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Vector Spaces of Generalized Euler Integrals

Daniele Agostini, Claudia Fevola, Anna-Laura Sattelberger, Simon Telen

TL;DR

This work unifies several perspectives on vector spaces generated by generalized Euler integrals, showing that, for generic data, the dimension of the spaces obtained by varying integration parameters, Mellin-transform parameters, or GKZ-system parameters all coincide and equal $(-1)^n\chi(X)$ for the very affine variety $X=(\mathbb{C}^*)^n\setminus V(f_1\cdots f_\ell)$. It builds a coherent bridge between twisted de Rham theory, Mellin-transform methods, and $A$-hypergeometric (GKZ) systems, proving that the corresponding solution spaces are isomorphic and governed by the topology of $X$ (via Euler characteristics) or its Newton-polytope under suitable non-degeneracy assumptions. The paper derives explicit IBP-type relations from the twisted de Rham framework, connects these to Bernstein–Sato theory, and places the discussion in a computational context with both symbolic and numerical approaches, including fast GKZ-rank computations and numerical homotopy methods for counting critical points. Together these results yield a versatile toolkit for master integrals across physics and mathematics, with practical algorithms for computing relations and dimensions. The combination of D-module theory, GKZ systems, and numerical nonlinear algebra provides a robust path from integrals to linear relations and dimensional invariants, illuminating the role of topology in the structure of Feynman-type integrals and their generalizations.

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

Vector Spaces of Generalized Euler Integrals

TL;DR

This work unifies several perspectives on vector spaces generated by generalized Euler integrals, showing that, for generic data, the dimension of the spaces obtained by varying integration parameters, Mellin-transform parameters, or GKZ-system parameters all coincide and equal for the very affine variety . It builds a coherent bridge between twisted de Rham theory, Mellin-transform methods, and -hypergeometric (GKZ) systems, proving that the corresponding solution spaces are isomorphic and governed by the topology of (via Euler characteristics) or its Newton-polytope under suitable non-degeneracy assumptions. The paper derives explicit IBP-type relations from the twisted de Rham framework, connects these to Bernstein–Sato theory, and places the discussion in a computational context with both symbolic and numerical approaches, including fast GKZ-rank computations and numerical homotopy methods for counting critical points. Together these results yield a versatile toolkit for master integrals across physics and mathematics, with practical algorithms for computing relations and dimensions. The combination of D-module theory, GKZ systems, and numerical nonlinear algebra provides a robust path from integrals to linear relations and dimensional invariants, illuminating the role of topology in the structure of Feynman-type integrals and their generalizations.

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 -modules. We present an overview and uncover new relations between these approaches. We also provide new algorithmic tools.
Paper Structure (7 sections, 18 theorems, 108 equations, 3 figures)

This paper contains 7 sections, 18 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Twisted de Rham cohomology
  3. Mellin transform
  4. GKZ systems
  5. Numerical nonlinear algebra
  6. Vanishing of cohomology groups (Appendix by Saiei-Jaeyong Matsubara-Heo)
  7. Julia code

Key Result

Theorem 1.1

Let $X \subset (\mathop{\mathrm{\mathbb{C}}}\nolimits^*)^n$ be the very affine variety eq:X_intro, where $f_j$ are Laurent polynomials with fixed monomial supports and generic coefficients. Let $V_\Gamma, V_{s, \nu}, V_{c^*}, H_A(\kappa)$ be as defined above, with generic choices of parameters each. where $\chi(X)$ denotes the topological Euler characteristic of $X.$

Figures (3)

  • Figure 1: Twisted cycles in $\mathbb{P}^1 \setminus \{0,1,2,\infty \}$ are singular cycles on an elliptic curve.
  • Figure 2: The polytopes $\mathop{\mathrm{NP}}\nolimits(h)$ and ${\mathop{\mathrm{Conv}}\nolimits}(\{0\} \cup \mathop{\mathrm{NP}}\nolimits(h))$ from \ref{['ex:lines']} (left) and \ref{['ex:elliptic']} (right).
  • Figure 3: Estimating $\phi_{AB}(x_i)$ using a numerical ODE solver (yellow) with initial condition at $x_1 = A.$ Results are improved by adding Newton iterations in each step (green).

Theorems & Definitions (58)

  • Theorem 1.1
  • Example 2.1: $\ell = 2, n = 1$
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Example 2.7: $n = 2, \ell = 1$
  • ...and 48 more