Horn system restricted to hypersurface singularity
References
- Vsevolod Chestnov, Saiei J. Matsubara-Heo, Henrik J. Munch, Nobuki Takayama,
Restrictions of Pfaffian Systems for Feynman Integrals,
doi:10.1007/JHEP11(2023)202
- 2023-03-28-mt_mm-for-web.pdf
- 2023-08-23-rest-for-web.pdf
- 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