# Vector Spaces of Generalized Euler Integrals

@inproceedings{Agostini2022VectorSO, title={Vector Spaces of Generalized Euler Integrals}, author={Daniele Agostini and Claudia Fevola and Anna-Laura Sattelberger and Simon Telen}, year={2022} }

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 uncover new relations between these approaches. We also provide new algorithmic tools.

## One Citation

### FeynGKZ: a Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

- Mathematics
- 2022

In the Lee-Pomeransky representation, Feynman integrals can be identiﬁed as a sub-set of Euler-Mellin integrals, which are known to satisfy Gel ' fand-Kapranov-Zelevinsky (GKZ) system of partial…

## References

SHOWING 1-10 OF 50 REFERENCES

### AN ALGORITHM OF COMPUTING COHOMOLOGY INTERSECTION NUMBER OF HYPERGEOMETRIC INTEGRALS

- MathematicsNagoya Mathematical Journal
- 2021

Abstract We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic…

### Introduction to Compact Riemann Surfaces

- Mathematics
- 2011

The theory of Riemann surfaces is a classical field of mathematics where geometry and analysis play equally important roles. The purpose of these notes is to present some basic facts of this theory…

### Likelihood Degenerations

- Mathematics
- 2021

Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the…

### Global Analysis of GG Systems

- Mathematics
- 2020

This paper deals with some analytic aspects of GG system introduced by I.M.Gelfand and M.I.Graev: We compute the dimension of the solution space of GG system over the field of functions meromorphic…

### On twisted de Rham cohomology

- MathematicsNagoya Mathematical Journal
- 1997

Abstract. Consider the complex of differential forms on an open affine subvariety U of AN with differential where d is the usual exterior derivative and ø is a fixed 1-form on U. For certain U and ø,…

### Scattering Amplitudes from Intersection Theory.

- MathematicsPhysical review letters
- 2018

Pic Picard-Lefschetz theory is used to prove a new formula for intersection numbers of twisted cocycles associated with a given arrangement of hyperplanes, which become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.

### Computing cohomology intersection numbers of GKZ hypergeometric systems

- MathematicsProceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)
- 2022

In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in \cite{GM}, \cite{MH}, \cite{MT} and \cite{MT2}.…

### D-Modules and Holonomic Functions

- MathematicsArXiv
- 2019

This lecture notes address the more general case when the coefficients are polynomials, and focuses on left ideals, or D-ideals, which represent holonomic functions in several variables by the linear differential equations they satisfy.

### Feynman integrals and intersection theory

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we…

### Critical points and number of master integrals

- Mathematics
- 2013

A bstractWe consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the…