_{4}$:

$\displaystyle\partial_{4}\mathrm{Std}=\partial_{4}\left[\begin{array}[]{c}\partial_{4}\\\
1\end{array}\right]=\left[\begin{array}[]{c}\partial_{4}^{2}\\\
\partial_{4}\end{array}\right]=P_{4}\cdot\left[\begin{array}[]{c}\partial_{4}\\\
1\end{array}\right]\,.$ (5.61)

To compute $P_{4}$, we need to resolve the higher-order monomial
$\partial_{4}^{2}$ in terms of $\partial_{4}$ and $1$. But $\partial_{4}^{2}$
does not appear in the generators $d_{i}$ from (5.2), so the seeding set
$\mathrm{Der}_{Q=0}=\\{1\\}$ is insufficient. It turns out that the degree
$Q=1$ monomials
$\mathrm{Der}_{1}=\\{1,\partial_{1},\partial_{2},\partial_{3},\partial_{4}\\}$
lead to a solvable system.

It is rather tedious to compute the Macaulay matrix by hand given this seed.
Instead, we use the newly developed package mt_mm written by Nobuki Takayama
and Saiei J. Matsubara-Heo in the CAS called asir. asir can be built from
source by cloning the git repository  https://github.com/openxm-org/OpenXM.
Source code and documentation for mt_mm can be found in the directories

            OpenXM/src/asir-contrib/packages/src/mt_mm
            OpenXM/doc/asir-contrib/en.

For the case at hand, the Macaulay matrix is obtained by entering the
following commands into asir111 We call the generators Di rather than di since
most lower-case objects are name-protected in asir. :

⬇

D1 = z1*dz1 + z2*dz2 + z3*dz3 + z4*dz4 - b1;

D2 = z2*dz2 + z4*dz4 - b2;

D3 = z3*dz3 + z4*dz4 - b3;

D4 = dz1*dz4 - dz2*dz3;

Ideal = [D1, D2, D3, D4];

Std = [dz4, 1];

mt_mm.find_macaulay(Ideal, Std, [z1, z2, z3, z4]);

The output is a list
$\\{M_{\mathrm{Ext}},M_{\mathrm{Std}},\mathrm{Ext},\mathrm{Std}\\}.$
Explicitly, we obtain the following data:

$M_{\mathrm{Ext}}=\left[\begin{array}[]{c||ccccccccccccccccccccc}\mathrm{Ext}\to&\partial_{1}\partial_{2}\partial_{3}&\partial_{2}^{2}\partial_{3}&\partial_{2}\partial_{3}^{2}&\partial_{1}^{2}\partial_{4}&\partial_{1}\partial_{2}\partial_{4}&\partial_{1}\partial_{3}\partial_{4}&\partial_{2}\partial_{3}\partial_{4}&\partial_{1}\partial_{4}^{2}&\partial_{1}^{2}&\partial_{1}\partial_{2}&\partial_{2}^{2}&\partial_{1}\partial_{3}&\partial_{2}\partial_{3}&\partial_{3}^{2}&\partial_{1}\partial_{4}&\partial_{2}\partial_{4}&\partial_{3}\partial_{4}&\partial_{4}^{2}&\partial_{1}&\partial_{2}&\partial_{3}\\\
\hline\cr\hline\cr
1d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{1}&z_{2}&z_{3}\\\
\hline\cr
1d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{2}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr
1d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{3}\\\
\hline\cr
1d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&-1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{1}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{1}&z_{2}&z_{3}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{1}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{2}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{1}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{3}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{1}d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&-1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{2}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{1}&z_{2}&z_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1-\beta_{1}\\\
\hline\cr\partial_{2}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-\beta_{3}\\\
\hline\cr\partial_{2}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1-\beta_{3}\\\
\hline\cr\partial_{2}d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{3}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{1}&z_{2}&\scalebox{0.7}{$\cdot$}&z_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1-\beta_{1}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{3}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1-\beta_{3}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{3}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-\beta_{3}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{3}d_{4}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{4}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{1}&z_{2}&\scalebox{0.7}{$\cdot$}&z_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1-\beta_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{4}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-\beta_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{4}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&z_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-\beta_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{4}d_{4}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&-1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\end{array}\right]$

$M_{\mathrm{Std}}=\left[\begin{array}[]{c||cc}\mathrm{Std}\to&\partial_{4}&1\\\
\hline\cr\hline\cr 1d_{1}&z_{4}&-\beta_{1}\\\ \hline\cr
1d_{2}&z_{4}&-\beta_{3}\\\ \hline\cr 1d_{3}&z_{4}&-\beta_{3}\\\ \hline\cr
1d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{1}d_{1}&1-\beta_{1}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{1}d_{2}&1-\beta_{3}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{1}d_{3}&1-\beta_{3}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{1}d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{2}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{2}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{2}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{2}d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{3}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{3}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{3}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{3}d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\hline\cr\partial_{4}d_{1}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{4}d_{2}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{4}d_{3}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\hline\cr\partial_{4}d_{4}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\end{array}\right]\,.$

The 21 columns in $M_{\mathrm{Ext}}$ are labeled by the exterior monomials
$\mathrm{Ext}$. The two columns in $M_{\mathrm{Std}}$ are labeled by
$\mathrm{Std}$. The $20$ rows in $M_{\mathrm{Ext}}$ and $M_{\mathrm{Std}}$ are
labeled by the seeds $\mathrm{Der}_{1}$ acting on the generators $d_{i}$.

Rows of the matrix $M_{\mathrm{Ext}}$ dictate relations among the external
monomials. One notices, however, that some of the rows are linearly dependent
on each other (this stems from an overzealous choice of the set
$\big{\\{}\partial^{q}d_{k}\big{\\}}$ from (5.16), as only a subset of the
seeds would suffice). Row reduction of $M_{\mathrm{Ext}}$ reveals that the
$17$ rows numbered by $\\{1,\ldots,14,16,17,20\\}$ are independent. Let
$M_{\mathrm{Ext}}^{\prime}$ be the $17\times 21$ matrix built from these rows.
The $17\times 2$ matrix $M_{\mathrm{Std}}^{\prime}$ is defined in a similar
manner. This row reduction can naturally be performed by fixing parameters to
numbers for more complicated examples.

Next we construct the two matrices $C_{\mathrm{Ext}}^{(4)}$ and
$C_{\mathrm{Std}}^{(4)}$ according to (5.11), finding that

$\displaystyle C_{\mathrm{Ext}}^{(4)}$
$\displaystyle=\left[\begin{array}[]{ccccccccccccccccccccc}\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
\end{array}\right]$ (5.64) $\displaystyle C_{\mathrm{Std}}^{(4)}$
$\displaystyle=\left[\begin{array}[]{cc}\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
1&\scalebox{0.7}{$\cdot$}\end{array}\right]\,.$ (5.67)

Now we can employ the central equations (5.13) and (5.14): The two-step
process is

  1. 1.

Define an unknown matrix $C^{(4)}$ of size $R\times S$ where
$R=|\mathrm{Std}|=2$ and $S=\\#\\{\text{rows in
}M_{\mathrm{Ext}}^{\prime}\\}=17.$ This matrix is determined by solving

$\displaystyle C_{\mathrm{Ext}}^{(4)}-C^{(4)}\cdot
M_{\mathrm{Ext}}^{\prime}=0\,,$ (5.68)

which can be done instantaneously with e.g. Mathematica.

  2. 2.

The solution for $C^{(4)}$ is inserted into

$\displaystyle C_{\mathrm{Std}}^{(4)}-C^{(4)}\cdot
M_{\mathrm{Std}}^{\prime}=P_{4}$ (5.69)

to obtain the Pfaffian matrix $P_{4}$.

The resulting Pfaffian system in direction $z_{4}$ works out to

$\displaystyle\partial_{4}\left[\begin{array}[]{c}\partial_{4}\\\
1\end{array}\right]=\left[\begin{array}[]{cc}\frac{-\beta_{1}+\beta_{3}+\beta_{3}-1}{z_{4}}+\frac{\beta_{1}z_{1}}{z_{1}z_{4}-z_{2}z_{3}}&\frac{\beta_{3}\beta_{3}z_{1}}{z_{4}(z_{2}z_{3}-z_{1}z_{4})}\\\
1&0\\\ \end{array}\right]\cdot\left[\begin{array}[]{c}\partial_{4}\\\
1\end{array}\right]\,.$ (5.76)

In particular, the first row of this Pfaffian equation says that

$\displaystyle\partial_{4}^{2}=\frac{z_{2}z_{3}(\beta_{1}-\beta_{3}-\beta_{3}+1)+z_{1}z_{4}(\beta_{3}+\beta_{3}-1)}{z_{4}(z_{1}z_{4}-z_{2}z_{3})}\partial_{4}-\frac{\beta_{3}\beta_{3}z_{1}}{z_{4}(z_{1}z_{4}-z_{2}z_{3})}1\quad\text{mod}\
\mathcal{I}\,,$

which, in physicist terms, could be thought of as an "IBP reduction" of
$\partial_{4}^{2}$ in terms of the "MIs" $\partial_{4}$ and $1$.

Pfaffian matrices in the directions $\\{z_{1},z_{2},z_{3}\\}$ are found in a
similar manner. $\blacksquare$

### Chapter 6GKZ Systems

> Among the Euler type integrals … there are the integrals $\int\Pi
> P_{i}(t_{1},\ldots,t_{n})^{\alpha_{i}}\ \times$ $t_{1}^{\beta_{1}}\ldots
> t_{n}^{\beta_{n}}\,\mathrm{d}t_{1}\ldots\mathrm{d}t_{n}$, where $P_{i}$ are
> polynomials, i.e. practically all integrals which arise in quantum field
> theory.
>
> Gel’fand, Kapranov, Zelevinsky (’89) [41]

A GKZ system is a particular $\mathcal{D}$-module $\mathcal{D}/H_{A}(\beta)$,
where $H_{A}(\beta)$ is an annihilating ideal built from an integer matrix $A$
and a collection of complex parameters $\beta_{i}$. We have secretly already
shown examples of generators for GKZ systems in equations (4.2), (5.21) and
(5.2).

Why study this specific $\mathcal{D}$-module? As indicated by the quote above,
GKZ systems can be related to FIs when they are written in parametric
representations rather than in momentum space (the connection between GKZ
systems and FIs will be explained in the next chapter). What’s more, it is
easy to determine the annihilating ideal $H_{A}(\beta)$ as well as a basis of
standard monomials, which is computationally advantageous.

####  Section 6.1Euler integrals

To define the GKZ system, let us begin by considering the class of functions
annihilated by the generators of $H_{A}(\beta)$. These functions are called
_Euler integrals_ [93], written as111 The most general definition contains
several polynomials $\\{g_{1},\ldots,g_{l}\\}$ in the integrand, but for our
inquiry a single polynomial $g(z|x)$ will suffice.

$\displaystyle
f_{\beta}(z)=\int_{\Gamma}g(z|x)^{\beta_{0}}x_{1}^{-\beta_{1}}\cdots
x_{n}^{-\beta_{n}}\frac{\mathrm{d}x}{x}\quad\text{with}\quad\frac{\mathrm{d}x}{x}:=\frac{\mathrm{d}x_{1}\wedge\cdots\wedge\mathrm{d}x_{n}}{x_{1}\cdots
x_{n}}\,.$ (6.1)

Let us dwell on the building blocks of this integral.

  1. 1.

$\Gamma$ is some chosen integration cycle in $\mathbb{C}^{n}$. For the purpose
of this thesis we may pick $\Gamma=(0,\infty)^{n}$ (though to be rigorous one
should also specify a Riemann sheet data, as the integrand is multivalued; see
[93, Section 3.1]).

  2. 2.

$g(z|x)$ is a Laurent polynomial in the integration variables
$x=(x_{1},\ldots,x_{n})$ with monomial coefficients $z=(z_{1},\ldots,z_{N}).$
In multivariate exponent notation

$\displaystyle
g(z|x)=\sum_{i=1}^{N}z_{i}\,x^{\alpha_{i}}\,,\quad\alpha_{i}\in\mathbb{Z}^{n}\,.$
(6.2)

Crucially, each monomial coefficient $z_{i}$ is regarded is an _independent_
variable of the function $f_{\beta}(z)$.

  3. 3.

The exponents $\beta=(\beta_{0},\ldots,\beta_{n})\in\mathbb{C}^{n+1}$ are
complex parameters. We generally assume that they are _generic_ real numbers,
meaning that they should not be integral. This will be formulated more
precisely as a _non-resonance_ condition later on.

By appending222 This is called the _Cayley trick_ or the _Cayley
configuration_.  a "1" in front of each multi-index $\alpha_{i}$ from (6.2),
we define a collection of $N$ column vectors
$a_{i}=\big{[}1,\alpha_{i}\big{]}^{T}\in\mathbb{Z}^{n+1}.$ These vectors are
assumed to span $\mathbb{Z}^{n+1}$. Further, they give rise to an $(n+1)\times
N$ dimensional matrix

$\displaystyle
A=\left[\begin{array}[]{c|c|c}a_{1}&\ldots&a_{N}\end{array}\right]\,.$ (6.4)

The polynomial $g$ in the integrand of (6.1) thus specifies the integer matrix
$A$, and vice versa. The (left) kernel of $A$ is defined as

$\displaystyle\mathrm{ker}\left(A\right)=\left\\{(u_{1},\,\ldots,\,u_{N})\in\mathbb{Z}^{N}\
\Big{|}\ \sum_{i=1}^{N}u_{i}\,a_{i}=\textbf{0}\right\\}\,.$ (6.5)

This can alternatively be viewed as a linear space spanned by finitely many
integer vectors $u$ satisfying $A\cdot u=\mathbf{0}$. Given a vector
$u\in\mathrm{ker}\left(A\right)$, we shall decompose it into positive and
negative components by writing $u=u^{+}-u^{-}$ for
$u^{+},\,u^{-}\in\mathbb{Z}^{N}_{\geq 0}.$

####  Section 6.2GKZ $\mathcal{D}$-module

The main characters of this chapter are the following operators parametrized
by the integer matrix $A$ and the complex parameters $\beta$:

$\displaystyle E\quad$ $\displaystyle=\quad A\cdot\theta-\beta$ (6.6)
$\displaystyle\Box_{u}\quad$
$\displaystyle=\quad\partial^{u^{+}}-\partial^{u^{-}}\,,\quad\forall\,u\in\mathrm{ker}\left(A\right)\,.$
(6.7)

The vector $\theta=\big{[}\theta_{1},\,\ldots,\,\theta_{N}\big{]}^{T}$ is a
collection of _Euler operators_ $\theta_{i}=z_{i}\,\partial_{i}\,.$ The $n+1$
operators $E_{i}$ are often called the _homogeneity operators_ (cf. Section
(6.5)). The ideal generated by the operators $\Box_{u}$ is called the _toric
ideal_ 333 The name stems from its relation to _toric geometry_ , a field of
algebraic geometry that studies varieties containing a torus as an open dense
subset [85]. .

Quite remarkably, $E$ and $\Box_{u}$ can be shown [42] to be the complete set
of annihilating operators for the Euler integral (6.1):

$\displaystyle E_{i}\bullet f_{\beta}(z)\quad$ $\displaystyle=\quad 0\,,\quad
i=0,\ldots,n$ (6.8) $\displaystyle\Box_{u}\bullet f_{\beta}(z)\quad$
$\displaystyle=\quad 0\,,\quad\forall\,u\in\mathrm{ker}\left(A\right)\,.$
(6.9)

The _GKZ system_ [41, 94] associated to an Euler integral is defined as the
$\mathcal{D}$-module $\mathcal{D}/H_{A}(\beta)$, where $H_{A}(\beta)$ is the
annihilating ideal generated by $E$ and $\Box_{u}$:

$\displaystyle H_{A}(\beta)\quad=\quad\sum_{i=0}^{n}\mathcal{D}\,E_{i}\ \
\hskip 5.69046pt+\hskip
5.69046pt\sum_{u\,\in\,\mathrm{ker}\left(A\right)}\mathcal{D}\,\Box_{u}\,.$
(6.10)

Given some complicated integral, it might well be very hard to find the
complete set of annihilating operators. But for the case of Euler integrals
having indeterminate $z$ variables, we can apparently immediately write down
all of those operators! In the $\mathcal{D}$-module philosophy, we may thus
study the GKZ system as a proxy for studying the Euler integral itself.

The GKZ system is holonomic [95], so it is endowed with the structure of a
finite-dimensional vector space. This means that we can use the Macaulay
matrix method to construct first-order Pfaffian systems for Euler integrals
given the higher-order operators that generate $H_{A}(\beta)$. This will be
topic of Chapter (7).

