相关论文: What can be computed in algebraic geometry?

We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…

This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.

A priori, the set of birational transformations of an algebraic variety is just a group. We survey the possible algebraic structures that we may add to it, using in particular parametrised family of birational transformations.

In this article, we briefly describe various tools and approaches that algebraic geometry has to offer to straighten bent objects. Throughout this article we will consider a specific example of a bent or curved piece of paper which in our…

Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…

In this paper we prove that the only algebraic constant mean curvature (cmc) surfaces in R^3 of order less than four are the planes, the spheres and the cylinders. The method used heavily depends on the efficiency of algorithms to compute…

We review the main features of a mathematical framework encompassing some of the salient quantum mechanical and geometrical aspects of Hall systems with finite size and general boundary conditions. Geometrical as well as algebraic…

This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on…

Gendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on…

We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.

The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…

We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…

We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.

Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…

This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in…

Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…