Related papers: Intersection Numbers, Polynomial Division and Rela…
A growing body of evidence suggests that the complexity of Feynman integrals is best understood through geometry. Recent mathematical developments [Kontsevich and Soibelman, arXiv:2402.07343] have illuminated the role of exponential…
We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $\tilde{O}(n^{1/2})$ rounds. In contrast, the previous…
The study of Feynman rules is much facilitated by the two Symanzik polynomials, homogeneous polynomials based on edge variables for a given Feynman graph. We review here the role of a recently discovered third graph polynomial based on…
Using the parallel/orthogonal space method, we calculate the planar two-loop three-point diagram and two rotated reduced planar two-loop three-point diagrams. Together with the crossed topology, these diagrams are the most complicated ones…
We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories. The relevant…
Autonomous intersection management has the potential to reduce road traffic congestion and energy consumption. To realize this potential, efficient algorithms are needed. However, most existing studies locally optimize one intersection at a…
Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of…
We invent an automated method for computing the divergent part of Feynman integrals in dimensional regularization. Our method exploits simplifications from four-dimensional integration-by-parts identities. Leveraging algorithms from the…
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…
From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric…
We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a…
We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of…
We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary…
We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that…
We present a unified deterministic approach for experimental design problems using the method of interlacing polynomials. Our framework recovers the best-known approximation guarantees for the well-studied D/A/E-design problems with simple…
We present a method to evaluate numerically Feynman diagrams directly from their Feynman parameters representation. We first disentangle overlapping singularities using sector decomposition. Threshold singularities are treated with an…
We describe a constructive procedure to separate overlapping infrared divergences in multi-loop integrals. Working with a parametric representation in D=4-2*epsilon dimensions, adequate subtractions lead to a Laurent series in epsilon,…
Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…