Rational function solutions and intersection numbers
Paper
Saiei-Jaeyeong Matsubara-Heo and Nobuki Takayama,
Algorithms for Pfaffian systems and cohomology intersection numbers
of hypergeometric integrals
, [March 27, 2020].
Programs
Errata [2020.11.25]: the program saito-b.rr before 2020.11.13
contained a bug and it was fixed on 2020.11.13.
Step-up operators of the buggy version is constant multiples of correct
step-up operators.
Then, all connection matrices of examples given in the paper and in our webpage
need corrections by the Gauge transformations by constant matrices.
This page is being updated and the old buggy page is here.
Errata [2020.11.17] of the paper.
Our
Risa/Asir program for Algorithm 1
[a constructor of Pfaffian equation], for constructing secondary equations,
and generating a Maple input for
IntegrableConnections by
M.Barkatou, T.Cluzeau, C.El.Bacha, J.-A.Weil.,
is
here [Our package for Risa/Asir, comments are in UTF-8 and Japanese].
A manual is here
This program is also included in the asir-contrib project.
You can download the latest package by
asir_contrib_update(|update=1);
in the Risa/Asir.
Example 4
$$A =
\left(
\begin{array}{cc|cc|cc}
1& 1& 0& 0& 0& 0 \\ \hline
0& 0& 1& 1& 0& 0 \\ \hline
0& 0& 0& 0& 1& 1 \\ \hline
1& 0& 0& 1& 0& 0 \\
0& 0& 1& 0& 0& 1 \\
\end{array}
\right)
$$
- The connection matrix $P$ with respect to $x_1$
[$P$ of the Pfaffian system $\frac{\partial}{\partial x_1} - P$]
for $x_2=-1, x_3 = \cdots = x_6=1$
can be obtained by
test13（）
in mt_gkz/check-by-hgpoly.rr.
$P$ is
$$
\left[
\begin{array}{ccc}
\frac{ {b}_{4} {x}_{1}+ {b}_{2}+ {b}_{3} - {b}_{4}- {b}_{5}}{ {x}_{1}^{ 2} - {x}_{1}}& \frac{ {b}_{3} {b}_{1}+ {b}_{3} {b}_{2}- {b}_{4} {b}_{3}}{ {b}_{1} {x}_{1}^{ 2} - {b}_{1} {x}_{1}}& \frac{ {b}_{2}^{ 2} + ( - {b}_{4} - {b}_{5}- 1) {b}_{2}}{ {b}_{1} {x}_{1}^{ 2} - {b}_{1} {x}_{1}} \\
\frac{ ( {b}_{2}+ {b}_{3}- {b}_{5}) {b}_{1}}{ {b}_{3} {x}_{1}- {b}_{3}}& \frac{ {b}_{1} {x}_{1}+ {b}_{2}- {b}_{4}}{ {x}_{1}^{ 2} - {x}_{1}}& \frac{ {b}_{2}^{ 2} + ( - {b}_{4} - {b}_{5}- 1) {b}_{2}}{ {b}_{3} {x}_{1}^{ 2} - {b}_{3} {x}_{1}} \\
\frac{ ( - {b}_{2} - {b}_{3}+ {b}_{5}) {b}_{1}}{ {b}_{2} {x}_{1}- {b}_{2}}& \frac{ - {b}_{3} {b}_{1} - {b}_{3} {b}_{2}+ {b}_{4} {b}_{3}}{ {b}_{2} {x}_{1}- {b}_{2}}& \frac{ - {b}_{2}+ {b}_{4}+ {b}_{5}+ 1}{ {x}_{1}- 1} \\
\end{array}
\right]
$$
- 3F2-ci2.rr , the intersection matrix
of twisted cohomology group, which is obtained by solving the secondary
equation by 3F2-ci2.ml.
The file tmp-test2.ml is generated by genfile3f2.rr and renamed to 3F2-ci2.ml.
The format of 3F2-ci2.rr is for Risa/Asir. $b_i$ is $\beta_i$ in the paper.
