A triangulation of a point configuration is regular if it can be given by a height function, that is every point gets lifted to a certain height and projecting the lower convex hull gives the triangulation. Checking regularity of a triangulation usually is done by solving a linear program. However when checking many flip-connected triangulations for regularity, one can instead ask which flips preserve regularity. When traversing the flip graph for enumerating all regular triangulations, this allows for vast reduction of the linear programs needing to be solved. At the same time the remaining linear programs will be much smaller.
The software package DetGB is a collection of functions we develop in the computer algebra system Maple for computing Gröbner bases of determinantal ideals and thus facilitates the study, analysis, and visualization of determinantal ideals in a computational way.
In this paper, we study the Fully Homomorphic Encryption system proposed by Craig Gentry, Amit Sahai, and Brent Waters in [3]. We present a restated version of the proposed cryptosystem, and consider the ciphertexts resulting from the use of this system, to find patterns in them. For this task, we train a machine learning pipeline, utilizing Topological Data Analysis. We show that for secure parameters chosen according to the available literature, this machine learning approach can simply guess what the plain text should have been in a majority of cases (accuracy > 70%), in very good time complexity (< O(n3)). This attack was ineffective against the base Learning With Errors (LWE) problem, which hints at a possible flaw somewhere in the reduction from LWE to the cryptosystem.
We present a practical algorithm to computing the automorphism group of a variety. Our algorithm is based on combinatorial tools from graph theory, in particular the notion of a canonical form. Performance issues and relations to other approaches will be discussed.
This paper describes work towards an approach to using massively parallel methods for computing syzygies and free resolutions of finitely generated modules over polynomial rings over fields. Our primary focus here is Schreyer’s resolution. Our method exploits the inherent parallelism of the algorithm, primarily utilizing Petri nets, within the GPI-Space [10] framework as our language for parallel workflows. GPI-Space is a task-based workflow management system that employs Petri nets as its coordination layer, while the computation is carried out by the computer algebra system Singular [9]. We outline how the algorithm is modeled through a Petri net, explaining the coordination of tasks and data structures within the parallel computing environment.
We describe the methods used in the GAP package GRAPE for proper vertex-colouring a graph, including the determination of a minimum vertex-colouring and hence the chromatic number. These methods are designed to exploit the automorphism group of the graph.
Mathematical software designers, numerical analysts, and other researchers often need high quality tables of function values, but most current libraries and systems that produce such tables offer limited information about accuracy. To address this void, the National Institute of Standards and Technology (NIST) Applied and Computational Mathematics Division (ACMD) and the University of Antwerp Computational Mathematics (CMA) Research Group are collaborating to build the DLMF Standard Reference Tables on Demand (DLMF Tables) web service. DLMF Tables will provide a standard of comparison for testing numerical software by computing, on demand, special functions to user-defined accuracy with guaranteed error bounds.
We introduce the concept of monodromy coordinates for representing solutions to large polynomial systems. Representing solutions this way provides a time-memory trade-off in a monodromy solving algorithm. We describe an algorithm, which interpolates the usual monodromy solving algorithm, for computing such a representation and analyze its space and time complexity.
There is a strong tradition of computation in number theory, with notable data-driven insights including the prime number theorem and the conjecture of Birch and Swinnerton-Dyer. A huge arithmetic online database known as the LMFDB went live in the mid-2010 s, to which we began applying machine learning methodologies in 2020. This led to a data scientific perspective on old problems, and the discovery of surprising new structures in arithmetic statistics known as murmurations. In this extended abstract, we will apply unsupervised learning techniques to a small dataset taken from the LMFDB, chosen so as to demonstrate one approach to generalising the original experiments.
I analyse differences in style between traditional prose mathematics writing and computer-formalised mathematics writing, presenting five case studies. I note two aspects where good style seems to differ between the two: in their incorporation of computation and of abstraction. I argue that this reflects a different mathematical aesthetic for formalised mathematics.
We describe a generic JSON based file format which is suitable for computations in computer algebra. This is implemented in the computer algebra system OSCAR, but we also indicate how it can be used in a different context.
