Algorithms for minimal Picard–Fuchs operators of Feynman integrals

In even space-time dimensions the multi-loop Feynman integrals are of rational function in projective space. By using an algorithm that extends Griffiths--Dwork reduction for case hypersurfaces with singularities, we derive Fuchsian linear differential equations, Picard--Fuchs respect to kinematic parameters a large class massive integrals. With this approach obtain operator high multiplicities and loop orders. Using recent factorisation algorithms give minimal order most cases studied paper. Amongst our results generic two-point $n-1$-loop sunset integral two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting conjecture relative periods Calabi--Yau $n-2$. We have checked explicitly till six loops. As well, particular 11 five-point tardigrade non-planar two-loop four mass configurations, suggesting it arises from $K3$ surface Picard number 11. determine as well operators graphs various dimensions, finding either Liouvillian or elliptic solutions.