Example 6.1. Consider the Euler integral

$\displaystyle
f_{\beta}(z)=\int_{\Gamma}\big{(}z_{1}x_{1}+z_{2}x_{2}+z_{3}x_{3}+z_{4}x_{1}x_{2}+z_{5}x_{3}^{2}\big{)}^{\beta_{0}}x_{1}^{-\beta_{1}}x_{2}^{-\beta_{3}}x_{3}^{-\beta_{3}}\frac{\mathrm{d}x}{x}\,.$
(6.11)

We seek to construct its GKZ system. The first step is to input the exponent
multi-indices (with $1$’s appended) into the columns of the $A$-matrix (6.4):

$\displaystyle A=\left[\begin{array}[]{ccccc}1&1&1&1&1\\\ \hline\cr
1&0&0&1&0\\\ 0&1&0&1&0\\\ 0&0&1&0&2\end{array}\right]\,.$ (6.16)

The 4th column, for instance, represents the monomial
$z_{4}x^{(1,1,0)}=z_{4}x_{1}x_{2}.$ The operators $E_{i}$ from (6.6) can now
be written as

$\displaystyle
E=A\cdot\theta-\beta=\left[\begin{array}[]{c}\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}+\theta_{5}-\beta_{0}\\\
\theta_{1}+\theta_{4}-\beta_{1}\\\ \theta_{2}+\theta_{4}-\beta_{2}\\\
\theta_{3}+2\theta_{5}-\beta_{3}\\\ \end{array}\right]\,.$ (6.21)

The toric part (6.7) of the GKZ system requires knowing
$\mathrm{ker}\left(A\right)$. This can be found by solving a linear system, or
by using one’s favorite CAS (e.g. the command NullSpace in Mathematica). One
finds that $\mathrm{ker}\left(A\right)=\mathrm{span}(u),$ where $u$ is a
vector that we decompose as

$\displaystyle u\quad=\quad\left[\begin{array}[]{c}1\\\ 1\\\ -2\\\ -1\\\
1\end{array}\right]\quad=\quad
u^{+}-u^{-}\quad=\quad\left[\begin{array}[]{c}1\\\ 1\\\ 0\\\ 0\\\
1\end{array}\right]-\left[\begin{array}[]{c}0\\\ 0\\\ 2\\\ 1\\\
0\end{array}\right]\,.$ (6.37)

The toric operator is therefore

$\displaystyle\Box_{u}=\partial^{u^{+}}-\partial^{u^{-}}=\partial_{1}\partial_{2}\partial_{5}-\partial_{3}^{2}\partial_{4}\,.$
(6.38)

In summary, the annihilating ideal for the Euler integral (6.11) is generated
as $H_{A}(\beta)=\langle E,\Box_{u}\rangle,$ where $E$ and $\Box_{u}$ are
given by (6.21) and (6.38). $\blacksquare$

####  Section 6.3Twisted de Rham cohomology

The GKZ system can be related to another useful vector space for studying
Euler integrals, namely the _twisted de Rham cohomology group_. The elements
of this group are, loosely speaking, Euler inte _grands_. By defining this
space we will discover concrete formulas for representing the Euler integral
(6.1) as a differential operator inside a $\mathcal{D}$-module.

Twisted de Rham cohomology has recently garnered a lot of attention in the
study of FIs because it offers a novel and potentially more powerful framework
for IBP reduction. See e.g. [96, 21, 97, 98, 99, 100, 101, 102, 103, 104, 105,
106, 107, 108] and references therein.

##### 6.3.1 "Traditional" de Rham cohomology

Before discussing the "twisted" de Rham cohomology group, it is instructive to
first summarize the "traditional" theory. The story begins with the celebrated
theorem by Stokes:

$\displaystyle\int_{\Gamma}\mathrm{d}\omega=\int_{\partial\Gamma}\omega\,.$
(6.39)

$\omega=h(x)\mathrm{d}^{k}x$ is a differential $k$-form, where $h(x)$ is some
rational function, and $\mathrm{d}$ is the total derivative on integration
variables acting as

$\displaystyle\mathrm{d}\omega=\frac{\partial h(x)}{\partial
x_{1}}\mathrm{d}x_{1}\wedge\mathrm{d}^{k}x+\cdots+\frac{\partial
h(x)}{\partial x_{n}}\mathrm{d}x_{n}\wedge\mathrm{d}^{k}x\,.$ (6.40)

Note that $\mathrm{d}$ sends a $k$-form to a $(k+1)$-form.

The notation $\partial\Gamma$ denotes the boundary of the $k$-dimensional
integration contour $\Gamma$ (see the classical textbook [109] for a rigorous
definition). Note that the boundary is $(k-1)$-dimensional.

We focus on a particular class of integrals $\int_{\Gamma}\omega$ for which

$\displaystyle\begin{array}[]{cccccl}&\Gamma\text{ is a \emph{cycle}
}&\iff&\hskip 14.22636pt\partial\Gamma&=&0\\\ &\omega\text{ is a \emph{co-
cycle} }&\iff&\hskip 14.22636pt\mathrm{d}\omega&=&0\,.\end{array}$

Consider now the following two scenarios:

  1. 1.

Fix $\omega$ and shift $\Gamma$ by the boundary of another contour,
$\Gamma\to\Gamma+\partial\gamma$. It follows from Stokes’ theorem that

$\displaystyle\int_{\Gamma+\partial\gamma}\omega=\int_{\Gamma}\omega+\int_{\partial\gamma}\omega=\int_{\Gamma}\omega+\cancelto{0}{\int_{\gamma}\mathrm{d}\omega}=\int_{\Gamma}\omega\,.$
(6.41)

So, the value of the integral does not depend on any particular contour, but
rather an _equivalence class_ $[\Gamma]$ of contours, where
$\Gamma\sim\Gamma^{\prime}$ precisely when
$\Gamma=\Gamma^{\prime}+\partial\gamma$.

  2. 2.

Fix $\Gamma$ and shift $\omega$ by the total derivative of another
differential form, $\omega\to\omega+\mathrm{d}\psi$. Stokes’ theorem implies
that

$\displaystyle\int_{\Gamma}(\omega+\mathrm{d}\psi)=\int_{\Gamma}\omega+\int_{\Gamma}\mathrm{d}\psi=\int_{\Gamma}\omega+\cancelto{0}{\int_{\partial\Gamma}\psi}=\int_{\Gamma}\omega\,.$
(6.42)

The numerical value of the integral is thus determined by an _equivalence
class_ $[\omega]$ of differential forms, where $\omega\sim\omega^{\prime}$ iff
$\omega=\omega^{\prime}+\mathrm{d}\psi$.

It is therefore natural to study contours up to boundary terms, and
differential forms up to total derivatives. Let’s formalize these notions.

Define

$\displaystyle C_{k}=\big{\\{}\text{formal sums of $k$-dimensional cycles}\
\Gamma\big{\\}}\,.$ (6.43)

The boundary map $\partial_{k}:C_{k}\to C_{k-1}$ sends $k$-cycles to
$(k-1)$-cycles. An important result is that "the boundary of a boundary is
zero":

$\displaystyle\partial_{k}\circ\partial_{k+1}=0\quad\implies\quad\mathrm{Image}(\partial_{k+1})\
\subset\ \mathrm{Kernel}(\partial_{k})\,.$ (6.44)

It thus makes sense to define the $k$th _homology group_ as the quotient space

$\displaystyle
H_{k}=\frac{\mathrm{Kernel}(\partial_{k})}{\mathrm{Image}(\partial_{k+1})}\,.$
(6.45)

The numerator indicates that we focus on $k$-cycles $\Gamma$ for which
$\partial_{k}\Gamma=0$, and the denominator means that we mod out by boundary
terms of the form $\partial_{k+1}\gamma$. So this space indeed consists of the
equivalence classes $[\Gamma]$ discussed above.

Furthermore, let

$\displaystyle\Omega^{k}=\big{\\{}\text{formal sums of co-cycle $k$-forms}\
\omega\big{\\}}\,.$ (6.46)

The total derivative $\mathrm{d}_{k}:\Omega^{k}\to\Omega^{k+1}$ sends
$k$-forms to $(k+1)$-forms. It holds that

$\displaystyle\mathrm{d}_{k}\circ\mathrm{d}_{k-1}=0\quad\implies\quad\mathrm{Image}(\mathrm{d}_{k-1})\
\subset\ \mathrm{Kernel}(\mathrm{d}_{k})\,.$ (6.47)

The $k$th _de Rham cohomology group_ is then defined by the quotient

$\displaystyle
H^{k}=\frac{\mathrm{Kernel}(\mathrm{d}_{k})}{\mathrm{Image}(\mathrm{d}_{k-1})}\,.$
(6.48)

The numerator consists of co-cycles satisfying $\mathrm{d}_{k}\omega=0$, and
the denominator mods out by total derivatives $\mathrm{d}_{k-1}\psi$. This
space therefore consists of the equivalence classes $[\omega]$ mentioned
earlier.

##### 6.3.2 Twisted de Rham cohomology

Now consider integrals of the form

$\displaystyle\int_{\Gamma}U\,\omega_{q}\quad\text{where}\quad
U=g(z|x)^{\beta_{0}}\,x^{-\beta^{\prime}}\quad\text{and}\quad\omega_{q}=g(z|x)^{-q_{0}}x^{q^{\prime}}\frac{\mathrm{d}x}{x}\,,$
(6.49)

with $x^{\beta^{\prime}}=x_{1}^{\beta_{1}}\cdots x_{n}^{\beta_{n}}.$ We
collect the integer exponents of $\omega_{q}$ into a vector

$\displaystyle q=(q_{0},\,q^{\prime})\in\mathbb{Z}\times\mathbb{Z}^{n}\,.$
(6.50)

In "traditional" de Rham cohomology, we restricted ourselves to integrals of
differential forms with rational coefficient functions. Here there is a new
ingredient: the _twist_ $U$. This function is multivalued because the
exponents $\beta=(\beta_{0},\beta^{\prime})$ are non-integer by stipulation.

In this setting, Stokes’ theorem (6.39) says

$\displaystyle\int_{\Gamma}\mathrm{d}\big{[}U\,\omega_{q}\big{]}=\int_{\partial_{U}\Gamma}U\,\omega_{q}\,.$
(6.51)

We have upgraded the usual boundary operator to $\partial_{U}$ due to the
multivaluedness of $U$. $\partial_{U}$ keeps track of the branch of $U$
because $\Gamma$ might cross some branch cuts in its path; we refer to [93,
Chapter 3.1] for a proper definition.

On the LHS, we can expand the integrand to get

$\displaystyle\mathrm{d}\big{[}U\,\omega_{q}\big{]}$
$\displaystyle=U\mathrm{d}\omega_{q}+\mathrm{d}U\wedge\omega_{q}$ (6.52)
$\displaystyle=U\underbrace{\left[\mathrm{d}+\frac{\mathrm{d}U}{U}\wedge\right]}_{\nabla}\omega_{q}\,.$
(6.53)

In the last line we defined the _covariant derivative_ $\nabla$. Inserting the
twist $U$ from (6.49), it can be represented as

$\displaystyle\nabla=\mathrm{d}+\beta_{0}\frac{\mathrm{d}g(z|x)}{g(z|x)}\wedge-\sum_{i=1}^{n}\beta_{i}\frac{\mathrm{d}x_{i}}{x_{i}}\wedge\,.$
(6.54)

We choose to focus on integrals $\int_{\Gamma}U\omega_{q}$ such that

$\displaystyle\begin{array}[]{cccccl}&\Gamma\text{ is a \emph{twisted
cycle}}&\iff&\hskip 14.22636pt\partial_{U}\Gamma&=&0\\\ &\omega_{q}\text{ is a
\emph{twisted co-cycle}}&\iff&\hskip
14.22636pt\nabla\omega_{q}&=&0\,.\end{array}$

Following the same logic as before, we infer that the integral in question
only depends on the equivalence classes $[\Gamma]$ and $[\omega_{q}]$, where
$\Gamma\sim\Gamma^{\prime}\iff\Gamma=\Gamma^{\prime}+\partial_{U}\gamma$ and
$\omega_{q}\sim\omega_{q}^{\prime}\iff\omega_{q}=\omega_{q}^{\prime}+\nabla\psi$.

We won’t need the twisted homology group in what follows, so we skip that
definition. To define the _twisted de Rham cohomology group_ , we need to be
careful in distinguishing between the $x$\- and $z$-variables appearing in the
integrand. Let

$\displaystyle X=\left\\{(z,x)\,\in\,\mathbb{A}^{N}\times(\mathbb{C}^{*})^{n}\
\big{|}\ g(z|x)\neq 0\right\\}\quad\text{and}\quad Y=\mathbb{A}^{N}\,,$ (6.55)

where $\mathbb{A}$ is the complex affine line (essentially $\mathbb{C}$
without a distinguished choice of origin), and $\mathbb{C}^{*}$ is the complex
torus (which equals $\mathbb{C}\setminus\\{0\\}$ as a set). The space of
holomorphic functions on $X$ (technically a _sheaf_) is written as

$\displaystyle\mathcal{O}(X)=\mathbb{C}\Big{[}z_{1},\ldots,z_{N},x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1},\frac{1}{g}\Big{]}\,.$ (6.56)

For $1\leq k\leq n$, $\omega_{q}$ from (6.49) should be thought of as a
differential $k$-form in the $x$-variables (i.e. we allow for
$\mathrm{d}^{k}x$ but not $\mathrm{d}^{k}z$), with a coefficient function that
lives in $\mathcal{O}(X)$. These objects are called _relative $k$-forms_ in
the literature [23, Section 1.5] [110, Section 4.1], and they live in the
space444 The subscript $X/Y$ does not denote a quotient. This is merely the
standard notation for the space of relative $k$-forms.

$\displaystyle\Omega^{k}_{X/Y}\quad=\bigoplus_{\begin{subarray}{c}K\,\subset\,\\{1,\ldots,n\\}\\\
|K|\,=\,k\end{subarray}}\mathcal{O}(X)\,\mathrm{d}x^{K}\,.$ (6.57)

The covariant derivative $\nabla_{k}:\Omega^{k}_{X/Y}\to\Omega^{k+1}_{X/Y}$
thus sends a relative $k$-form to a relative $(k+1)$-form. A calculation shows
that

$\displaystyle\nabla_{k}\circ\nabla_{k-1}=0\quad\implies\quad\mathrm{Image}(\nabla_{k-1})\
\subset\ \mathrm{Kernel}(\nabla_{k})\,.$ (6.58)

The $k$th _twisted de Rham cohomology group_ can at last be defined as

$\displaystyle\mathbb{H}^{k}_{A}(\beta)=\frac{\mathrm{Kernel}(\nabla_{k})}{\mathrm{Image}(\nabla_{k-1})}\,.$
(6.59)

We have emphasized the dependence on the $A$-matrix and the $\beta$-parameter
vector, as they are used to build the integrand (6.49).

$\mathbb{H}^{k}_{A}(\beta)$ is endowed with the structure of a finite-
dimensional vector space [111]. When $\beta$ is generic enough (loosely
speaking, it should not be integral), then the dimension of
$\mathbb{H}^{k}_{A}(\beta)$ equals number of roots of the _likelihood
equations_ [112, 113]

$\displaystyle\frac{\beta_{0}}{g(z|x)}\frac{\partial g(z|x)}{\partial
x_{i}}-\frac{\beta_{i}}{x_{i}}=0\quad\text{for}\quad i=1,\ldots,n\,.$ (6.60)

The integer obtained in this fashion in fact equals the _Euler characteristic_
$\chi$ of the space $(\mathbb{C}^{*})^{n}\setminus\\{x_{1}\cdots x_{n}\cdot
g(z|x)=0\\}.$ For FIs, this integer even corresponds to the number of MIs
[114]555 Albeit without taking symmetry relations into account, which arise
when different momentum routings result in the same FI. ! What is more, this
counting works whether or not the $z$’s take on special values. In the GKZ
case, these variables are required to be independent and indeterminate. But
one is often interested in situations where some of the $z$’s are equal to
each other, and others equal unity. The dimension of
$\mathbb{H}^{k}_{A}(\beta)$ will generally change when the $z$’s are fixed to
special values, but the counting via (6.60) will still be valid.

