Horn system restricted to hypersurface singularity

References

  1. Vsevolod Chestnov, Saiei J. Matsubara-Heo, Henrik J. Munch, Nobuki Takayama, Restrictions of Pfaffian Systems for Feynman Integrals, doi:10.1007/JHEP11(2023)202
  2. 2023-03-28-mt_mm-for-web.pdf
  3. 2023-08-23-rest-for-web.pdf
  4. mt_mm package manual

Data format

Pfaffian systems for Horn's functions are restricted to hypersurface singularities. The ODE on the hypersurface singularity $f(x,y)=0$ is given by two square matrices
$P_1$ and $P_2$
whose elements lie in the quotient ring $K[x,y]/f$. If we parametrize the singularity $f(x,y)=0$ as
$x=x(t), y=y(t)$
then the ODE is given as
$\frac{dF}{dt}= \left( P_1 x'(t) +P_2 y'(t) \right) F $
where $F$ is a column vector valued function.

The following data are given for some special parameters. For other parameters, please use the programs (まだ bug あり?) 2024-01-06-Hn.rr and 2024-01-06-rest-hs.rr .

Ref: Appell $F_4$ data

$F_4$ data

(以下まだ bug あり?)

$H_{1}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}+ {e}- 1)- ( {x} \partial_{{x}}- {y} \partial_{{y}}+ {a}) ( {x} \partial_{{x}}+ {y} \partial_{{y}}+ {b})$, $ \partial_{{y}} ( {y} \partial_{{y}}- {x} \partial_{{x}}- {a})+ ( {x} \partial_{{x}}+ {y} \partial_{{y}}+ {b}) ( {y} \partial_{{y}}+ {c})$

Hypersurface singularity:
$f= ( - 1) ( 4 {y} {x}- {y}^{ 2} - 2 {y}- 1)=0$

Rank of the restricted system=3
Pfaffian matrices

$H_{2}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}+ {e}- 1)- ( {x} \partial_{{x}}- {y} \partial_{{y}}+ {a}) ( {x} \partial_{{x}}+ {b})$, $ \partial_{{y}} ( {y} \partial_{{y}}- {x} \partial_{{x}}- {a})+ ( {y} \partial_{{y}}+ {c}) ( {y} \partial_{{y}}+ \partial)$

Hypersurface singularity:
$f= ( {y} {x}- {y}- 1)=0$

Rank of the restricted system=3
Pfaffian matrices

$H_{3}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}+ {y} \partial_{{y}}+ {c}- 1)- ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}) ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}+ 1)$, $ \partial_{{y}} ( {x} \partial_{{x}}+ {y} \partial_{{y}}+ {c}- 1)- ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}) ( {y} \partial_{{y}}+ {b})$

Hypersurface singularity:
$f= ( {x}+ {y}^{ 2} - {y})=0$

Rank of the restricted system=3
Pfaffian matrices

$H_{4}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}+ {c}- 1)- ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}) ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}+ 1)$, $ \partial_{{y}} ( {y} \partial_{{y}}+ \partial- 1)- ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}) ( {y} \partial_{{y}}+ {b})$

Hypersurface singularity:
$f= ( - 1) ( 4 {x}- {y}^{ 2} + 2 {y}- 1)=0$

Rank of the restricted system=3
Pfaffian matrices

$H_{5}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}- {y} \partial_{{y}}- {b})+ ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}) ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}+ 1)$, $ \partial_{{y}} ( {y} \partial_{{y}}+ {c}- 1)- ( 2 {x} \partial_{{x}}+ {y} \partial_{{y}}+ {a}) ( {y} \partial_{{y}}- {x} \partial_{{x}}+ {b})$

Hypersurface singularity:
$f= ( 16 {x}^{ 2} + ( 27 {y}^{ 2} - 36 {y}+ 8) {x}- {y}+ 1)=0$

Rank of the restricted system=3
Pfaffian matrices

$H_{6}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}- {y} \partial_{{y}}- {b})+ ( 2 {x} \partial_{{x}}- {y} \partial_{{y}}+ {a}) ( 2 {x} \partial_{{x}}- {y} \partial_{{y}}+ {a}+ 1)$, $ \partial_{{y}} ( {y} \partial_{{y}}- 2 {x} \partial_{{x}}- {a})+ ( {y} \partial_{{y}}- {x} \partial_{{x}}+ {b}) ( {y} \partial_{{y}}+ {c})$

Hypersurface singularity:
$f= ( {y}^{ 2} {x}- {y}- 1)=0$

Rank of the restricted system=3
Pfaffian matrices

$H_{7}$

Differential operators:
$ \partial_{{x}} ( {x} \partial_{{x}}+ \partial- 1)- ( 2 {x} \partial_{{x}}- {y} \partial_{{y}}+ {a}) ( 2 {x} \partial_{{x}}- {y} \partial_{{y}}+ {a}+ 1)$, $ \partial_{{y}} ( {y} \partial_{{y}}- 2 {x} \partial_{{x}}- {a})+ ( {y} \partial_{{y}}+ {b}) ( {y} \partial_{{y}}+ {c})$

Hypersurface singularity:
$f= ( 4 {y}^{ 2} {x}- {y}^{ 2} - 2 {y}- 1)=0$

Rank of the restricted system=3
Pfaffian matrices