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
A preliminary version of 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 [comments are in UTF-8 and Japanese].
This program will become a package of asir-contrib project in a near future.
A manual is not ready for now, but it contains sample inputs in the file check.rr.
Movie of running demo1.rr [2 min].
Example 4
$$A =
\left(
\begin{array}{cc|cc|cc}
1& 1& 0& 0& 0& 0 \\ \hline
0& 0& 1& 1& 0& 0 \\ \hline
0& 0& 0& 0& 1& 1 \\ \hline
1& 0& 0& 1& 0& 0 \\
0& 0& 1& 0& 0& 1 \\
\end{array}
\right)
$$
- 3F2-ci.rr , the intersection matrix
of twisted cohomology group.
The format is for Risa/Asir. $b_i$ is $\beta_i$ in the paper.
A human readble form of the matrix is
$$
\left[\begin{array}{ccc}
r_{11}& \frac{ {b}_{4}+ {b}_{5}} { {b}_{5} {b}_{4} ( {b}_{2}- {b}_{4}- {b}_{5})}& \frac{ {b}_{4} ( {b}_{1}+ {b}_{2}- {b}_{4} - {b}_{4}) {z}_{1}- {b}_{5} {b}_{3}} { {b}_{5} ( {b}_{4}+ 1) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}+ 1)} \\
\frac{ {b}_{4}+ {b}_{5}} { {b}_{5} {b}_{4} ( {b}_{2}- {b}_{4}- {b}_{5})}& r_{22} & \frac{ - ( {b}_{4} {b}_{1} {z}_{1}- {b}_{5} {b}_{2}- {b}_{5} {b}_{3}+ {b}_{5} {b}_{4}+ {b}_{5}^{ 2} )} { {b}_{5} ( {b}_{4}+ 1) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}+ 1)} \\
\frac{ {b}_{4} ({b}_{1}+ {b}_{2}- {b}_{4} - {b}_{5}) {z}_{1}- {b}_{5} {b}_{3}} { {b}_{5} ( {b}_{4}- 1) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}- 1)}& \frac{ - {b}_{4} {b}_{1} {z}_{1}+ {b}_{5} ( {b}_{2}+ {b}_{3}- {b}_{4}- {b}_{5})} { {b}_{5} ( {b}_{4}- 1) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}- 1)}& r_{33} \\
\end{array}\right]
$$
where
$$
r_{11} = -\frac{ ( {b}_{4} {b}_{2}+ ( {b}_{4}+ {b}_{5}) {b}_{3}) {b}_{1}+ {b}_{4} {b}_{2}^{ 2} + ( {b}_{4} {b}_{3}- {b}_{4}^{ 2} - {b}_{5} {b}_{4}) {b}_{2}+ ( - {b}_{4}^{ 2} - {b}_{5} {b}_{4}) {b}_{3}} { {b}_{5} {b}_{4} {b}_{1}( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}+ {b}_{3}- {b}_{5}) }
$$
$$
r_{22}=
- \frac{ ( {b}_{5} {b}_{2}+ ( {b}_{4}+ {b}_{5}) {b}_{3}- {b}_{5} {b}_{4}- {b}_{5}^{ 2} ) {b}_{1}+ {b}_{5} {b}_{2}^{ 2} + ( {b}_{5} {b}_{3}- {b}_{5} {b}_{4}- {b}_{5}^{ 2} ) {b}_{2}} { {b}_{5} {b}_{4} {b}_{3} ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{1}+ {b}_{2}- {b}_{4})}
$$
$$
r_{33}=
-\frac{b_4\left\{ ({b_1}b_2-{b_1}b_5+{b_2}^{2}-b_2{b_4}-2b_2b_5+{b_4}b_5+{b_5}^{2})b_1b_4{z_1}^{2}
-2b_1b_3b_4b_5z_1
+({b_2}^{2}+b_2{b_3}-2b_2b_4-b_2{b_5}-{b_3}b_4+{b_4}^{2}+b_4b_5)b_3b_5 \right\}
}
{ {b}_{5} {b}_{2} ( {b}_{4}- 1) ( {b}_{4}+ 1) ( {b}_{2}- {b}_{4}- {b}_{5}) ( {b}_{2}- {b}_{4}- {b}_{5}- 1) ( {b}_{2}- {b}_{4}- {b}_{5}+ 1)}
$$
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-pf.rr , coefficient matrices $P_2$ and $P_3$
of Example 5.
It was obtained by
test11_b()
in check.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-num.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-param.ml , Maple input of Example 5
with parameters $b_1, b_2, b_3$.
We could not find a rational solution
on a machine with Intel Xeon CPU E5-4650 2.70GHz and 256GB memory.
It is shown in our paper that the solution is a rational function in $b_i$'s.
It will be a future challenge to solve this problem, e.g., by a rational reconstruction
method. Note that when we specialize parameters to numbers, IntegrableConnections
finds a solution in a few seconds.
Last update: March 27, 2020