The Atiyah–Bott localization formula is a powerful tool for calculating the degree of equivariant classes of the moduli space of rational stable maps M_0,m(X,β ) , where X denotes a smooth toric variety, m is a non-negative integer, and β is an effective 1-cycle. Implementation of the formula entails intricate computational challenges, involving graph theory, colorings, partitions, and other discrete objects. Furthermore, the computed solution is a large summation of rational numbers, underscoring the imperative nature of computational efficiency. This formula has been applied in very specific cases for computing Gromov–Witten invariants, addressing enumerative problems, and determining the small quantum ring of X, among other applications. A comprehensive implementation as a Julia package has been recently presented by the author. We show the features of the package with a particular emphasis to the noteworthy contribution of the package Oscar.jl. Finally, we delve into the fundamental prerequisites for extending the implementation to encompass algebraic GKM manifolds.
Computational birational geometry is one of the key playing fields in an algorithmic approach to algebraic geometry, since birational maps are the fundamental way to relate algebraic varieties (or schemes). An important application is an algorithmic approach to the Minimal Model Program (MMP), which aims to classify algebraic varieties with mild singularities by finding simple birational models of such varieties in their birational equivalence class. This note presents work towards parallel methods to solve problems in birational geometry. Making use of a representation of algebraic schemes in terms of charts allows for a parallel computational approach for handling both the varieties and rational maps between them. In this note, we illustrate this approach on examples.
Linear codes related to applications in Galois Geometry often require a certain divisibility of the occurring weights. In this paper we present an algorithmic framework for the classification of linear codes over finite fields with restricted sets of weights. The underlying algorithms are based on lattice point enumeration and integer linear programming. We present new enumeration and non-existence results for projective two-weight codes, divisible codes, and additive F4-codes.
A method for developing a GeoGebra-based web app is introduced, utilizing React for the user interface and WebAssembly for computation. React enhances UI components and improves error detection, overcoming the limitations of GeoGebra's script input. Additionally, WebAssembly facilitates the use of C functions within web browsers, thereby enhancing GeoGebra's computational capabilities. The effective collaboration between React and WebAssembly is demonstrated through the implementation of the 'Lights Out' puzzle.
We present a new methodology for utilising machine learning technology in symbolic computation research. We explain how a well known human-designed heuristic to make the choice of variable ordering in cylindrical algebraic decomposition may be represented as a constrained neural network. This allows us to then use machine learning methods to further optimise the heuristic, leading to new networks of similar size, representing new heuristics of similar complexity as the original human-designed one. We present this as a form of ante-hoc explainability for use in computer algebra development.
Families of ℓ-adic Galois representations are an important tool in modern algebraic number theory. Andrew Wiles [1] used families of representations associated with elliptic curves to prove Fermat’s Last Theorem and the Langlands philosophy conjectures a deep connection to the theory of automorphic forms [2]. We formalise the definition of families of ℓ-adic Galois representations as well as the definition of compatibility on them for the first time in an interactive theorem prover and discuss the formalisation process.
We present a framework for generalizing tropical homotopy continuation for computing stable intersections of tropical hypersurfaces, tropical linear spaces, and tropical inverted linear spaces. We also report on beginning implementations in OSCAR.
Computer Algebra Systems (e.g. Maple) are used in research, education, and industrial settings. One of their key functionalities is symbolic integration, where there are many sub-algorithms to choose from that can affect the form of the output integral, and the runtime. Choosing the right sub-algorithm for a given problem is challenging: we hypothesise that Machine Learning (ML) can guide this sub-algorithm choice. A key consideration of our methodology is how to represent the mathematics to the ML model: we hypothesise that a representation which encodes the tree structure of mathematical expressions would be well suited. We trained both an LSTM and a TreeLSTM model for sub-algorithm prediction and compared them to Maple’s existing approach. Our TreeLSTM performs much better than the LSTM, highlighting the benefit of using an informed representation of mathematical expressions. It is able to produce better outputs than Maple’s current state-of-the-art meta-algorithm, giving a strong basis for further research.
Modeling-Simulation-Optimization workflows play a fundamental role in applied mathematics. The Mathematical Research Data Initiative, MaRDI, responded to this by developing a FAIR and machine-interpretable template for a comprehensive documentation of such workflows. MaRDMO, a Plugin for the Research Data Management Organiser, enables scientists from diverse fields to document and publish their workflows on the MaRDI Portal seamlessly using the MaRDI template. Central to these workflows are mathematical models. MaRDI addresses them with the MathModDB ontology, offering a structured formal model description. Here, we showcase the interaction between MaRDMO and the MathModDB Knowledge Graph through an algebraic modeling workflow from the Digital Humanities. This demonstration underscores the versatility of both services beyond their original numerical domain.
