fuchsia 16.7.25

Description:

fuchsia 16.7.25

Fuchsia reduces differential equations for Feynman master integrals to canonical form.
In concrete terms, let us say we have a system of differential equations of this form:

∂f(x,ϵ)/∂x = 𝕄(x,ϵ) f(x,ϵ)

where 𝕄(x,ϵ) is a given matrix of rational functions in x and ϵ, i.e, a free variable and an infinitesimal parameter.
Our ultimately goal is to find a column vector of unknown functions f(x,ϵ) as a Laurent series in ϵ, which satisfies our equations.
With the help of Fuchsia we can find a transformation matrix 𝕋(x,ϵ) which turns our system to the equivalent Fuchsian system of this form:

∂g(x,ϵ)/∂x = ϵ 𝕊(x) g(x,ϵ)

where 𝕊(x) = ∑ᵢ 𝕊ᵢ/(x-xᵢ) and f(x,ϵ) = 𝕋(x,ϵ) g(x,ϵ).
Such a transformation is useful, because we can easily solve the equivalent system for g(x,ϵ) (see [1]) and then, multiplying it by 𝕋(x,ϵ), find f(x,ϵ).
You can learn about the algorithm used in Fuchsia to find such transformations from Roman Lee’s paper [2].
Fuchsia is available both as a command line utility and as a (Python) library for SageMath [3].
It will run on most Unix-like operating systems.
Documentation with more information, installation and usage details is here [4].


[1] https://arxiv.org/abs/1304.1806
[2] https://arxiv.org/abs/1411.0911
[3] http://www.sagemath.org/
[4] http://www.gituliar.net/fuchsia/fuchsia.pdf