$\mathbb{H}^{k}_{A}(\beta)$ can also be endowed with the structure of a
$\mathcal{D}$-module. For generic $z$ and $\beta$, it is in fact isomorphic to
the GKZ system as a $\mathcal{D}$-module! That is,

$\displaystyle\mathcal{D}_{N}/H_{A}(\beta)\quad\cong\quad\mathbb{H}^{k}_{A}(\beta)\,.$
(6.61)

Appendix (A.4) presents a rough sketch for the proof of this claim. Here we
simply take this fact as given, and in the next subsection we leverage it to
map elements from the RHS of the isomorphism to elements on the LHS.

##### 6.3.3 Representing Euler integrals inside $\mathcal{D}$-modules

The isomorphism (6.61) sends the equivalence class of operators
$[1]\in\mathcal{D}/H_{A}(\beta)$ to the equivalence class of differential
forms $\left[\frac{\mathrm{d}x}{x}\right]\in\mathbb{H}^{k}_{A}(\beta).$
Suppose we are given some other equivalence class $[\omega(z)]$. In this
section we shall construct an operator $d$ with the property

$\displaystyle d\bullet\left[\frac{\mathrm{d}x}{x}\right]=[\omega(z)]\,.$
(6.62)

Given a vector $q=(q_{0},q^{\prime})\in\mathbb{Z}\times\mathbb{Z}^{n},$ recall
the expression for $\omega_{q}$ from (6.49). For some fixed $\Gamma$, the
Euler integral associated to this differential form is denoted by

$\displaystyle\langle\omega_{q}\rangle=\int_{\Gamma}U\omega_{q}\,.$ (6.63)

By differentiating w.r.t. $z_{i}$, then $q$ gets shifted by the $i$th column
vector of the $A$-matrix:

$\displaystyle\partial_{i}\bullet\langle\omega_{q}\rangle=(\beta_{0}-q_{0})\langle\omega_{q+a_{i}}\rangle\,.$
(6.64)

There exists an operator which shifts $q$ in the opposite direction as well,
i.e. by $-a_{i}$. To set this up, the authors of [115] construct a so-called
_creation operator_ $C_{i}$ and a _b-function_ $b_{i}(\beta)$ such that

$\displaystyle C_{i}\,\partial_{i}-b_{i}(\beta)=0\quad\text{mod}\quad
H_{A}(\beta)\,.$ (6.65)

Using that $\langle\omega_{q}\rangle$ is a solution to $H_{A}(\beta-q)$, one
can then derive that

$\displaystyle C_{i}\bullet\langle\omega_{q}\rangle=b_{i}(\beta-
a_{i})\langle\omega_{q-a_{i}}\rangle\,.$ (6.66)

Suppose that an integral $\langle\omega_{q}\rangle$ is given. Since the
vectors $a_{i}$ span $\mathbb{Z}^{n+1}$ by stipulation, it is possible to
write the $q$-vector for $\langle\omega_{q}\rangle$ as

$\displaystyle q=\sum_{i=1}^{N}r_{i}\,a_{i}\,,\quad r_{i}\in\mathbb{Z}\,.$
(6.67)

The integers $r_{i}$ allow us to balance shifts by $\pm a_{i}$ in the
$q$-vector of $\langle\omega_{q}\rangle$ via (6.64) and (6.66). It thereby
becomes possible to erect the operator

$\displaystyle
d_{q}=\prod_{r_{i}<0}C_{i}^{-r_{i}}\prod_{r_{i}>0}\frac{1}{B(\beta)B^{\prime}(\beta)}\partial^{r_{i}}$
(6.68)

satisfying666 Contrary to what was promised in the early remarks of this
section, this equation is formulated as an action on a integral
$\langle\omega_{q}\rangle$ rather than on a differential form $[\omega(z)]$.
The action of a differential operator on $[\omega(z)]$ follows from (A.45),
but the formula for $d_{q}$ remains the same in that case.

$\displaystyle
d_{q}\bullet\langle\omega_{0}\rangle=\langle\omega_{q}\rangle\,,\quad\omega_{0}=\frac{\mathrm{d}x}{x}\,.$
(6.69)

The two functions $B(\beta)$ and $B^{\prime}(\beta)$ are built from the
prefactors of (6.64) and (6.66). Their explicit formulas are somewhat lengthy,
so we refer to [116] for details. The formula (6.68) has been implemented in
the asir package mt_gkz; see Example (6.3.3) below.

Note that the vector $r=[r_{1},\ldots,r_{N}]^{T}\in\mathbb{Z}^{N}$ from (6.67)
is not unique for fixed $q$. Given a collection of vectors
$u_{i}\in\mathbb{Z}^{N}$ which lie in $\mathrm{ker}\left(A\right)$, then it is
the case that $q=A\cdot\big{(}r+\sum_{i}c_{i}\,u_{i}\big{)}$ for any choice of
$c_{i}\in\mathbb{Z}$. This freedom can be exploited to simplify the operator
$d_{q}$ from (6.68). In particular, the creation operators $C_{i}$ tend to
have a large degrees in $\partial_{i}$, so for practical purposes it is useful
to adjust the coefficients $c_{i}$ such that $r_{i}>0$, in which case the
first product of (6.68) drops out.

Later on when we discuss applications to FIs, the $q$-vector will be used to
specify a particular integral within a family. In the Lee-Pomeransky
representation of a FI (shown in the future equation (7.3)), $q_{0}$ is
related to the spacetime dimension, and $q^{\prime}$ contains the propagator
powers.

Example 6.1. Consider the GKZ system for the 4-fold Euler integral

$\displaystyle
f_{\beta}(z)=\int_{\Gamma}\big{(}z_{1}x_{1}+z_{2}x_{2}+z_{3}x_{3}+z_{4}x_{4}+z_{5}x_{1}x_{3}+z_{6}x_{2}x_{4}\big{)}^{\beta_{0}}x^{-\beta^{\prime}}\frac{\mathrm{d}x}{x}\,.$
(6.70)

The goal of this example is to build the operator $d_{q}$ corresponding to the
differential form

$\displaystyle\omega_{q}=\left[\frac{x_{1}x_{2}}{g^{2}}\frac{\mathrm{d}x}{x}\right]\quad\text{with}\quad
q=(2,1,1,0,0)\,.$ (6.71)

We shall gain more insight from deriving $d_{q}$ by hand, rather than using
the closed formula (6.68). The idea is to take suitable derivatives of
$f_{\beta}(z)$ w.r.t. $z_{i}$. Since $\omega_{q}$ has a $g^{2}$ in the
denominator, we ought to take two derivatives to decrease
$\beta_{0}\to\beta_{0}-2$. The powers of $x_{1}$ and $x_{2}$ in $\omega_{q}$
are obtained by specifically choosing derivatives w.r.t. $z_{1}$ and $z_{2}$.
So we have

$\displaystyle\partial_{1}\partial_{2}\bullet
f_{\beta}(z)=\beta_{0}(\beta_{0}-1)\int_{\Gamma}g(z|x)^{\beta_{0}-2}x^{-\beta^{\prime}}x_{1}x_{2}\frac{\mathrm{d}x}{x}\,.$
(6.72)

Rearranging factors,

$\displaystyle\frac{\partial_{1}\partial_{2}}{\beta_{0}(\beta_{0}-1)}\bullet
f_{\beta}(z)=\int_{\Gamma}g(z|x)^{\beta_{0}}x^{-\beta^{\prime}}\left[\frac{x_{1}x_{2}}{g^{2}}\frac{\mathrm{d}x}{x}\right]\,.$
(6.73)

The $\mathcal{D}$-module representative for the differential form $\omega_{q}$
is therefore
$d_{q}=\frac{\partial_{1}\partial_{2}}{\beta_{0}(\beta_{0}-1)}\,.$

The same computation can be performed in asir via the following script:

⬇

import("mt_gkz.rr");

A = [

[1,1,1,1,1,1]

[1,1,0,0,0,0]

[0,0,1,1,0,0]

[1,0,0,0,1,0]

[0,0,1,0,0,1]

];

Ap = mt_gkz.make_Ap(A, 1);

Ap = vtol(Ap);

Beta = [b0,b1,b2,b3,b4];

Rvec = [1,1,0,0,0,0];

mt_gkz.ff(Rvec, A, Ap, Beta);

The first row of Ap consists of $1$’s, and the remaining rows contain only
$0$’s777 Ap is built from the vectors called $\mathbf{a}_{i}^{\prime}$ in
[116]. See the documentation of mt_gkz for additional details. . The
$r$-vector Rvec satisfies $q=A\cdot r$. The final command mt_gkz.ff(...)
outputs the result `(dx1*dx2)/(b0^2-b0)`, as expected. $\blacksquare$

To summarize, we now have three ways of representing _the same_ object:

  1. 1.

The Euler integral $\langle\omega_{q}\rangle=\int_{\Gamma}U\omega_{q},$
written in the notation of (6.49). This lives in some suitable space of
functions.

  2. 2.

The equivalence class of differential forms $[\omega_{q}]$. This lives in the
twisted cohomology group (6.59).

  3. 3.

The equivalence class of differential operators $[d_{q}]$ defined by formula
(6.68). It lives inside the $\mathcal{D}$-module
$\mathcal{D}_{N}/H_{A}(\beta)$.

####  Section 6.4Basis of standard monomials

As the GKZ $\mathcal{D}$-module is holonomic, it must have a finite collection
of standard monomials serving as a basis. There is a simple formula for the
size of this basis given a genericity condition on the parameter vector
$\beta=(\beta_{0},\ldots,\beta_{n})$. We require some definitions from
polyhedral geometry to state the formula.

  * •

Interpret the column vectors $a_{i}$ of the $A$-matrix as points in
$\mathbb{Z}^{n+1}$. The positive real _cone_ generated by these points is
defined as

$\displaystyle\mathbf{C}[A]=\left\\{\sum_{i=1}^{N}\lambda_{i}\,a_{i}\ \Big{|}\
\lambda_{i}\geq 0\right\\}\,.$ (6.74)

  * •

We say that $\beta$ is _non-resonant_ if any integer shift
$\beta+\mathbb{Z}^{n+1}$ contains no points on the _facets_ of
$\mathbf{C}[A]$. Facets are defined as codimension-1 faces of $\mathbf{C}[A]$.

Alternatively, we can say that $\beta$ is resonant when some integer translate
of the facet hyperplanes for $\mathbf{C}[A]$ can be made to "hit" $\beta$.

While this definition is somewhat technical, in practice it just means that
$\beta$ should be "generic enough", in the sense of not containing integer
values.

  * •

The _Newton polytope_ associated to $A$ is

$\displaystyle\mathbf{N}[A]=\left\\{\sum_{i=1}^{N}\lambda_{i}\,a_{i}\ \Big{|}\
\lambda_{i}\geq 0\quad\text{and}\quad\sum_{i=1}^{N}\lambda_{i}=1\right\\}\,.$
(6.75)

In other words, $\mathbf{N}[A]$ is the convex hull of $A$’s column vectors
$a_{i}\in\mathbb{Z}^{n+1}$.

When $\beta$ is non-resonant, it was proved in [95] that the holonomic rank of
the GKZ system, and hence also the number of standard monomials, is given by

$\displaystyle R=n!\,\times\,\mathrm{vol}(\mathbf{N}[A])\,,$ (6.76)

where $\mathrm{vol}(\mathbf{N}[A])$ is the volume of the Newton polytope (this
can be efficiently computed with software such as Polymake [117]). This
formula is quite remarkable. Recall that $R$ counts the number of independent
solutions to the PDEs (6.8)-(6.9). The latter is statement of complex
analysis, while the formula (6.76) is purely combinatorial.

For GKZ systems, not only is the number of basis elements known a priori,
there is even a fast algorithm to determine _explicit_ standard monomials due
to Hibi, Nishiyama and Takayama [118]. The idea of the algorithm is to compute
standard monomials for a simpler, auxiliary ideal. The construction can be
summarized as follows. Given the toric ideal of a GKZ system

$\displaystyle\mathcal{I}_{\Box}=\big{\langle}\Box_{u}\big{\rangle}_{\forall
u\,\in\,\mathrm{ker}\left(A\right)}\,,$ (6.77)

one must first build its ideal of initial terms
$\mathrm{in}_{\prec}(\mathcal{I}_{\Box})$ (recall (4.21)). Any generator $d$
of $\mathrm{in}_{\prec}(\mathcal{I}_{\Box})$ can be written in the form
$d=\prod_{i=1}^{N}\partial^{q_{i}}\text{ for }q_{i}\in\mathbb{Z}_{\geq 0}$ The
_distraction_ of $d$ is defined by

$\displaystyle\mathrm{dist}(d)=\prod_{i=1}^{N}\partial_{i}(\partial_{i}-1)\cdots(\partial_{i}-q_{i}-1)\,.$
(6.78)

The authors of [118] then consider the ideal $\mathcal{I}^{\prime}$ generated
as

$\displaystyle\mathcal{I}^{\prime}\quad=\quad\sum_{i=0}^{n}\mathcal{D}\,E_{i}\big{|}_{z_{1}=\cdots=z_{N}=1}\hskip
5.69046pt+\hskip 5.69046pt\sum_{\forall
d\,\in\,\mathrm{in}_{\prec}(\mathcal{I}_{\Box})}\mathcal{D}\,\mathrm{dist}(d)\,,$
(6.79)

where $E_{i}$ are the homogeneity operators from (6.6). Surprisingly, the set
of standard monomials for $\mathcal{D}/\mathcal{I}^{\prime}$ also constitutes
a basis for the GKZ system $\mathcal{D}/H_{A}(\beta)$. Hibi et al. thus obtain
standard monomials for the latter by a Gröbner basis computation in the
former. Crucially, $\mathcal{I}^{\prime}$ is an ideal in the _commutative_
subring $\mathbb{C}[\partial_{1},\ldots,\partial_{N}],$ which makes the
computation much more efficient than the execution of Buchberger’s algorithm
in a non-commutative Weyl algebra.

Example 6.1. To illustrate the efficiency of this algorithm, consider the
following $A$-matrix formed from a polynomial with $N=26$ terms:

$\displaystyle
A=\left[\begin{array}[]{cccccccccccccccccccccccccc}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\
1&1&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&1&1&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}\\\
1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&1&1&\scalebox{0.7}{$\cdot$}\\\
\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1\\\
\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1\\\
\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}\\\
\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&1&1&\scalebox{0.7}{$\cdot$}\\\
\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&\scalebox{0.7}{$\cdot$}&1\\\
\end{array}\right]\,.$

The algorithm of [118] has been implemented by Nobuki Takayama in asir via the
command

⬇

mt_gkz.cbase_by_euler(A);

In less than a minute on a laptop, this command gives a list of $R=238$
standard monomials for the GKZ system. The first few elements are

$\displaystyle\mathrm{Std}=\big{\\{}\partial_{26}\partial_{25}^{4},\,\partial_{26}^{4}\partial_{14},\,\partial_{25}^{2}\partial_{24}^{2},\,\partial_{25}^{3}\partial_{8},\,\partial_{25}^{3}\partial_{12},\,\ldots\big{\\}}\,.$
(6.80)

$\blacksquare$

####  Section 6.5Integrand rescaling

By cleverly rescaling the $z$-variables of the Euler integral (6.1), we show
here how to reduce the number of variables from $N$ to $N-(n+1)$. This
reduction is useful for practical computations.

The construction follows from a homogeneity property satisfied by Euler
integrals:

$\displaystyle
f_{\beta}(t^{a_{1}}z_{1},\,\ldots,\,t^{a_{N}}z_{N})=t_{0}^{\beta_{0}}\,t_{1}^{\beta_{1}}\cdots
t_{n}^{\beta_{n}}f_{\beta}(z)\,,$ (6.81)

where the vectors $a_{i}\in\mathbb{Z}^{n+1}$ come from the columns of the
$A$-matrix. By differentiating w.r.t. $t$ and using the chain rule, it can be
shown that this property is equivalent to the collection of PDEs $E_{i}\bullet
f_{\beta}(z)=0$ from (6.6). This is the reason for calling $E_{i}$ the
"homogeneity operators".

We can freeze $(n+1)$ of the $z$-variables to unity by judiciously choosing
the rescaling parameters $t\in\mathbb{C}^{n+1}$. This is best illustrated by
an example.

