ESPRIT for multidimensional general grids

We present a new method for complex frequency estimation in several variables, extending the classical (1d) ESPRIT-algorithm. We also consider how to work with data sampled on non-standard domains (i.e going beyond multi-rectangles)

Reconstructing Rational Functions with FireFly\texttt{FireFly}

We present the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use $\texttt{FireFly}$ in the context of integration-by-parts reductions and compare runtime and memory consumption to a fully algebraic approach with the program $\texttt{Kira}$.Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio

Multidimensional unstructured sparse recovery via eigenmatrix

This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems. Numerical results are provided to demonstrate the performance of the proposed method.Comment: arXiv admin note: substantial text overlap with arXiv:2311.1660

Deterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials

We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [von zur Gathen/Kaltofen, J. Comp. Sys. Sci., 1985] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal by introducing essential factorization schemes which can be thought of as a relaxation of the regular factorization notion

Rational Tracer: a Tool for Faster Rational Function Reconstruction

Rational Tracer (Ratracer) is a tool to simplify complicated arithmetic expressions using modular arithmetics and rational function reconstruction, with the main idea of separating the construction of expressions (via tracing, i.e. recording the list of operations) and their subsequent evaluation during rational reconstruction. Ratracer can simplify arithmetic expressions (provided as text files), solutions of linear equation systems (specifically targeting Integration-by-Parts (IBP) relations between Feynman integrals), and even more generally: arbitrary sequences of rational operations, defined in C++ using the provided library ratracer.h. Any of these can also be automatically expanded into series prior to reconstruction. This paper describes the usage of Ratracer specifically focusing on IBP reduction, and demonstrates its performance benefits by comparing with Kira+FireFly and Fire6. Specifically, Ratracer achieves a typical ~10x probe time and ~5x overall time speedup over Kira+FireFly, and even higher if only a few terms in $\varepsilon$ need to be reconstructed

Integral Reduction with Kira 2.0 and Finite Field Methods

We present the new version 2.0 of the Feynman integral reduction program Kira and describe the new features. The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly. This procedure can be parallelized on computer clusters with MPI. Furthermore, the support for user-provided systems of equations has been significantly improved. This mode provides the flexibility to integrate Kira into projects that employ specialized reduction formulas, direct reduction of amplitudes, or to problems involving linear system of equations not limited to relations among standard Feynman integrals. We show examples from state-of-the-art Feynman integral reduction problems and provide benchmarks of the new features, demonstrating significantly reduced main memory usage and improved performance w.r.t. previous versions of Kira