# 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

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

Corresponding author of this page: Claudia Fevola, claudia.fevola@mis.mpg.de

Software used: Julia (Version 1.7.1), Macaulay2 (Version 1.13), Singular (Version 4.1.2)

System setup used: MacBook Pro with macOS Monterey 12.0.1, Processor 2,7 GHz Intel Core i7, Memory 16 GB 2133 MHz LPDDR3, Graphics Intel Iris Graphics 550 1536 MB.