Example 6.1. This is a continuation of Example (1). After relabeling
$\beta_{i}\to\beta_{i-1}$ in the generators called $d_{i}$ in (5.2), we now
see that this is the GKZ system built from $A$-matrix

$\displaystyle A=\left[\begin{array}[]{cccc}1&1&1&1\\\ 0&1&0&1\\\ 0&0&1&1\\\
\end{array}\right]\,.$ (6.85)

The associated Euler integral is

$\displaystyle
f_{\beta}(z)=\int_{\Gamma}\big{(}z_{1}+z_{2}x_{1}+z_{3}x_{2}+z_{4}x_{1}x_{2}\big{)}^{\beta_{0}}x_{1}^{-\beta_{1}}x_{2}^{-\beta_{2}}\frac{\mathrm{d}x}{x}\,.$
(6.86)

In this case, the homogeneity property (6.81) reads

$\displaystyle
f_{\beta}(t_{0}z_{1},\,t_{0}t_{1}z_{2},\,t_{0}t_{2}z_{3},\,t_{0}t_{1}t_{2}z_{4})=t_{0}^{\beta_{0}}t_{1}^{\beta_{1}}t_{2}^{\beta_{2}}f_{\beta}(z_{1},z_{2},z_{3},z_{4})\,.$
(6.87)

Choosing

$\displaystyle t_{0}=\frac{1}{z_{1}}\,,\quad t_{1}=\frac{z_{1}}{z_{2}}\,,\quad
t_{2}=\frac{z_{1}}{z_{3}}\,,$ (6.88)

then (6.87) becomes

$\displaystyle
f_{\beta}(1,\,1,\,1,w)=z_{1}^{-\beta_{0}+\beta_{1}+\beta_{2}}z_{2}^{-\beta_{1}}z_{3}^{-\beta_{2}}f_{\beta}(z_{1},z_{2},z_{3},z_{4})\quad\text{with}\quad
w=\frac{z_{1}\,z_{4}}{z_{2}\,z_{3}}\,.$ (6.89)

A $4$-variable problem has thereby been reduced to one that depends on a
single cross-ratio variable $w$.

The three homogeneity operators $\\{E_{0},E_{1},E_{2}\\}$ have now been "gauge
fixed" away by this rescaling. The GKZ system then only depends on a single
toric operator:

$\displaystyle\Box_{u}\bullet
f_{\beta}(z)=(\partial_{1}\partial_{4}-\partial_{2}\partial_{3})\bullet
f_{\beta}(z)=0\,.$ (6.90)

Inserting the representation of $f_{\beta}$ from the RHS of (6.89), a short
calculation shows that the two terms in $\Box_{u}$ get mapped to

$\displaystyle\begin{array}[]{cccl}&\partial_{1}\partial_{4}&\to&-w^{-1}(\beta_{0}-\beta_{1}-\beta_{2}-w\partial_{w})(w\partial_{w})\\\
&\partial_{2}\partial_{3}&\to&(\beta_{1}+w\partial_{w})(\beta_{2}+w\partial_{w})\,.\end{array}$
(6.93)

The toric PDE (6.90) thus turns into the second-order ODE

$\displaystyle L\bullet f_{\beta}(1,1,1,w)=0\quad\text{with}$ (6.94)
$\displaystyle
L=w(1-w)\partial_{w}^{2}+\big{[}1-\beta_{0}+\beta_{1}+\beta_{2}-(1+\beta_{1}+\beta_{2})w\big{]}\partial_{w}-\beta_{1}\beta_{2}\,,$

which is recognized to be the DEQ for Gauss’ hypergeometric ${}_{2}F_{1}$
function. $\blacksquare$

The rescaling parameters (6.88) were here pulled out of a hat. In Appendix
(A.3), we present a general formula for how to choose these $t$’s.

####  Section 6.6Recurrence relations

This section describes a method, relying solely on products of Pfaffian
matrices, to determine recurrence relations among Euler integrals with
different values of $\beta$. (We remark that these matrix products can be
swiftly calculated using rational reconstruction.) One can think of these
recurrence relations as being a counterpart to IBP for FIs, though in this
case the IBPs come first and the Pfaffian systems last. That logic is reversed
by the method presented here, as we obtain Pfaffian matrices first (via the
Macaulay matrix method), and then use those Pfaffians to derive recurrence
relations/IBPs.

##### 6.6.1 Shifting $\beta$ by matrix factorials

For convenience, we shall work with a rescaled version of the Euler integral:

$\displaystyle f_{\beta}(z)\to
f(\beta)=\frac{1}{\Gamma(\beta_{0}+1)}\int_{\Gamma}g(z|x)^{\beta_{0}}x^{-\beta^{\prime}}\frac{\mathrm{d}x}{x}\,,$
(6.95)

where we chose to only emphasize the $\beta$-dependence of $f(\beta)$ since
$z$ will stay fixed throughout the discussion. The $\Gamma$-function prefactor
provides a clean shift relation when a derivative is taken w.r.t. $z_{i}$:

$\displaystyle\partial_{i}\bullet f(\beta)=f(\beta-a_{i})\,.$ (6.96)

Let $\mathrm{Std}$ be a basis of standard monomials for a GKZ system
$\mathcal{D}/H_{A}(\beta)$. We convert this $\mathcal{D}$-module basis into a
basis of functions by writing

$\displaystyle F(\beta)=\left[\begin{array}[]{c}\mathrm{Std}_{1}\bullet
f(\beta)\\\ \vdots\\\ \mathrm{Std}_{R}\bullet f(\beta)\end{array}\right]\,.$
(6.100)

Suppose we have computed the Pfaffian system for $F(\beta)$ via the Macaulay
matrix method. Combining the relation
$\partial_{i}\mathrm{Std}_{j}=\mathrm{Std}_{j}\partial_{i}$ with (6.96), we
deduce that

$\displaystyle\partial_{i}\bullet F(\beta)=P_{i}(\beta)\cdot F(\beta)=F(\beta-
a_{i})\,.$ (6.101)

In other words, matrix multiplication by $P_{i}(\beta)$ induces a difference
equation for $F(\beta)$.

It is possible to shift $\beta$ in the opposite direction as well. Assuming
$\beta$ is non-resonant, then $P_{i}(\beta+a_{i})$ is invertible. Setting

$\displaystyle Q_{i}(\beta)=P_{i}(\beta+a_{i})^{-1}\,,$ (6.102)

we thereby obtain an "inverse derivative operator" $\partial_{i}^{-1}$ acting
as

$\displaystyle\partial_{i}^{-1}\bullet F(\beta)=Q_{i}(\beta)\cdot
F(\beta)=F(\beta+a_{i})\,.$ (6.103)

Let $b\in\mathbb{Z}_{\geq 0}$. If $b>0$, we write the _falling matrix
factorial_ as

$\displaystyle\big{[}P_{i}(\beta)\big{]}_{b}=P_{i}\big{(}\beta-(b-1)a_{i}\big{)}\cdot
P_{i}\big{(}\beta-(b-2)a_{i}\big{)}\cdots P_{i}(\beta-a_{i})\cdot
P_{i}(\beta)\,,$ (6.104)

and the _rising matrix factorial_ as

$\displaystyle\big{(}Q_{i}(\beta)\big{)}_{b}=Q_{i}\big{(}\beta+(b-1)a_{i}\big{)}\cdot
Q_{i}\big{(}\beta+(b-2)a_{i}\big{)}\cdots Q_{i}(\beta+a_{i})\cdot
Q_{i}(\beta)\,.$ (6.105)

If $b=0$, then
$\big{[}P_{i}(\beta)\big{]}_{0}=\big{(}Q_{i}(\beta)\big{)}_{0}=\mathbf{1}.$

When the subscript $b$ is promoted to an integer vector $v\in\mathbb{Z}_{\geq
0}^{N}$, these definitions are extended to

$\displaystyle\big{[}P(\beta)\big{]}_{v}=$
$\displaystyle\prod_{i=1}^{N}\left[P_{i}\left(\beta-{\textstyle\sum_{j=i+1}^{N}}\,v_{j}\,a_{j}\right)\right]_{v_{i}}$
(6.106) $\displaystyle\big{(}Q(\beta)\big{)}_{v}=$
$\displaystyle\prod_{i=1}^{N}\left[Q_{i}\left(\beta-{\textstyle\sum_{j=i+1}^{N}}\,v_{j}\,a_{j}\right)\right]_{v_{i}}\,,$
(6.107)

with the convention that $\sum_{j=i+1}^{N}v_{j}a_{j}=0$ when $i=N$.

Now, by induction over $b>0$, it is short proof (see [82, Lemma 6.3]) to show
that

$\displaystyle\begin{array}[]{ccccl}\partial^{b}_{i}\bullet
F(\beta)&=&\big{[}P_{i}(\beta)\big{]}_{b}\cdot F(\beta)&=&F(\beta-ba_{i})\\\
\partial^{-b}_{i}\bullet F(\beta)&=&\big{(}Q_{i}(\beta)\big{)}_{b}\cdot
F(\beta)&=&F(\beta+ba_{i})\,.\end{array}$ (6.110)

Repeatedly applying these relations for each $v_{i}$ in a given vector
$v\in\mathbb{Z}_{\geq 0}^{N}$, we obtain the general shift relations

$\displaystyle\begin{array}[]{ccccl}\partial^{v}_{i}\bullet
F(\beta)&=&\big{[}P_{i}(\beta)\big{]}_{v}\cdot
F(\beta)&=&F\left(\beta-\sum_{i=1}^{N}v_{i}\,a_{i}\right)\\\\[5.0pt]
\partial^{-v}_{i}\bullet F(\beta)&=&\big{(}Q_{i}(\beta)\big{)}_{v}\cdot
F(\beta)&=&F\left(\beta+\sum_{i=1}^{N}v_{i}\,a_{i}\right)\,.\end{array}$
(6.113)

Note that these matrix factorial relations are not uniquely represented
because

$\displaystyle\begin{array}[]{cccc}F\big{(}(\beta-
a_{i})-a_{j}\big{)}&=&P_{i}(\beta-a_{j})\cdot P_{j}(\beta)\cdot F(\beta)\\\
&\text{and}&\\\ F\big{(}(\beta-a_{j})-a_{i}\big{)}&=&P_{j}(\beta-a_{i})\cdot
P_{i}(\beta)\cdot F(\beta)\end{array}$ (6.117)

imply the commutation relation

$\displaystyle P_{i}(\beta-a_{i})\cdot P_{j}(\beta)=P_{j}(\beta-a_{i})\cdot
P_{i}(\beta)\,,$ (6.118)

and likewise for $Q_{i}$.

##### 6.6.2 General $\beta$ shift algorithm

Here we employ the formulas derived in the previous subsection to state a
general algorithm for expressing a given Euler integral as a sum of basis
integrals. To begin, write the standard monomials in the form
$\mathrm{Std}_{i}=\partial^{q_{i}}$ for some $q_{i}\in\mathbb{Z}_{\geq
0}^{N}.$ Recalling that the $i$th component of $F(\beta)$ is of the form
$\mathrm{Std}_{i}\bullet f(\beta),$ then (6.96) implies that

$\displaystyle F(\beta)=\left[\begin{array}[]{c}f(\beta-A\cdot q_{1})\\\
\vdots\\\ f(\beta-A\cdot q_{R})\end{array}\right]\,.$ (6.122)

Given an integer vector $q_{0}\in\mathbb{Z}^{N}$, we seek a recurrence
relation for $f(\beta-A\cdot q_{0})$ of the form

$\displaystyle f(\beta-A\cdot
q_{0})=\sum_{i=1}^{R}c_{i}(\beta,z)\,f(\beta-A\cdot
q_{i})\quad\text{with}\quad c_{i}\in\mathbb{Q}(\beta,z)\,.$ (6.123)

Observe that $f(\beta-A\cdot q_{0})$ is the first element of the vector
$F\big{(}\beta-A\cdot(q_{0}-q_{1})\big{)}.$ Combining this fact with the shift
relations (6.113), we are lead to the following algorithm.

Input:

  * •

Vector $q_{0}\in\mathbb{Z}^{N}$.

  * •

Indeterminate vector $\beta$.

  * •

Monomial basis
$\mathrm{Std}=\big{\\{}\partial^{q_{1}},\ldots,\partial^{q_{R}}\big{\\}}$ for
$q_{i}\in\mathbb{Z}_{\geq 0}^{N}.$

Output: The recurrence relation (6.123).

1:Calculate $P_{i}(\beta)$, $i=1,\ldots,N,$ w.r.t. the basis $\mathrm{Std}$ by
calling Algorithm (1).

2:Decompose  $\displaystyle q_{0}-q_{1}=q^{+}-q^{-}=q$ for
$q^{\pm}\in\mathbb{Z}_{\geq 0}^{N}$.

3:Compute the matrix factorial
$\displaystyle\Big{[}P\Big{(}\beta-{\textstyle\sum_{q_{j}<0}}\,q_{j}\,a_{j}\Big{)}\Big{]}_{q^{+}}\
\cdot\ \big{(}Q(\beta)\big{)}_{q^{-}}\ \cdot\ F(\beta)\,.$

4:return the first element of step 3.

Algorithm 2 : Recurrence relations for Euler integrals

Since this algorithm avoids the use of derivatives, it is possible to fix the
$z$-variables to special values in the Pfaffian matrices. We give a simple
example of how to use this algorithm in Section (7.4).

### Chapter 7Generalized Feynman Integrals

In the physics literature, Tullio Regge observed already in 1968 that FIs
often evaluate to _hypergeometric functions_ [119] (for instance the
${}_{2}F_{1}$ function that we saw in the previous chapter). In the
mathematics literature, although a large zoo of hypergeometric111 Let
$F(z)=\sum_{n=0}^{\infty}c_{n}z^{n}$. We say that $F$ is _hypergeometric_ if
$c_{n+1}/c_{n}$ is a rational function of $n$. The generalization to
multivariate and logarithmic series is immediate.  functions (carrying famous
names such as Gauss, Kummer, Mellin, Barnes, Appell, Kampé de Fériet,
Lauricella, Horn and Meijer) were known in the last half of the previous
century, there was arguably no overarching framework that tied them all
together. The work of Gel’fand, Kapranov and Zelevinsky in the 1990s did
exactly that: it gave a general construction which subsumed the whole zoo of
hypergeometric functions as special cases.

Following in the footsteps of Regge, it is then natural to wonder whether GKZ
systems can say something about FIs. The first proper study in this direction
appears to have been by Nasrollahpoursamami in 2016 [28, 120]. There has since
then been a flurry of works studying this topic [121, 122, 123, 124, 125, 126,
127, 128, 113, 82, 83, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139,
140, 141, 142, 143]. There is now even a practical program, named FeynGKZ
[123], that computes series expansions for FIs based on the GKZ framework.

The connection between FIs and GKZ systems is not so immediate in the momentum
space representation. The connection becomes more clear in parametric
representations such as the _Lee-Pomeransky representation_ (LPr) [43]. The FI
then looks more like the Euler integral defined in (6.1), which is a solution
to the GKZ system. The difference is that the $z$-variables take on special
values in the LPr, so the class of Euler integrals is too general. Though the
proper function space for FIs is unknown (cf. Section (2.1.3)), the Euler
integral can thus be thought of as an "upper bound" on the complexity of this
space.

In what follows, we use the technology developed in the preceding chapters to
study some simple FIs in the GKZ framework. We especially focus on Pfaffian
systems for MIs.

####  Section 7.1Lee-Pomeransky representation as a GKZ system

Fix a Feynman graph $G$. There are two special _Symanzik polynomials_
associated to $G$, given by

$\displaystyle\mathcal{U}(x)$
$\displaystyle=\sum_{T}\prod_{e\,\notin\,T}x_{e}$ (7.1)
$\displaystyle\mathcal{F}(x)$
$\displaystyle=-\sum_{F}p(F)^{2}\prod_{e\,\notin\,F}\hskip 5.69046pt+\hskip
5.69046pt\mathcal{U}(x)\sum_{i=1}^{n}m_{i}^{2}x_{i}\,.$ (7.2)

$T$ and $F$ are sets of so-called _spanning trees_ and _spanning forests_
respectively, and $p(F)^{2}$ is the Minkowski square of the momentum flowing
into certain forest components. We shall not dwell on these definitions here,
as there are many exhaustive expositions already, see e.g. [44, Chapter 3].
The parameters $m_{i}$ denote the internal masses associated to the $n$
internal edges of $G$.