A human readble form of the matrix is
$$
\left[\begin{array}{ccc}
r_{11}& \frac{ ( {b}_{4}+ {b}_{5})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5})}& \frac{ ( ( {b}_{4} {b}_{1}+ {b}_{4} {b}_{2}- {b}_{4}^{ 2} - {b}_{5} {b}_{4}) {z}_{1}- {b}_{5} {b}_{3})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}+ 1)} \\
\frac{ ( {b}_{4}+ {b}_{5})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5})}& r_{22}& \frac{ ( - 1) ( {b}_{4} {b}_{1} {z}_{1}- {b}_{5} {b}_{2}- {b}_{5} {b}_{3}+ {b}_{5} {b}_{4}+ {b}_{5}^{ 2} )} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}+ 1)} \\
\frac{ ( ( {b}_{4} {b}_{1}+ {b}_{4} {b}_{2}- {b}_{4}^{ 2} - {b}_{5} {b}_{4}) {z}_{1}- {b}_{5} {b}_{3})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}- 1)}& \frac{ ( - 1) ( {b}_{4} {b}_{1} {z}_{1}- {b}_{5} {b}_{2}- {b}_{5} {b}_{3}+ {b}_{5} {b}_{4}+ {b}_{5}^{ 2} )} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}- 1)}& r_{33} \\
\end{array}\right]
$$
where
$$
r_{11} =
\frac{ ( - 1) ( ( {b}_{4} {b}_{2}+ ( {b}_{4}+ {b}_{5}) {b}_{3}) {b}_{1}+ {b}_{4} {b}_{2}^{ 2} + ( {b}_{4} {b}_{3}- {b}_{4}^{ 2} - {b}_{5} {b}_{4}) {b}_{2}+ ( - {b}_{4}^{ 2} - {b}_{5} {b}_{4}) {b}_{3})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}+ {b}_{3}- {b}_{5}) ( {b}_{1})}
$$
$$
r_{22}=
\frac{ ( - 1) ( ( {b}_{5} {b}_{2}+ ( {b}_{4}+ {b}_{5}) {b}_{3}- {b}_{5} {b}_{4}- {b}_{5}^{ 2} ) {b}_{1}+ {b}_{5} {b}_{2}^{ 2} + ( {b}_{5} {b}_{3}- {b}_{5} {b}_{4}- {b}_{5}^{ 2} ) {b}_{2})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{3}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{1}+ {b}_{2}- {b}_{4})}
$$
$$
r_{33}=
\frac{ ( - 1) ( ( ( {b}_{4} {b}_{2}- {b}_{5} {b}_{4}) {b}_{1}^{ 2} + ( {b}_{4} {b}_{2}^{ 2} + ( - {b}_{4}^{ 2} - 2 {b}_{5} {b}_{4}) {b}_{2}+ {b}_{5} {b}_{4}^{ 2} + {b}_{5}^{ 2} {b}_{4}) {b}_{1}) {z}_{1}^{ 2} - 2 {b}_{5} {b}_{4} {b}_{3} {b}_{1} {z}_{1}+ {b}_{5} {b}_{3} {b}_{2}^{ 2} + ( {b}_{5} {b}_{3}^{ 2} + ( - 2 {b}_{5} {b}_{4}- {b}_{5}^{ 2} ) {b}_{3}) {b}_{2}- {b}_{5} {b}_{4} {b}_{3}^{ 2} + ( {b}_{5} {b}_{4}^{ 2} + {b}_{5}^{ 2} {b}_{4}) {b}_{3})} { ( {b}_{5}) ( {b}_{4}) ( {b}_{2}) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}- 1) ( {b}_{2}- {b}_{4}- {b}_{5}+ 1)}
$$
Example 5
$$
A = \left(\begin{array}{ccccc}
1& 1& 1& 1& 1 \\ \hline
0& 1& 0& 2& 0 \\
0& 0& 1& 0& 2 \\
\end{array}\right)
$$
- LLC-pf2.rr , coefficient matrices $P_2$ and $P_3$
of Example 5.
It was obtained by
test_LLC()
in check-by-hgpoly.rr.
The format is for Risa/Asir, but any computer algebra system
can parse it with small modifications.
Note that x2 and x3 are z2 and z3 of the paper respectively.
- LLC-num2.ml , Maple input of Example 5 for
IntegrableConnections by
M.Barkatou, T.Cluzeau, C.El.Bacha, J.-A.Weil.
It finds a rational solution of the secondary equation for $P_2$ and $P_3$
when $b_1=\frac{1}{2}, b_2 = \frac{1}{3}, b_3=\frac{1}{5}$
in a few seconds.
The solution is a constant multiple of the intersection matrix.
- LLC-param2.ml , Maple input of Example 5
with parameters $b_1, b_2, b_3$.
We can find a rational solution on a machine with Intel Xeon CPU E5-4650 2.70GHz and 256GB memory in about 30 seconds.
The cohomology intersection matrix is
CIM-LLC2.rr .
These Maple inputs are generated by
genfileLLC2.rr .
Last update: Dec 10, 2020