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We investigate the reduction of Feynman integrals to master integrals using Gr\"obner bases in a rational double-shift algebra Y in which the integration-by-parts (IBP) relations form a left ideal. The problem of reducing a given family of integrals to master integrals can then be solved once and for all by computing the Gr\"obner basis of the left ideal formed by the IBP relations. We demonstrate this explicitly for several examples. We introduce so-called first-order normal-form IBP relations which we obtain by reducing the shift operators in Y modulo the Gr\"obner basis of the left ideal of IBP relations. For more complicated cases, where the Gr\"obner basis is computationally expensive, we develop an ansatz based on linear algebra over a function field to obtain the normal-form IBP relations., Comment: 25 pages, 4 figures, 12 ancillary files; v2: clarified text and improved notation

We report on an approach to integration-by-parts reduction based on Gr\"obner bases. We establish the underlying noncommutative rational double-shift algebra wherein the integration-by-parts relations form a left ideal. We describe in detail the one-loop massless box as an example where we achieved the full reduction to master integrals by means of the Gr\"obner basis approach, and report on the performance of the implementation. We also identify potential bottlenecks in more complicated examples and elaborate on interesting further directions., Comment: 11 pages, 2 figures, contribution to the proceedings of "Loops and Legs in Quantum Field Theory - LL2022, 25-30 April, 2022, Ettal, Germany''. v2: Minor revision, reference added

In this paper, we show how to compute using Fitting ideals the nonfree locus of the moduli space of arrangements of a rank $3$ simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our approach relies on the so-called Ziegler restriction and Yoshinaga's freeness criterion for multiarrangements. We use these computations to verify Terao's freeness conjecture for rank $3$ central arrangements with up to $14$ hyperplanes in any characteristic.

Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns out that many measurement assemblages used in quantum information theory as well as for studying the foundations of quantum mechanics are entirely determined by symmetry; moreover, starting from a certain symmetry group, novel types of measurement sets can be constructed. The insight gained from symmetry allows us to easily determine whether the measurements in the set are incompatible under noisy conditions, i.e., whether they can be regarded as genuinely distinct ones. In addition, symmetry enables us to identify finite sets of measurements having a high sensitivity to reveal the quantumness of distributed quantum states., Comment: 6+12 pages, 4 figures. Comments are welcome!

The goal of this paper is to introduce a new constructive geometric proof of the affine version of Chevalley's Theorem. This proof is algorithmic and a verbatim implementation resulted in an efficient code for computing the constructible image of rational maps between affine varieties. Our approach extends the known descriptions of uniform matrix product states to $\operatorname{uMPS}(2,2,5)$, Comment: More elaborate comparison to other approaches (Section 5)

In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics distinct from $2$ and $5$. Another database query proves that Terao's freeness conjecture is true for rank $3$ arrangements with $14$ hyperplanes in any characteristic., Comment: Improved exposition

This short paper presents saturation-based algorithms for homogenization and elimination. This algorithm can compute elimination ideals by using syzygies and ideal membership test, hence it works with any} monomial order, in particular without the use of block-elimination orders. The used saturation is a translation of the geometric fact that the projective closure of an affine scheme has no components in the hyperplane at infinity., Comment: Minor improvements

The ideal transform of a graded module $M$ is known to compute the module of twisted global sections of the sheafification of $M$ over a relative projective space. We introduce a second description motivated by the relative BGG-correspondence. However, our approach avoids the full BGG-correspondence by replacing the Tate resolution with the computationally more efficient purely linear saturation and the Castelnuovo-Mumford regularity with the often enough much smaller linear regularity. This paper provides elementary, constructive, and unified proofs that these two descriptions compute the (truncated) modules of twisted global sections. The main argument relies on an established characterization of Gabriel monads., Comment: Changed title, added examples, simplified arguments

The purpose of this paper is to develop an efficient computational model for Abelian categories of coherent sheaves over certain classes of varieties. These categories are naturally described as Serre quotient categories. Hence, our approach relies on describing general Serre quotient categories in a constructive way which leads to an efficient computer implementation., Comment: updated bibliography