Starting from the momentum space representation of a FI, it is possible to
derive an equivalent parametric representation called the _Lee-Pomeransky
representation_ (LPr) [43] (see also [44, Section 2.5]). We denote it by

$\displaystyle
I(\mathrm{D}_{0}|\nu)=\frac{\Gamma\left(\frac{\mathrm{D}}{2}\right)}{\Gamma\left(\frac{\mathrm{D}}{2}-\omega\right)\Gamma(\nu)}\int_{0}^{\infty}\mathcal{G}(x)^{-\frac{\mathrm{D}}{2}}\,x^{\nu}\,\frac{\mathrm{d}x}{x}\,,$
(7.3)

where

  * •

$\mathcal{G}$ is the _Lee-Pomeransky_ (LP) _polynomial_

$\displaystyle\mathcal{G}(x)=\mathcal{U}(x)+\mathcal{F}(x)\,.$ (7.4)

  * •

$\omega$ is the _superficial degree of divergence_

$\displaystyle\omega=\sum_{i=1}^{n}\nu_{e}-\mathrm{D}L/2$ (7.5)

written in terms of edge weights
$\nu=(\nu_{1},\ldots,\nu_{n})\in\mathbb{Z}^{n}$, the spacetime dimension
$\mathrm{D}=\mathrm{D}_{0}-2\epsilon$, and the number of loops $L$. We assume
$\mathrm{D}_{0}\in 2\cdot\mathbb{Z}_{>0}$ is even for simplicity.

  * •

$\Gamma(\nu)$ is shorthand for $\prod_{i=1}^{n}\Gamma(\nu_{i})$.

##### 7.1.1 Generalized Feynman integral

Let $0<\delta\ll 1$. The two small parameters $(\epsilon,\delta)$ correspond
to analytic regulators in the works of Speer [144, 145]. Fix the exponents
$\beta$ of the Euler integral $f_{\beta}(z)$ from (6.1) to222 Writing
$-\epsilon\delta$ instead of, say, $\delta$ is merely a matter of convention.
This convention was chosen so as to more easily compare results from Section
(7.2) with [101].

$\displaystyle\beta=(\epsilon,\,-\epsilon\delta,\,\ldots,-\epsilon\delta)\hskip
5.69046pt-\hskip 5.69046pt(\mathrm{D}_{0}/2,\,\nu_{1},\ldots,\,\nu_{n})\,.$
(7.6)

The _generalized FI_ (GFI) is now defined to be

$\displaystyle I(\mathrm{D}_{0}|\nu)=c(\mathrm{D}_{0}|\nu)\times
f_{\beta}(z)\,.$ (7.7)

  * •

Inserting the special choice of $\beta$ from (7.6), the Euler integral reads

$\displaystyle
f_{\beta}(z)=\int_{(0,\infty)^{n}}\mathcal{G}(z|x)^{\epsilon-\mathrm{D}_{0}/2}\,x_{1}^{\nu_{1}+\epsilon\delta}\cdots
x_{n}^{\nu_{n}+\epsilon\delta}\,\frac{\mathrm{d}x}{x}\,,$ (7.8)

where the integration contour has been fixed to $\Gamma=(0,\infty)^{n}.$

  * •

Rewriting the prefactor in (7.3) with this $\beta$, we have

$\displaystyle
c(\mathrm{D}_{0}|\nu)=\frac{\Gamma\big{(}\mathrm{D}_{0}/2-\epsilon\big{)}}{\Gamma\Big{(}(L+1)(\mathrm{D}_{0}/2-\epsilon)-|\nu|-n\epsilon\delta\Big{)}\prod_{i=1}^{n}\Gamma\big{(}\nu_{i}+\epsilon\delta\big{)}}\,,$
(7.9)

with $|\nu|=\nu_{1}+\ldots+\nu_{n}$.

  * •

The monomials $x^{\alpha_{i}}$ in $\mathcal{G}$ come from the LP polynomial
(7.3):

$\displaystyle\mathcal{G}(z|x)=\mathcal{U}(z|x)+\mathcal{F}(z|x)=\sum_{i=1}^{N}z_{i}\,x^{\alpha_{i}}\,.$
(7.10)

But, in contrast to the original LP polynomial, each monomial coefficient
$z_{i}$ is here taken to be indeterminate.

There is a GKZ system $\mathcal{D}/H_{A}(\beta)$ associated to such a GFI.
According to the isomorphism (6.61), $I(\mathrm{D}_{0}|\nu)$ is also
represented by a twisted cohomology group $\mathbb{H}^{k}_{A}(\beta)$. In the
notation of Section (6.3.3), given a twist
$U=\mathcal{G}(z|x)^{\epsilon}\,x^{\epsilon\delta}$ and a differential form

$\displaystyle[\omega_{\mathrm{D}_{0}/2,\nu}]=c(\mathrm{D}_{0}|\nu)\times\mathcal{G}^{-\mathrm{D}_{0}/2}\,x^{\nu}\,\frac{\mathrm{d}x}{x}\,,$
(7.11)

we then have that
$I(\mathrm{D}_{0}|\nu)=\langle\omega_{\mathrm{D}_{0}/2,\nu}\rangle.$ The
$r$-vector from (6.67) therefore satisfies $A\cdot
r=\big{(}\mathrm{D}_{0}/2,\nu\big{)}.$ The columns $a_{i}=[1,\alpha_{i}]^{T}$
of this $A$-matrix come from the multi-indices $\alpha_{i}$ in (7.10).

Now, making the identifications

  * •

$\delta\to 0$

  * •

$z_{i}\to\mathbb{Z}_{>0}\ \text{or}\ \big{\\{}\text{kinematic variables}\
(m_{i}^{2},p_{i}^{2},p_{i}\cdot p_{j})\big{\\}}$

then a GFI agrees exactly with the LPr. The $\delta\to 0$ limit is not so
troublesome (this parameter is inserted to ensure non-resonance of the GKZ
system). The more pressing issue is how to deal with each $z_{i}$ being
indeterminate in the GFI. By the homogeneity rescaling described in Section
(6.5), we can at least rescale $(n+1)$ of the $z$-variables to unity in order
to match with some or all of the monomial coefficients in the
$\mathcal{U}$-polynomial. In doing so, it is possible to exactly match the GFI
with a large class of 1-loop graphs [82, Section 5.1] \- in particular those
which are fully massless, or contain off-shell legs whose squared momenta are
independent, $p_{i}^{2}\neq p_{j}^{2}\neq 0$.

In general, however, there are _more_ GKZ variables than there are independent
monomial coefficients in the proper LP polynomial. This is especially true at
$L=2$ loops and beyond. In those cases one must take a kind of "limit", called
a _restriction_ , of the GKZ system. We study such restrictions in detail in
Chapters (8) and (9). For the remaining part of this chapter though, we only
study examples where the number of GKZ variables can be made to match with the
proper FI.

####  Section 7.2Macaulay matrix example: 1-loop massless box

Here we compute the Pfaffian system for the 1-loop massless box diagram using
the Macaulay matrix method from Chapter (5). This example was studied in the
LPr using twisted cohomology in [101].

The kinematic configuration is

$\displaystyle\sum_{i=1}^{4}p_{i}=0\,,\quad
p_{1}^{2}=\cdots=p_{4}^{2}=0\,,\quad s=2p_{1}\cdot p_{2}\,,\quad t=2p_{2}\cdot
p_{3}\,.$

In momentum space, one has the following four inverse propagators:

$\displaystyle\begin{array}[]{lllllll}&D_{1}&=&-\ell^{2}\,,&D_{2}&=&-(\ell-
p_{1})^{2}\,,\\\ &D_{3}&=&-(\ell-p_{1}-p_{2})^{2}\,,&D_{4}&=&-(\ell-
p_{1}-p_{2}-p_{3})^{2}\,.\end{array}$ (7.14)

The corresponding GFI reads

$\displaystyle
I(\mathrm{D}_{0}|\nu)=c(\mathrm{D}_{0}|\nu)\int_{(0,\infty)^{4}}\mathcal{G}(z|x)^{\epsilon-\mathrm{D}_{0}/2}\,x_{1}^{\nu_{1}+\epsilon\delta}\cdots
x_{4}^{\nu_{4}+\epsilon\delta}\,\frac{\mathrm{d}x}{x}\,,$ (7.15)

where

$\displaystyle\mathcal{G}(z|x)=\sum_{i=1}^{4}z_{i}x_{i}+z_{5}x_{1}x_{3}+z_{6}x_{2}x_{4}\,.$
(7.16)

Upon rescaling the integration variables in the proper LPr as $x\to x/(-s),$
then its monomial coefficients become

$\displaystyle z_{1}=\ldots=z_{5}=1\,,\quad z_{6}=\frac{t}{s}=z\,.$ (7.17)

So the proper LP has one independent coefficient $z$. On the other hand, the
GKZ system has $N=6$ variables. Using homogeneity, we can rescale $n+1=4+1=5$
of those variables to unity. To match with (7.17), we choose to rescale
$\\{z_{1},\ldots,z_{5}\\}$. Including an overall $s$-dependent prefactor
because of the $x$-rescaling above,

$\displaystyle
I(\mathrm{D}_{0}|\nu)\to(-s)^{\mathrm{D}_{0}/2-\epsilon-4\epsilon\delta-|\nu|}\times
I(\mathrm{D}_{0}|\nu)\,,$ (7.18)

then our GFI matches exactly with the LPr in the limit $\delta\to 0$.

Having "gauged away" the Euler homogeneity operators, the GKZ system now only
contains the toric part. Using (A.37) or the asir command mt_gkz.gkz_b, the
single toric operator works out to

$\displaystyle\mathcal{I}_{\Box}=$ $\displaystyle\
z^{3}[1+z]\partial^{3}\hskip 5.69046pt+\hskip
5.69046ptz^{2}\big{[}3(1+z)+\epsilon(2+z+6\delta(1+z))\big{]}\partial^{2}$
(7.19) $\displaystyle\hskip 5.69046pt+\hskip
5.69046ptz\big{[}(1+\epsilon+3\delta\epsilon)^{2}+z(1+\epsilon+6\delta\epsilon+\epsilon^{2}\delta(2+9\delta))\big{]}\partial\hskip
5.69046pt+\hskip 5.69046ptz\epsilon^{3}\delta^{2}[1+4\delta]\,,$

where $\partial=\partial_{z}$.

##### 7.2.1 Basis

Since the number of GKZ variables matches with the proper LPr, the formula
$R=n!\times\mathrm{vol}(\mathbf{N}[A])$ ought to correctly count the number of
MIs. In polymake, this volume can be computed as

⬇

$P = new Polytope(POINTS =>

[[1,1,0,0,0],

[1,0,1,0,0],

[1,0,0,1,0],

[1,0,0,0,1],

[1,1,0,1,0],

[1,0,1,0,1]]

);

print $P -> VOLUME;

The matrix above is the transpose of the $A$-matrix associated to the LP
polynomial (7.16), due to the conventions of the software. The script outputs
the volume $1/8$, so we expect $R=4!\times 1/8=3$ MIs. This indeed holds true,
as can be verified by an IBP calculation.

Taking inspiration from [146], we choose the following basis of canonical
integrals near $4$ dimensions:

$\displaystyle
e^{(I)}=(-s)^{\epsilon+1}\left[\begin{array}[]{c}z\,I(4|0102)\\\ I(4|1020)\\\
\epsilon\,z\,(-s)\,I(4|1111)\end{array}\right]\,.$ (7.23)

Using (7.11), this corresponds to the basis of differential forms

$\displaystyle
e^{(\text{dR})}=\mathrm{c}\cdot\left[\begin{array}[]{c}\frac{x_{4}}{x_{1}\,x_{2}\,\mathcal{G}^{2}}\mathrm{d}x\\\\[4.0pt]
\frac{x_{3}}{x_{2}\,x_{4}\,\mathcal{G}^{2}}\mathrm{d}x\\\\[4.0pt]
\frac{1}{\mathcal{G}^{2}}\mathrm{d}x\end{array}\right]\,,$ (7.27)

where $\mathrm{c}$ is a $3\times 3$ diagonal matrix containing the
$\Gamma$-function constants $c(\mathrm{D}_{0}|\nu)$ multiplied by the
prefactors from (7.23). The formulas (6.68) and (A.37) turn these differential
forms into a $\mathcal{D}$-module basis

$\displaystyle
e^{(\mathcal{D})}=\mathrm{c}\cdot\left[\begin{array}[]{c}\frac{\epsilon\delta-1}{\epsilon(\epsilon-1)}\partial-\frac{z}{\epsilon(\epsilon-1)}\partial^{2}\\\\[4.0pt]
\frac{\epsilon(1-3\delta)(4\delta-1)}{\epsilon-1}+\frac{z(7\epsilon\delta-2\epsilon-1)}{\epsilon(\epsilon-1)}\partial-\frac{z^{2}}{\epsilon(\epsilon-1)}\partial^{2}\\\\[4.0pt]
\frac{4\epsilon\delta-\epsilon-1}{\epsilon(\epsilon-1)}\partial-\frac{z}{\epsilon(\epsilon-1)}\partial^{3}\end{array}\right]\,.$
(7.31)

##### 7.2.2 Macaulay matrix

Our goal is to obtain the Pfaffian system $\partial e^{(\mathcal{D})}=P\cdot
e^{(\mathcal{D})}.$ Using the Macaulay matrix method, we first construct the
Pfaffian system in the basis of standard monomials,

$\displaystyle\partial\mathrm{Std}=\partial\,\left[\begin{array}[]{c}\partial^{2}\\\
\partial\\\ 1\end{array}\right]=\left[\begin{array}[]{c}\partial^{3}\\\
\partial^{2}\\\
\partial\end{array}\right]=P^{(\mathrm{Std})}\cdot\left[\begin{array}[]{c}\partial^{2}\\\
\partial\\\ 1\end{array}\right]\,,$ (7.41)

whereafter we gauge transform to the $e^{(\mathcal{D})}$ system. With the
toric operator (7.19) and the basis $\mathrm{Std}$ as input, the Macaulay data
works out to

$\displaystyle M_{\mathrm{Ext}}$
$\displaystyle=\left[\begin{array}[]{c}z^{2}(z+1)\end{array}\right]$ (7.43)
$\displaystyle M_{\mathrm{Std}}$
$\displaystyle=\left[\begin{array}[]{c}z\big{[}\epsilon(6\delta+2)+z(6\epsilon\delta+\epsilon+3)+3\big{]}\\\\[3.0pt]
\big{(}3\epsilon\delta+\epsilon+1\big{)}^{2}+z\big{[}\epsilon^{2}\delta(9\delta+2)+6\epsilon\delta+\epsilon+1\big{]}\\\\[3.0pt]
\epsilon^{3}\delta^{2}(4\delta+1)\end{array}\right]^{T}$ (7.47) $\displaystyle
C_{\mathrm{Ext}}$
$\displaystyle=\left[\begin{array}[]{ccc}1&0&0\end{array}\right]\,,\quad
C_{\mathrm{Std}}=\left[\begin{array}[]{ccc}0&0&0\\\ 1&0&0\\\
0&1&0\end{array}\right]$ (7.52) $\displaystyle\mathrm{Mons}$
$\displaystyle=\mathrm{Ext}\ \sqcup\ \mathrm{Std}=\\{\partial^{3}\\}\ \sqcup\
\\{\partial^{2},\,\partial,\,1\\}\,.$

Letting $C=\big{[}c_{11}\ c_{12}\ c_{13}\big{]}$ be an unknown matrix, we then
solve $C_{\mathrm{Ext}}-C\cdot M_{\mathrm{Ext}}=0$ to obtain

$\displaystyle
C=\left[\begin{array}[]{ccc}\frac{1}{z^{2}(z+1)}&0&0\end{array}\right]\,.$
(7.54)

The Pfaffian matrix in the basis of standard monomials is then

$\displaystyle P^{(\mathrm{Std})}$ $\displaystyle=C_{\mathrm{Std}}-C\cdot
M_{\mathrm{Std}}$ (7.55)
$\displaystyle=\left[\begin{array}[]{ccc}P^{(\mathrm{Std})}_{11}&P^{(\mathrm{Std})}_{12}&P^{(\mathrm{Std})}_{13}\\\
1&0&0\\\ 0&1&0\end{array}\right]\,,$ (7.59)

with

$\displaystyle P^{(\mathrm{Std})}_{11}$
$\displaystyle=-\frac{\epsilon\big{[}6\delta(z+1)+z+2\big{]}+3(z+1)}{z(z+1)}$
$\displaystyle P^{(\mathrm{Std})}_{12}$
$\displaystyle=-\frac{\big{(}3\epsilon\delta+\epsilon+1\big{)}^{2}+z\big{[}\epsilon^{2}\delta(9\delta+2)+6\epsilon\delta+\epsilon+1\big{]}}{z^{2}(z+1)}$
(7.60) $\displaystyle P^{(\mathrm{Std})}_{13}$
$\displaystyle=-\frac{\epsilon^{3}\delta^{2}(4\delta+1)}{z^{2}(z+1)}\,.$

Finally, using the basis change method described in Appendix (A.1) we build a
matrix $G$ such that $e^{(\mathcal{D})}=G\cdot e^{(\mathrm{Std})},$ leading to
the gauge transformation

$\displaystyle P$ $\displaystyle=\left(\partial\bullet G+G\cdot
P^{(\mathrm{Std})}\right)\cdot G^{-1}$ (7.61)
$\displaystyle=\epsilon\left[\begin{array}[]{ccc}-\frac{1}{z}&0&0\\\ 0&0&0\\\
-\frac{2}{z(z+1)}&\frac{2}{z+1}&-\frac{1}{z(z+1)}\end{array}\right]\,.$ (7.65)

The limit $\delta\to 0$ has already been taken here. This matrix is canonical
as expected, and is in agreement with an independent LiteRed computation.

