# Feynman integrals as A-hypergeometric functions

## TL;DR

This paper demonstrates that Feynman integrals can be represented as solutions to GKZ hypergeometric systems, enabling new computational approaches using algebraic and geometric methods.

## Contribution

It introduces a novel framework linking Feynman integrals to A-hypergeometric functions via GKZ systems and develops algorithms for their evaluation.

## Key findings

- Feynman integrals are solutions of GKZ systems.
- Canonical series algorithms can evaluate integrals with arbitrary powers.
- The approach connects algebraic geometry with quantum field theory calculations.

## Abstract

We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel'fand-Kapranov-Zelevinsky (GKZ) system. In order to define such GKZ system, we consider the polynomial obtained from the Symanzik polynomials $g=\mathcal{U}+\mathcal{F}$ as having indeterminate coefficients. Noncompact integration cycles can be determined from the coamoeba---the argument mapping---of the algebraic variety associated with $g$. In general, we add a deformation to $g$ in order to define integrals of generic graphs as linear combinations of their canonical series. We evaluate several Feynman integrals with arbitrary non-integer powers in the propagators using the canonical series algorithm.