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.

Example 4

$$A = \left( \begin{array}{cc|cc|cc} 1& 1& 0& 0& 0& 0 \\ \hline 0& 0& 1& 1& 0& 0 \\ \hline 0& 0& 0& 0& 1& 1 \\ \hline 1& 0& 0& 1& 0& 0 \\ 0& 0& 1& 0& 0& 1 \\ \end{array} \right) $$
  1. 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

$$ A = \left(\begin{array}{ccccc} 1& 1& 1& 1& 1 \\ \hline 0& 1& 0& 2& 0 \\ 0& 0& 1& 0& 2 \\ \end{array}\right) $$
  1. 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.
  2. 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.
  3. 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