####  Section 7.3Macaulay matrix example: 1-loop pentagon with off-shell leg

Here we compute a Pfaffian system for the 1-loop massless pentagon integral
with one massive leg. The kinematics are

$\displaystyle\sum_{i=1}^{5}p_{i}=0\,,\quad p_{1}^{2}=\cdots$
$\displaystyle=p_{4}^{2}=0\,,\quad p_{5}^{2}=p^{2}$
$\displaystyle\text{and}\quad s_{ij}$ $\displaystyle=2p_{i}\cdot p_{j}\,.$

Note that the identity $(p_{1}+p_{2}+p_{3}+p_{4})^{2}=(-p_{5})^{2}=p^{2}$ can
be used to eliminate one of the Mandelstam variables $s_{ij}$.

The inverse propagators are given by

$\displaystyle\begin{array}[]{llllll}D_{1}&=&-\ell^{2}\,,\quad D_{2}=-(\ell-
p_{1})^{2}\,,&D_{3}&=&-(\ell-p_{1}-p_{2})^{2}\\\ D_{4}&=&-(\ell-
p_{1}-p_{2}-p_{3})^{2}\,,&D_{5}&=&-(\ell-
p_{1}-p_{2}-p_{3}-p_{4})^{2}\,.\end{array}$ (7.68)

The associated GFI is

$\displaystyle
I(\mathrm{D}_{0}|\nu)=c(\mathrm{D}_{0}|\nu)(-s_{12})^{\mathrm{D}_{0}/2-\epsilon-|\nu|-5\epsilon\delta}\int_{(0,\infty)^{5}}\mathcal{G}(z|x)^{\epsilon-\mathrm{D}_{0}/2}\,x_{1}^{\nu_{1}+\epsilon\delta}\cdots
x_{5}^{\nu_{5}+\epsilon\delta}\,\frac{\mathrm{d}x}{x}\,,$ (7.69)

with LP monomials given by

$\displaystyle\mathcal{G}(z|x)=\sum_{i=1}^{5}z_{i}\,x_{i}+z_{7}\,x_{1}\,x_{4}+z_{8}\,x_{1}\,x_{5}+z_{9}\,x_{2}\,x_{4}+z_{10}\,x_{1}\,x_{5}+z_{11}\,x_{3}\,x_{5}\,,$
(7.70) $\displaystyle\text{corresponding to}\quad
A=\left[\begin{array}[]{ccccccccccc}1&1&1&1&1&1&1&1&1&1&1\\\
1&0&0&0&0&1&1&1&0&0&0\\\ 0&1&0&0&0&0&0&0&1&1&0\\\ 0&0&1&0&0&1&0&0&0&0&1\\\
0&0&0&1&0&0&1&0&1&0&0\\\ 0&0&0&0&1&0&0&1&0&1&1\end{array}\right]\,.$ (7.77)

The $s_{12}$-dependent prefactor in (7.69) results from rescaling integration
variables by $x_{i}\to x_{i}/(-s_{12}).$

The proper LP polynomial has the monomial coefficients

$\displaystyle\begin{array}[]{lllll}&z_{1}&=&\ldots\ \ =\ \ z_{6}\ \ =\ \ 1\\\
&z_{7}&=&1+y_{2}+y_{4}\\\ &z_{8}&=&y_{1}\\\ &z_{9}&=&y_{4}\\\
&z_{10}&=&-1+y_{1}-y_{2}-y_{3}\\\
&z_{11}&=&-1+y_{1}-y_{2}-y_{3}-y_{4}-y_{5}\end{array}$ (7.84)

in terms of ratios

$\displaystyle y_{1}=\frac{p^{2}}{s_{12}}\ ,\ y_{2}=\frac{s_{13}}{s_{12}}\ ,\
y_{3}=\frac{s_{14}}{s_{12}}\ ,\ y_{4}=\frac{s_{23}}{s_{12}}\ ,\
y_{5}=\frac{s_{24}}{s_{12}}\ .$ (7.85)

There are 5 monomial coefficients that differ from unity. We can exactly match
them with GKZ variables using homogeneity. More precisely, the GKZ system
originally has $N=11$ variables, but $n+1=5+1=6$ variables can be rescaled as
$z_{1}=\ldots=z_{6}=1,$ leaving us with 5 variables
$\\{z_{7},\ldots,z_{11}\\}$.

##### 7.3.1 Basis

Computing the volume of the Newton polytope with polymake, we expect
$R=5!\times 13/120=13$ MIs. With the asir command mt_gkz.cbase_by_euler, we
instantly identify a basis of $13$ standard monomials

$\displaystyle
e^{(\mathrm{Std})}=\left[\begin{array}[]{c}\partial_{9}\partial_{11}^{2}\\\
\partial_{9}^{2}\\\ \partial_{10}^{2}\\\ \partial_{8}\partial_{11}\\\
\partial_{9}\partial_{11}\\\ \partial_{10}\partial_{11}\\\
\partial_{11}^{2}\\\ \partial_{7}\\\ \partial_{8}\\\ \partial_{9}\\\
\partial_{10}\\\ \partial_{11}\\\ 1\end{array}\right]\,.$ (7.99)

For the sake of verifying our results against an independent IBP calculation,
we also determine an integral basis using LiteRed:

$\displaystyle
e^{(I)}=(-s_{12})^{\epsilon}\left[\begin{array}[]{c}I(2|0,0,1,0,1)\\\
I(2|0,1,0,0,1)\\\ I(2|0,1,0,1,0)\\\ I(2|1,0,0,0,1)\\\ I(2|1,0,0,1,0)\\\
I(2|1,0,1,0,0)\\\ \epsilon I(4|1,0,1,0,1)\\\
\epsilon(-s_{12})I(4|0,1,1,1,1)\\\ \epsilon(2\epsilon-1)I(6|1,0,1,1,1)\\\
\epsilon(-s_{12})I(4|1,1,0,1,1)\\\ \epsilon(-s_{12})I(4|1,1,1,0,1)\\\
\epsilon(-s_{12})I(4|1,1,1,1,0)\\\
\epsilon^{2}(-s_{12})I(6|1,1,1,1,1)\end{array}\right]\,.$ (7.113)

This basis has been slightly dressed up with prefactors to simplify its
Pfaffian system. Moreover, the dimension $\mathrm{D}_{0}$ of each
$I(\mathrm{D}_{0}|\nu)$ is chosen to ensure that its associated $r$-vector
only contains non-negative integers, as per the remark above Example (6.3.3).
The $\mathcal{D}$-module basis corresponding to (7.113) is computed by
inserting these $r$-vectors into the asir command mt_gkz.rvec_red2, with the
result

$\displaystyle
e^{(\mathcal{D})}=\mathrm{c}\cdot\left[\begin{array}[]{c}\partial_{11}\\\\[3.0pt]
\partial_{10}\\\\[3.0pt] \partial_{9}\\\\[3.0pt] \partial_{8}\\\\[3.0pt]
\partial_{7}\\\\[3.0pt]
\epsilon(5\delta+1)+z_{7}\partial_{7}+z_{8}\partial_{8}+z_{9}\partial_{9}+z_{10}\partial_{10}+z_{11}\partial_{11}\\\\[3.0pt]
(4\epsilon\delta+\epsilon+1)\partial_{11}+z_{11}\partial_{11}^{2}+z_{9}\partial_{9}\partial_{11}+z_{10}\partial_{10}\partial_{11}\\\\[3.0pt]
\partial_{9}\partial_{11}\\\\[3.0pt] e^{(\mathcal{D})}_{9}\\\\[3.0pt]
\partial_{7}\partial_{10}\\\\[3.0pt]
(5\epsilon\delta+\epsilon+1)\partial_{10}+z_{10}\partial_{10}^{2}+z_{7}\partial_{7}\partial_{10}+z_{8}\partial_{8}\partial_{10}+z_{9}\partial_{9}\partial_{10}+z_{11}\partial_{11}\partial_{10}\\\\[3.0pt]
(5\epsilon\delta+\epsilon+1)\partial_{9}+z_{9}\partial_{9}^{2}+z_{7}\partial_{7}\partial_{9}+z_{8}\partial_{8}\partial_{9}+z_{10}\partial_{10}\partial_{9}+z_{11}\partial_{11}\partial_{9}\\\\[3.0pt]
(4\epsilon\delta+\epsilon+2)\partial_{11}\partial_{9}+z_{9}\partial_{11}\partial_{9}^{2}+z_{10}\partial_{10}\partial_{11}\partial_{9}+z_{11}\partial_{11}^{2}\partial_{9}\end{array}\right]\,,$
(7.127)

where

$\displaystyle e^{(\mathcal{D})}_{9}=$ $\displaystyle\
\epsilon\delta\left(4\epsilon\delta+1\right)\partial_{11}+z_{11}\epsilon\delta\partial_{11}^{2}+z_{7}\left(4\epsilon\delta+\epsilon+1\right)\partial_{7}\partial_{11}+z_{9}\left(5\epsilon\delta+\epsilon+2\right)\partial_{9}\partial_{11}$
$\displaystyle+z_{10}\epsilon\delta\partial_{10}\partial_{11}+z_{7}z_{11}\partial_{7}\partial_{11}^{2}+z_{9}z_{11}\partial_{9}\partial_{11}^{2}+z_{9}^{2}\partial_{9}^{2}\partial_{11}$
(7.128)
$\displaystyle+z_{7}z_{9}\partial_{7}\partial_{9}\partial_{11}+z_{7}z_{10}\partial_{7}\partial_{10}\partial_{11}+z_{9}z_{10}\partial_{9}\partial_{10}\partial_{11}\,,$

and $\mathrm{c}$ is a $13\times 13$ diagonal matrix containing prefactors (it
can be downloaded from a mathematica notebook in [147]).

##### 7.3.2 Macaulay matrix

The Macaulay matrix data is obtained in less than a second on a laptop via the
asir command mt_mm.find_macaulay. The matrix $M_{\mathrm{Ext}}$ has dimensions
$189\times 113$, meaning that there are $113$ exterior monomials. After fixing
every parameter in $M_{\mathrm{Ext}}$ to a number, row reduction reveals that
only $133$ out of the $189$ rows are independent. The matrix containing these
independent rows is denoted by $M^{\prime}_{\mathrm{Ext}}$. Starting from the
$189\times 13$ matrix $M_{\mathrm{Std}}$, we similarly define the $133\times
13$ matrix $M^{\prime}_{\mathrm{Std}}$.

For each $i\in\\{7,\ldots,11\\}$ we proceed to solve the equation

$\displaystyle C^{(i)}_{\mathrm{Ext}}-C^{(i)}\cdot
M^{\prime}_{\mathrm{Ext}}=0$ (7.129)

for the unknown matrices $C^{(i)}$. This only takes a few minutes on a laptop
with FiniteFlow. The Pfaffian matrix in direction $i$ is finally obtained from

$\displaystyle C^{(i)}_{\mathrm{Std}}-C^{(i)}\cdot
M^{\prime}_{\mathrm{Std}}=P^{(\mathrm{Std})}_{i}\,.$ (7.130)

To compare it with LiteRed, we perform a gauge transformation from the basis
$e^{(\mathrm{Std})}$ in (7.99) to $e^{(\mathcal{D})}$ in (7.127). After
sending $\delta\to 0$, we find perfect agreement.

####  Section 7.4Recurrence relation example: 1-loop bubble

Here we showcase Algorithm (2) by computing an IBP relation for a simple
bubble diagram. The inverse propagators are

$\displaystyle D_{1}=-\ell^{2}+m^{2}\,,\quad D_{2}=-(\ell+p)^{2}\,,$

where $p$ is an incoming external momentum. The GFI takes the form

$\displaystyle
I(\mathrm{D}_{0}|\nu_{1},\nu_{2})=c(\mathrm{D}_{0}|\nu_{1},\nu_{2})\int_{(0,\infty)^{2}}\mathcal{G}(z|x)^{\epsilon-\mathrm{D}_{0}/2}\,x_{1}^{\nu_{1}+\epsilon\delta}\,x_{2}^{\nu_{2}+\epsilon\delta}\,\frac{\mathrm{d}x}{x}\,,$
(7.131)

with the LP monomials

$\displaystyle\mathcal{G}(z|x)=z_{1}\,x_{1}+z_{2}\,x_{2}+z_{3}\,x_{1}\,x_{2}+z_{4}\,x_{1}^{2}\,,\quad
A=\left[\begin{array}[]{cccc}1&1&1&1\\\ 1&0&1&2\\\
0&1&1&0\end{array}\right]\,.$ (7.135)

We avoid the use of homogeneity in order to follow the steps of Algorithm (2)
as closely as possible333 Algorithm (2) assumes that the input $\mathrm{Std}$
consists of single monomials. Rewriting the basis in rescaled variables using
(A.37) would generally lead to operators with several terms, slightly
modifying the algorithm. . Instead, we simply specify

$\displaystyle z_{1}=z_{2}=1\,,\quad z_{3}=m^{2}-p^{2},\quad z_{4}=m^{2}$
(7.136)

at the end of the computation to match with the physical LPr.

Fixing a basis of MIs

$\displaystyle e=\left[\begin{array}[]{c}I(4|1,1)\\\
I(4|2,0)\end{array}\right]\,,$ (7.139)

we now proceed to find $c_{1,2}$ in the decomposition444 Here we pick an
example where all the $\nu_{i}$ are non-negative. For negative $\nu_{i}$, one
has to slightly tweak the algorithm with the shift vector $\alpha$ from [82,
Equation 6.28].

$\displaystyle e_{0}=I(4|1,2)=c_{1}e_{1}+c_{2}e_{2}$ (7.140)

using Algorithm (2).

###### Step 0: Input.

Set
$\beta=\\{\epsilon,-\epsilon\delta,-\epsilon\delta\\}\in\mathbb{C}^{3}\setminus\mathbb{Z}^{3}.$
According to the formula (6.123), Algorithm (2) pertains to Euler integrals of
the form

$\displaystyle f(\beta-A\cdot
q)=I(\mathrm{D}_{0}|\nu)/c(\mathrm{D}_{0}|\nu)\,.$ (7.141)

By the definition of $I(\mathrm{D}_{0}|\nu)$ from (7.7), we thus require that
the argument of $f$ be

$\displaystyle\beta-A\cdot q=\beta-\big{(}\mathrm{D}_{0}/2,\nu)\,.$ (7.142)

(Such a $q$-vector plays a similar role as the $r$-vector from (6.67).) For
the integrals in the example at hand, we have

$\displaystyle\begin{array}[]{c||c|c|c}&q&A\cdot
q&\mathrm{Std}_{q}=\partial^{q}\\\ \hline\cr\hline\cr
I(4|1,2)&q_{0}=(0,1,1,0)&(2,1,2)&\partial_{2}\partial_{3}\\\ \hline\cr
I(4|1,1)&q_{1}=(1,1,0,0)&(2,1,1)&\partial_{1}\partial_{2}\\\ \hline\cr
I(4|2,0)&q_{2}=(2,0,0,0)&(2,2,0)&\partial_{1}^{2}\end{array}$ (7.147)

In the final column, we provided the $\mathcal{D}$-module representatives of
the integrals (up to prefactors).

###### Step 1.

Calling Algorithm (1), we obtain the $2\times 2$ Pfaffian matrices
$\\{P_{1},P_{2},P_{3},P_{4}\\}$ in the basis
$\mathrm{Std}=\\{\partial^{q_{1}},\,\partial^{q_{2}}\\}.$

###### Step 2.

We decompose

$\displaystyle q_{0}-q_{1}$ $\displaystyle=(-1,0,1,0)$
$\displaystyle=(0,0,1,0)-(1,0,0,0)$ (7.148) $\displaystyle=q^{+}-q^{-}$
$\displaystyle=q\,.$

###### Step 3.

This step instructs us to compute the matrix factorial corresponding to the
operator $\partial_{3}^{+1}\,\partial_{1}^{-1},$ where $\partial_{3}$ stems
from $q^{+}$ and $\partial_{1}$ stems from $q^{-}$. Explicitly, we compute

$\displaystyle P_{3}(\beta+a_{1})\cdot Q_{1}(\beta)\cdot e\,.$ (7.149)

###### Step 4: Output.

The sought after recurrence relation is encoded in the first element of
(7.149). After sending $\delta\to 0$ and replacing the $z_{i}$ variables as in
(7.136), we obtain the result

$\displaystyle
I(4|1,2)=\frac{(1-2\epsilon)(p^{2}+m^{2})}{(p^{2}-m^{2})^{2}}I(4|1,1)-\frac{2m^{2}}{(p^{2}-m^{2})^{2}}I(4|2,0)\,,$
(7.150)

which is in agreement with LiteRed.

