# Vector Spaces of Generalized Euler Integrals

Daniele Agostini, Claudia Fevola, Anna-Laura Sattelberger, and Simon Telen: Vector Spaces of Generalized Euler Integrals

ABSTRACT: We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $$D$$-modules. We present an overview and discuss relations between these approaches. We also provide new algorithmic tools.

Example 3.7 in the article illustrates how to compute the $$s$$-parametric annihilator of $$f^s$$ for a univariate polynomial $$f \in \mathbb{C}[x]$$. This is done using the library dmod_lib in Singular. The code can be downloaded here singular_notebook.ipynb.zip. It is illustrated in the following notebook:

In Example 4.2 we compute the GKZ system associated to the polynomial $$f = -c_1 xy^2 + c_2 xy^3 + c_3 x^2y - c_4 x^2y^3 - c_5 x^3y + c_6 x^3y^2.$$ The code in the subpage linked below provides a way to perform this computation in Macaulay2:

Section 5 presents ideas for the use of numerical methods for computing the dimension of the vector spaces of generalized Euler integrals we are interested in. Moreover, it provides a method for obtaining $$\mathbb{C}$$-linear relations among the generators of the vector space

$V_\Gamma \, := \, \hbox{Span}_{\mathbb{C}} \left \{ [\Gamma] \longmapsto \int_\Gamma \,f^{s + a} \,x^{\nu + b} \, \frac{\mathrm{d} x}{x} \right \}_{(a,b) \, \in \, \mathbb{Z}^\ell \! \times \!\, \mathbb{Z}^n},$

presented in the introduction, when $$n=1$$. Our algorithm is implemented in julia. The code can be downloaded here julia_notebook.ipynb.zip. It is illustrated in the following notebook:

Project page created: 18/08/2022

Project contributors: Daniele Agostini, Claudia Fevola, Anna-Laura Sattelberger, Simon Telen