Related papers: Commutative Algebra and Algebraic Geometry using O…
We study an interplay between operator algebras and geometry of rational elliptic curves. Namely, let $\mathcal{O}_B$ be the Cuntz-Krieger algebra given by square matrix $B=(b-1, ~1, ~b-2, ~1)$, where $b$ is an integer greater or equal to…
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…
We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus.
Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points,…
The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively…
Composed image retrieval (CIR) requires complex reasoning over heterogeneous visual and textual constraints. Existing approaches largely fall into two paradigms: unified embedding retrieval, which suffers from single-model myopia, and…
This article presents some aspects and experience in the use of algebraic manipulation software applied to general relativity. Some years ago certain results were reported using computer algebra platforms, but the growing popularity of…
The main mathematical focus of this paper is a class of parametrised polynomial systems that we refer to as being tropically transverse. We show how their generic number of solutions can be expressed as the mixed volume of a modified…
We study the Onsager algebra from the ideal theoretic point of view. A complete classification of closed ideals and the structure of quotient algebras are obtained. We also discuss the solvable algebra aspect of the Onsager algebra through…
In classical invariant theory, the Gr\"obner base of the ideal of syzygies and the normal forms of polynomials of invariants are two core contents. To improve the performance of invariant theory in symbolic computing of classical geometry,…
Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational article is to introduce the notion of a curved operad, and to develop the operadic calculus…
We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…
We prove a theorem on algebraic osculation and we apply our result to the Computer Algebra problem of polynomial factorization. We consider X a smooth completion of the complex plane and D an effective divisor supported on the boundary of…
We study algebraic varieties associated with the camera resectioning problem. We characterize these resectioning varieties' multigraded vanishing ideals using Gr\"obner basis techniques. As an application, we derive and re-interpret…
The main idea in this paper is merging two techniques that have been recently developed. On the one hand, we consider MCCGS, standing for Minimal Canonical Comprehensive Groebner Systems, a recently introduced computational tool yielding…
This thesis is divided into two parts. The first one is composed of recollections on operad theory, model categories, simplicial homotopy theory, rational homotopy theory, Maurer-Cartan spaces, and deformation theory. The second part deals…
In this paper we study generic M(atrix) theory compactifications that are specified by a set of quotient conditions. A procedure is proposed, which both associates an algebra to each compactification and leads deductively to general…
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.