### Chapter 8Restriction of $\mathcal{D}$-modules

As emphasized in the previous chapter, the GKZ system does not perfectly
capture the nature of FIs in the LPr. The former requires all monomial
coefficients $z_{i}$ of the integrand polynomial $\mathcal{G}(z|x)$ to be
indeterminate, but the latter has some of the $z_{i}$ fixed to special values.
The question is then whether there exists a kind of _limit_ of the GKZ system,
at the level of $\mathcal{D}$-modules, which will exactly identify it with the
annihilating ideal of a proper FI.

To motivate the need for a restriction procedure, let us consider an example.

Example 8.1. Let

$\displaystyle
I(\mathrm{D}_{0}|\nu)\propto\int_{(0,\infty)^{5}}\mathcal{G}(z|x)^{\epsilon-\mathrm{D}_{0}/2}\,x_{1}^{\nu_{1}+\epsilon\delta}\cdots
x_{5}^{\nu_{5}+\epsilon\delta}\,\frac{\mathrm{d}x}{x}$

denote the GFI associated to the 2-loop massless N-box diagram. The kinematics
are

$\displaystyle p_{1}^{2}=\cdots=p_{4}^{2}=0\,\quad s=2p_{1}\cdot p_{2}\,\quad
t=2p_{2}\cdot p_{3}\,,$ (8.1)

and the LP monomials read

$\displaystyle\mathcal{G}(z|x)$
$\displaystyle=z_{1}x_{1}x_{2}+z_{2}x_{1}x_{4}+z_{3}x_{1}x_{5}+z_{4}x_{2}x_{3}+z_{5}x_{2}x_{4}+z_{6}x_{2}x_{5}$
(8.2)
$\displaystyle+z_{7}x_{3}x_{4}+z_{8}x_{3}x_{5}+z_{9}x_{1}x_{2}x_{4}+z_{10}x_{2}x_{3}x_{5}\,.$

The holonomic rank of the GKZ system is $R=9$, as dictated by the formula
(6.76), and it stays fixed when six of the $z_{i}$ variables are rescaled to
unity via homogeneity.

After rescaling integration variables by $x\to x/s$, the proper LPr has
monomial coefficients confined to the space

$\displaystyle
Y^{\prime}=\big{\\{}(z_{1},\ldots,z_{10})\in\mathbb{A}^{10}\,\big{|}\,z_{1}=\ldots=z_{9}=1\big{\\}}\,,$
(8.3)

with $z_{10}=t/s$ being generic. Counting the dimension of the twisted
cohomology group $\mathbb{H}^{k}_{A}(\beta)$ for $z\in Y^{\prime}$ via the
likelihood equation (6.60), one finds that
$\dim(\mathbb{H}^{k}_{A}(\beta))=3.$ This is the correct number of MIs, as as
verified by an IBP calculation.

As $R=9>3$, we infer that the GKZ system is "too big". We therefore seek some
way to recover the space $\mathbb{H}^{k}_{A}(\beta)$, having special values
for the $z$’s, via some limiting procedure of the $\mathcal{D}/H_{A}(\beta)$.
$\blacksquare$

"Limits" of $\mathcal{D}$-module are known under the name of _restrictions_
[85, Chapter 5]. Standard techniques for computing restrictions require huge
computational resources, and can generally not be applied to FIs of interest.
In this chapter, we shall develop two new methods for computing restrictions
that are more efficient. The first one takes the Pfaffian system associated to
a basis of Euler integrals as input. The second one works at the level of the
Macaulay matrix. These two methods have different strengths and weaknesses, so
it depends on the context whether to apply one or the other.

####  Section 8.1Preliminaries

We begin with some extra $\mathcal{D}$-module formalism. More details and
examples besides what is presented here can be found in [83, Appendix A].

##### 8.1.1 Tensor products

Tensor products of vector spaces are familiar to physicists from the study of
quantum mechanics. It is possible to define tensor products of
$\mathcal{D}$-modules as well. Let $\mathcal{M}$ and $\mathcal{N}$ be two
$\mathcal{D}$-modules. We proceed now to define the tensor product
$\mathcal{N}\otimes_{\mathcal{D}}\mathcal{M}.$

Let $n\otimes m$ denote the tuple $(n,m)$ where $n\in\mathcal{N}$ and
$m\in\mathcal{M}$, and define the space
$S(\mathcal{N}\otimes\mathcal{M})=\big{\\{}\text{finite, formal sums}\
\sum_{i}n_{i}\otimes m_{i}\big{\\}}\,.$ We specify an equivalence relation on
$S$:

  1. 1.

Linearity in the left slot:

$\displaystyle n_{1}\otimes m+n_{2}\otimes m\quad\sim\quad(n_{1}+n_{2})\otimes
m\,.$ (8.4)

  2. 2.

Linearity in the right slot:

$\displaystyle n\otimes m_{1}+n\otimes m_{2}\quad\sim\quad
n\otimes(m_{1}+m_{2})\,.$ (8.5)

  3. 3.

Moving factors "across" the $\otimes$ sign:

$\displaystyle(n\,d)\otimes m\quad\sim\quad n\otimes(d\,m)\,,$ (8.6)

for $d\in\mathcal{D}$. This is the reason for the subscript in
$\otimes_{\mathcal{D}}$.

The $\mathcal{D}$-module tensor product is then defined by the space
$S(\mathcal{N}\otimes\mathcal{M})$ modulo this equivalence relation, that is
$\mathcal{N}\otimes_{\mathcal{D}}\mathcal{M}:=S(\mathcal{N}\otimes\mathcal{M})/\sim.$

##### 8.1.2 Definition of restriction

The variables $z=(z_{1},\ldots,z_{N})$ parametrize an affine space
$Y=\mathbb{A}^{N}$. We write $\mathcal{D}_{Y},\mathcal{R}_{Y}$ etc. for spaces
of operators defined on $Y$. Consider an affine hyperplane $Y^{\prime}\subset
Y$ of lower dimension - this will, later on, represent the proper space of
kinematic variables. By an affine linear change of coordinates, we can define
$Y^{\prime}$ as the vanishing locus

$\displaystyle Y^{\prime}=\\{z\in Y\,|\,z_{1}=\cdots=z_{N^{\prime}}=0\\}\,.$
(8.7)

######  Definition 8.1.

Denote by $\iota:Y^{\prime}\hookrightarrow Y$ the natural inclusion map. Let
$\mathcal{M}$ be a $\mathcal{D}_{Y}$-module. The _restriction module_ of
$\mathcal{M}$ is defined as the $\mathcal{D}_{Y^{\prime}}$-module

$\displaystyle\iota^{*}\mathcal{M}:=\frac{\mathcal{M}}{z_{1}\mathcal{M}+\ldots+z_{N^{\prime}}\mathcal{M}}\,.$
(8.8)

The operation $\iota^{*}$ can be thought of as a pullback from $Y$ to
$Y^{\prime}$. See [85, Chapter 5] [86, Chapter 6] for details.

A typical use case for the restriction module is the following. Suppose the
$\mathcal{D}_{Y}$-module $\mathcal{M}=\mathcal{D}_{Y}/\mathcal{I}$ is
constructed from the annihilating ideal $\mathcal{I}$ of some function $f(z)$
for $z\in Y$. By computing
$\iota^{*}\mathcal{M}=\mathcal{D}_{Y^{\prime}}/\mathcal{I}^{\prime}$ for some
$\mathcal{I}^{\prime}$, then $\mathcal{I}^{\prime}$ is the annihilating ideal
for $f(z)$ where now $z\in Y^{\prime}$. Naturally, it is not necessary that
$\mathcal{M}$ be given by a quotient of operator spaces. As in Example (4.2),
the $\mathcal{D}_{Y}$-module $\mathcal{M}$ can also be equal, as a set, to a
suitable space of functions.

Example 8.1. Consider the GKZ system associated to the Euler integral

$\displaystyle
f_{\beta}(z_{1},z_{2},z_{3})=\int_{(0,\infty)^{2}}(z_{1}x_{1}+z_{2}x_{2}+z_{3}x_{1}x_{2})^{\beta_{0}}x_{1}^{-\beta_{1}}x_{2}^{-\beta_{2}}\frac{\mathrm{d}x}{x}\,.$
(8.9)

The annihilating ideal $\mathcal{I}=H_{A}(\beta)$ is solely generated by the
homogeneity operators

$\displaystyle E_{0}$
$\displaystyle=z_{1}\partial_{1}+z_{2}\partial_{2}+z_{3}\partial_{3}-\beta_{0}$
$\displaystyle E_{1}$
$\displaystyle=z_{1}\partial_{1}+z_{3}\partial_{3}-\beta_{1}$ (8.10)
$\displaystyle E_{2}$
$\displaystyle=z_{2}\partial_{2}+z_{3}\partial_{3}-\beta_{2}\,.$

We are interested in the annihilating ideal for the function
$f_{\beta}(1,1,z_{3})$. By an affine linear coordinate change $\\{z_{1}\to
z_{1}+1\,,z_{2}\to z_{2}+1\,,z_{3}\to z_{3}\\},$ we hence have the affine
subspace given by $Y^{\prime}=\\{z_{1}=z_{2}=0\\}.$

A variant of the restriction algorithm presented in Oaku’s seminal work [148]
has been implemented in the asir package nk_restriction [149]. Inputting the
data above, one finds the annihilating ideal

$\displaystyle\mathcal{I}^{\prime}=\langle-
z_{3}\partial_{3}-\beta_{0}+\beta_{1}+\beta_{3}\rangle\,.$ (8.11)

One can check that this indeed annihilates the function

$\displaystyle
f_{\beta}(1,1,z_{3})=\frac{\Gamma(-\beta_{0}+\beta_{1})\Gamma(-\beta_{0}+\beta_{2})\Gamma(\beta_{0}-\beta_{1}-\beta_{2})}{\Gamma(-\beta_{0})}z_{3}^{-\beta_{0}+\beta_{1}+\beta_{2}}\,.$
(8.12)

$\blacksquare$

The discussion above was directed at $\mathcal{D}$-modules, i.e. when the
action is given by operators with polynomial coefficient functions. We are,
more broadly, interested in $\mathcal{R}$-modules having rational function
coefficients, because this is what appears in Pfaffian systems. An
$\mathcal{R}$-module naturally arises from a given $\mathcal{D}$-module
$\mathcal{M}$ by taking a tensor product:
$\mathcal{R}\otimes_{\mathcal{D}_{Y}}\mathcal{M}.$

######  Definition 8.2.

The _rational restriction_ of a $\mathcal{D}_{Y}$-module $\mathcal{M}$ onto
$Y^{\prime}$ is defined by

$\displaystyle\mathcal{R}_{Y^{\prime}}\otimes_{\mathcal{D}_{Y^{\prime}}}\iota^{*}\mathcal{M}\,,$
(8.13)

where $\iota^{*}\mathcal{M}$ is the restriction module (8.8).

The restriction algorithms presented in this chapter will output Pfaffian
systems that live in the space (8.13).

##### 8.1.3 Normal form

Let $\partial_{i}\bullet e(z)=P_{i}\cdot e(z)$ be a Pfaffian system that is
_regular singular_ (cf. Section (4.2.1)). Loosely speaking, this means that
the solutions vector $e(z)$ is free of essential singularities such
$\exp(1/z)$ at $z=0$. It is expected that FIs fall into this category, as the
poles in $\epsilon$-factorized Pfaffian systems always appear to be simple.

We focus on a single pole of the system, which can be written as
$Y^{\prime}=\\{z_{1}=0\\}$ by a change of coordinates. Theorems by Deligne
[150, 151] ensure that the Pfaffian system can be brought into _normal form_
w.r.t. $Y^{\prime}$. Namely, there exists another set of Pfaffian matrices
$\widetilde{P}_{i}(z)$ such that

  1. 1.

$\widetilde{P}_{1}$ has a simple pole on $Y^{\prime}$ and
$\\{\widetilde{P}_{2},\ldots,\widetilde{P}_{N}\\}$ are finite on $Y^{\prime}$:

$\displaystyle\widetilde{P}_{1}(z)$
$\displaystyle=\sum_{n=-1}^{\infty}\widetilde{P}_{1,n}(z^{\prime})z_{1}^{n}\,,\quad\text{and}$
(8.14) $\displaystyle\widetilde{P}_{i}(z)$
$\displaystyle=\sum_{n=0}^{\infty}\widetilde{P}_{i,n}(z^{\prime})z_{1}^{n}\,,\hskip
8.5359pt\quad\text{for}\quad i=2,\ldots,N\,,$ (8.15)

where the truncated list of variables is

$\displaystyle z^{\prime}=(z_{2},\ldots,z_{N})\,.$ (8.16)

Note that the expansion of $\widetilde{P}_{1}$ starts at $n=-1$, and the
expansions for $\widetilde{P}_{i\,=\,2,\ldots,N}$ start at $n=0$.

  2. 2.

The spectrum of the _residue matrix_ $\widetilde{P}_{1,-1}(z^{\prime})$ is
_non-resonant_ , meaning that 1) no two distinct eigenvalues of
$\widetilde{P}_{1,-1}(z^{\prime})$ have an integral difference, and 2) the
only integral eigenvalue, if it exists, is $0$.

The matrices $\widetilde{P}_{i}(z)$ are related to the original ones
$P_{i}(z)$ via a gauge transformation. By the work of Moser [152], and later
Barkatou et al. [153, 154], there exist explicit _Moser reduction_ algorithms
for constructing such a gauge transformation (see [83, Appendix B] for a
review). Moser reduction has been implemented in software such as Fuchsia
[155] and Isolde [156].

It can be computationally expensive to enforce a non-resonant spectrum on a
Pfaffian system, as this step of the algorithm requires the calculation of
Jordan decompositions111 In particular, one is required to know the sizes of
the Jordan cells [83, Appendix B.2]. . However, if we only seek to perform
Moser reduction w.r.t. a single pole $z_{1}=0$ rather than many, then the
algorithm can readily be applied to rather large systems. Moreover, since the
Jordan decomposed matrix does not depend on the $z$-variables [157, Theorem
12.1], it is possible speed up the computation by fixing the variables to
generic numbers.

####  Section 8.2Restriction at the level of Pfaffian systems

In this section, we present a restriction algorithm which takes a Pfaffian
system on $Y=\mathbb{A}^{N}$ as input and outputs a new Pfaffian system that
holds true on an affine subspace $Y^{\prime}\subset Y$. As has been mentioned,
the original motivation was to calculate restrictions of GKZ systems. However,
the algorithm turns out to work for any regular holonomic
$\mathcal{D}$-module. So it can, rather serendipitously, also be applied to
Pfaffian systems for FIs obtained by conventional means, such as IBPs in
momentum space.

As it will turn out, the algorithm can also be used to compute logarithmic
series solutions to Pfaffian systems. This is helpful for calculating
asymptotic expansions for FIs, such as threshold and small-mass expansions.

##### 8.2.1 Restriction of a logarithmic connection

Suppose we are given a Pfaffian system on the "big" space $Y$. We want to
restrict it onto a "smaller" space $Y^{\prime}=\\{z_{1}=0\\}$. Further,
$z_{1}=0$ should constitute a pole in the Pfaffian system. There might poles
elsewhere on $Y$, but we imagine working "locally" around this single pole222
The notion of ”local” is here meant w.r.t. the Zariski topology. So to be
rigorous, we ought to formulate everything in terms of Zariski open subsets
$Y_{0}$ of $Y$ as in [83, Section 3]. Many such mathematical details are
neglected here in order to take the shortest path towards defining the
practical algorithms. . Assume that the Pfaffian system is in normal form
w.r.t. $Y^{\prime}$ (cf. Section (8.1.3)). Then it is possible to write it in
a peculiar way:

$\displaystyle z_{1}\partial_{1}\bullet e(z)$ $\displaystyle=P_{1}\cdot e(z)$
(8.17) $\displaystyle\partial_{i}\bullet e(z)$ $\displaystyle=P_{i}\cdot
e(z)\,,\quad i=2,\ldots,N\,,$ (8.18)

where $e(z)$ is an $R$-dimensional vector of functions. We would like to use
this Pfaffian system to construct a certain $\mathcal{D}$-module $\mathcal{M}$
consisting of _regular_ functions. The action of the derivative $\partial_{i}$
should be well-defined on such a space, in the sense of not creating unwanted
singularities. We have therefore guarded ourselves in two ways to ensure well-
definedness: 1) by working locally around $Y^{\prime}$ we can thereby ignore
singularities elsewhere, and 2) by moving $1/z_{1}$ from the RHS of (8.17) to
get $z_{1}\partial_{1}$ on the LHS. We are thence motivated to define a
subring of $\mathcal{D}_{Y}$ by

$\displaystyle\mathcal{D}_{Y}(\log Y^{\prime})=\mathbb{C}\langle
z_{1},\ldots,z_{N},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}z_{1}\partial_{1}},\partial_{2},\ldots,\partial_{N}\rangle\,,$
(8.19)

and the abelian group

$\displaystyle\mathcal{M}=\big{\\{}R\text{-dimensional vectors of functions
on}\ Y\ \text{that are regular near}\ Y^{\prime}\big{\\}}\,.$ (8.20)

The action of the ring $\mathcal{D}_{Y}(\log Y^{\prime})$ onto $\mathcal{M}$
is well-defined. The generators $z_{i}$ act by scalar multiplication, and the
generators $\\{z_{1}\partial_{1},\partial_{2},\ldots,\partial_{N}\\}$ act
according to the RHSs of the Pfaffian system (8.17)-(8.17). This turns
$\mathcal{M}$ into a $\mathcal{D}_{Y}(\log Y^{\prime})$-module. In technical
jargon, such an $\mathcal{M}$ is called a _logarithmic connection_ [110,
Section 11.4] [23, Section 5.2.2]. We now proceed to study the rational
restriction
$\mathcal{R}_{Y^{\prime}}\otimes_{\mathcal{D}_{Y^{\prime}}}\iota^{*}\mathcal{M}$
of the logarithmic connection $\mathcal{M}$.

To start, let $P_{1,-1}(z^{\prime})$ be the residue matrix (cf. (8.14))
associated to $P_{1}(z)$ defined in (8.17), where
$z^{\prime}=(z_{2},\ldots,z_{N})$. The entries of $P_{1,-1}(z^{\prime})$ live
in the field $\mathcal{K}_{Y^{\prime}}$ consisting of rational functions on
$Y^{\prime}$. The $R$ rows of $P_{1,-1}(z^{\prime})$ generate a subspace of
$\mathcal{K}_{Y^{\prime}}^{R}$ which we call $\mathcal{N}^{\prime}$. The
following quotient space is important for our construction of the rational
restriction:

$\displaystyle\mathcal{M}^{\prime}=\mathcal{K}_{Y^{\prime}}^{R}\,/\,\mathcal{N}^{\prime}\,.$
(8.21)

As a set, this consists of $R$-dimensional vectors of rational functions on
$Y^{\prime}$, and we identify two vectors if they differ by another vector
lying in the row space of $P_{1,-1}(z^{\prime})$. $\mathcal{M}^{\prime}$ can
actually be turned into an $\mathcal{R}_{Y^{\prime}}$-module. The action of
the derivatives $\\{\partial_{2},\ldots,\partial_{N}\\}$ on an $R$-dimensional
equivalence class $[m]\in\mathcal{M}^{\prime}$ is

$\displaystyle\partial_{i}\bullet[m]=P_{i,0}(z^{\prime})\cdot[m]\quad\text{for}\quad
i=2,\ldots,N\,,$ (8.22)

where the matrices $P_{i,0}(z^{\prime})$ come from the leading terms in the
$z_{1}$-expansion (8.15).

With all of this build-up, we can finally state the following crucial
isomorphism of $\mathcal{R}_{Y^{\prime}}$-modules333 The formula (8.23) shows
the rational restriction of $\mathcal{M}$ when it equals a space of functions
as a set. Recalling the discussion from Section (4.4), this isomorphism can
also be employed when $\mathcal{M}$ is of the form
$\mathcal{D}_{Y}/\mathcal{I}$ for some annihilating ideal of differential
operators $\mathcal{I}$. :

$\displaystyle\mathcal{R}_{Y^{\prime}}\otimes_{\mathcal{D}_{Y^{\prime}}}\iota^{*}\mathcal{M}\
\simeq\ \mathcal{M}^{\prime}\,.$ (8.23)

We refer to [83, Proposition 3.4] for many more mathematical details as well
as a proof.

The takeaway message up until now is this. We are given some Pfaffian matrices
$\\{P_{1}(z),\ldots,P_{N}(z)\\}$ on $Y$ which define $\mathcal{M}$. We would
like to know the Pfaffian matrices
$\\{Q_{2}(z^{\prime}),\ldots,Q_{N}(z^{\prime})\\}$ on the smaller space
$Y^{\prime}$. The matrices $Q_{i}(z^{\prime})$ are associated to the rational
restriction module of $\mathcal{M}$. Because of the isomorphism (8.23), the
matrices $Q_{i}(z^{\prime})$ are equally well associated to the quotient space
$\mathcal{M}^{\prime}$. Fortunately, we have a lot of information about
$\mathcal{M}^{\prime}$: it is completely specified by the residue matrix
$P_{1,-1}(z^{\prime})$! In the following subsection, we use this information
to provide an explicit formula for the Pfaffian matrices $Q_{i}(z^{\prime})$.

##### 8.2.2 Constructing the restricted Pfaffian system

Let $R^{\prime}$ denote the holonomic rank of $\mathcal{M}^{\prime}$. In
general $R^{\prime}<R$, i.e. the holonomic rank drops when we fix
$z$-variables to special values. Given an element
$m\in\mathcal{K}^{R}_{Y^{\prime}}$, we write $[m]$ for its equivalence class
in $\mathcal{M}^{\prime}$.

We define three sets of Pfaffian systems:

  1. 1.

We suppose the existence of a _special_ basis
$f=f(z^{\prime})=\\{f_{1},\ldots,f_{R}\\}$ obeying the Pfaffian system

$\displaystyle\partial_{i}\bullet f=\widetilde{Q}_{i}(z^{\prime})\cdot
f\quad\text{for}\quad i=2,\ldots,N\,.$ (8.24)

We do not yet know $f$ nor $\widetilde{Q}_{i}(z^{\prime})$.

  2. 2.

The basis $f$ is special for the following reason: we assume that
$\\{f_{R^{\prime}+1},\ldots,f_{R}\\}$ is a basis of $\mathcal{N}^{\prime}$.
The set $[f]=\\{[f_{1}],\ldots,[f_{R^{\prime}}]\\}$ is then a basis for
$\mathcal{M}^{\prime}$, with an associated Pfaffian system

$\displaystyle\partial_{i}\bullet[f]=Q_{i}(z^{\prime})\cdot[f]\quad\text{for}\quad
i=2,\ldots,N\,.$ (8.25)

By stipulation, the first $R^{\prime}\times R^{\prime}$ block of
$\widetilde{Q}_{i}(z^{\prime})$ equals $Q_{i}(z^{\prime})$. We do not yet know
$[f]$ nor $Q_{i}(z^{\prime})$.

  3. 3.

Let $e=e(z^{\prime})=\\{e_{1},\ldots,e_{R}\\}$ be the basis associated to the
Pfaffian system

$\displaystyle\partial_{i}\bullet e=P_{i,0}(z^{\prime})\cdot
e\quad\text{for}\quad i=2,\ldots,N\,.$ (8.26)

The matrices $P_{i,0}(z^{\prime})$ _are_ assumed to be known via the formula
(8.15). In contrast to $\widetilde{Q}_{i}(z^{\prime})$, the matrices
$P_{i,0}(z^{\prime})$ have no nice block structure.

We are hence in the following situation: we know $P_{i,0}(z^{\prime})$ (for
the basis $e$), but we do not know $\widetilde{Q}_{i}(z^{\prime})$ (for the
special basis $f$). We would like to know $\widetilde{Q}_{i}(z^{\prime})$,
because then we would automatically get the Pfaffian matrices
$Q_{i}(z^{\prime})$ for the rational restriction module
$\mathcal{M}^{\prime}$. We shall achieve this goal by building a gauge
transformation mapping $e$ to the special basis $f$. This is done by through
three new matrices $\\{\mathsf{B},\mathsf{R},\mathsf{M}\\}$, which we define
now.

  * •

Recall the basis $[f]=\\{[f_{1}],\ldots,[f_{R^{\prime}}]\\}$ of
$\mathcal{M}^{\prime}$. Each representative of these equivalence classes can
be expanded in the basis $e$ via some $R^{\prime}\times R$ matrix $\mathsf{B}$
having entries in $\mathcal{K}_{Y^{\prime}}$:

$\displaystyle f_{i}=\sum_{j=1}^{R}\mathsf{B}_{ij}\,e_{j}\quad\text{for}\quad
i=1,\ldots,R^{\prime}\,.$ (8.27)

The entries of $\mathsf{B}$ are not known at this point.

  * •

By the definition of $\mathcal{M}^{\prime}$, the residue matrix
$P_{1,-1}(z^{\prime})$ has exactly $R-R^{\prime}$ independent rows. Define an
$(R-R^{\prime})\times R$ matrix by

$\displaystyle\mathsf{R}=\texttt{RowReduce}\\!\left[P_{1,-1}(z^{\prime})\right]\,,$
(8.28)

where RowReduce includes the deletion of zero-rows. Then

$\displaystyle f_{i}=\sum_{j=1}^{R}\mathsf{R}_{ij}\,e_{j}\quad\text{for}\quad
i=R^{\prime}+1,\ldots,R$ (8.29)

forms a basis of $\mathcal{N}^{\prime}$.

  * •

Aligning the row of $\mathsf{B}$ and $\mathsf{R}$ produces an invertible
$R\times R$ matrix $\mathsf{M}$:

$\displaystyle\mathsf{M}=\left[\begin{array}[]{c}{\color[rgb]{0.57421875,0.09375,0.09375}\definecolor[named]{pgfstrokecolor}{rgb}{0.57421875,0.09375,0.09375}\mathsf{B}}\\\
\hline\cr{\color[rgb]{0.32421875,0.015625,0.4765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.32421875,0.015625,0.4765625}\mathsf{R}}\end{array}\right]=\vbox{\hbox{
\includegraphics[scale={1}]{Figs/M_mat.pdf} }}\ .$ (8.32)

Note that $\mathsf{R}$ is known by stipulation, and the matrix $\mathsf{B}$
can be anything as long as $\mathsf{M}$ becomes invertible. We shall give a
simple recipe for generating $\mathsf{B}$ later on.

In vector notation, we now have that $f=\mathsf{M}\cdot e$. This means that
the unknown matrices $\widetilde{Q}_{i}(z^{\prime})$ (associated to $f$) are
obtained from the known matrices $P_{i,0}(z^{\prime})$ (associated to $e$) by
gauge transforming with $\mathsf{M}^{-1}$. Because the sought after matrix
$Q_{i}(z^{\prime})$ is given by the first $R^{\prime}\times R^{\prime}$ block
of $\widetilde{Q}_{i}(z^{\prime})$, we can finally write the key formula

$\displaystyle\widetilde{Q}_{i}(z^{\prime})\quad=\quad\big{[}\partial_{i}\bullet\mathsf{M}\hskip
5.69046pt+\hskip 5.69046pt\mathsf{M}\cdot
P_{i,0}(z^{\prime})\big{]}\cdot\mathsf{M}^{-1}\quad=\quad\left[\begin{array}[]{c|c}Q_{i}(z^{\prime})&\star\\\
\hline\cr\mathbf{0}&\star\end{array}\right]\,.$ (8.35)

See [83, Appendix C.2.1] for an explanation of the zero block in the lower
left corner. Obtaining this block is a useful check for practical
computations.

Before closing this section, let us observe that the integer $N$ does not have
to be the "$N$" of the GKZ system. In fact, we did not use any notions related
to the GKZ system to derive (8.35), so the preceding discussion holds true for
any (regular singular) Pfaffian system!

##### 8.2.3 Holomorphic restriction of solutions

The previous section focused on the matrices associated to Pfaffian systems.
In the following two subsections, we draw our attention to the _solution
vectors_ of Pfaffian systems, in particular how to obtain actual series
representations using restriction. This topic can be viewed as singular
boundary value problem (see [157] [158] and references therein).

To emphasize the connection to MIs, let us use the notation $\vec{I}(z)$ for
the solution vector of a Pfaffian system

$\displaystyle\partial_{i}\bullet\vec{I}(z)=P_{i}(z)\cdot\vec{I}(z)\,,\quad
i=1,\ldots,N\,.$ (8.36)

Suppose that we are interested in solutions that are _holomorphic_ on
$Y^{\prime}=\\{z_{1}=0\\}$. This means that $\vec{I}(z)$ enjoys a Taylor
expansion

$\displaystyle\vec{I}(z)=\sum_{n=0}^{\infty}\vec{I}\,^{(n)}(z^{\prime})\,z_{1}^{n}\,,$
(8.37)

where as usual $z^{\prime}=(z_{2},\ldots,z_{N})$. The issue at hand is how
compute the expansion coefficient functions $\vec{I}\,^{(n)}$ up to some
appropriate order $\mathcal{O}(z_{1}^{k}).$

Assume that the matrices $P_{i}(z)$ are in normal form. Insert the
$z_{1}$-expansions of $P_{i}(z)$ and $\vec{I}(z)$, given by (8.14)-(8.15) and
(8.37), into the Pfaffian system (8.36). Relations among the $\vec{I}\,^{(n)}$
are then obtained by comparing powers of $z_{1}^{n}$ on both sides of the
equality [157, Chapter 4] [83, Appendix D.6]. Comparing the leading-order
terms in $z_{1}$, the following relations are found:

$\displaystyle\begin{array}[]{rl}&\partial_{i}\bullet\vec{I}\,^{(0)}(z^{\prime})\
=\ P_{i,0}(z^{\prime})\cdot\vec{I}\,^{(0)}(z^{\prime})\,,\quad
i=2,\ldots,N\\\\[10.0pt] &\text{Rank jump constraint:}\quad
P_{1,-1}(z^{\prime})\cdot\vec{I}\,^{(0)}(z^{\prime})\ =\ 0\,.\end{array}$
(8.40)

The higher-order terms in $z_{1}$ yield the recursion relation444 The matrix
in front of the sum is indeed invertible because the spectrum of
$P_{1,-1}(z^{\prime})$ is non-resonant. More precisely, the only potential
integral eigenvalue of $P_{1,-1}(z^{\prime})$ is $0$, and the diagonal is
shifted by some $n>0$, wherefore the determinant of the full expression is
non-zero.

$\displaystyle\vec{I}\,^{(n)}(z^{\prime})=\big{[}n\mathbf{1}-P_{1,-1}(z^{\prime})\big{]}^{-1}\
\cdot\ \sum_{i=0}^{n-1}P_{1,n-i-1}\cdot\vec{I}\,^{(i)}(z^{\prime})\,,\quad
n>0\,.$ (8.41)

The set of PDEs (8.40) is dubbed the _holomorphic restriction_ of the Pfaffian
system (8.36). The first equation of (8.40) shows that $\vec{I}\,^{(0)}$ plays
the role of the basis $e$ in (8.26). The second equation of (8.40) shows why
the residue matrix $P_{1,-1}(z^{\prime})$ played such a central role in the
construction of Section (8.2.1), wherein we modded out by the rows of
$P_{1,-1}(z^{\prime})$ to obtain the rational restriction module.

At the level of solutions for Pfaffian systems, we now have a clear
interpretation for the _rank jump_ from $R$ to $R^{\prime}$: the equation
$P_{1,-1}(z^{\prime})\cdot\vec{I}\,^{(0)}(z^{\prime})=0$ imposes constraints
among the entries of $\vec{I}\,^{(0)}(z^{\prime})$ in the limit $z_{1}\to 0$,
meaning that some of them become linearly dependent on each other. For FIs,
these relations turn out to be the new IBP relations among MIs which arise
when the limit $z_{1}\to 0$ is taken at the _integrand_ level. An example of
this phenomenon is shown in Section (9.2).

Observe that once $\vec{I}\,^{(0)}$ is determined from the PDEs (8.40), then
(8.41) fixes all higher-order terms $\vec{I}\,^{(n)}$ via simple matrix
multiplication (one can even use finite field reconstruction in this step for