相关论文: Toric Geometry in OSCAR
We review, from a didactic point of view, the definition of a toric section and the different shapes it can take. We'll then discuss some properties of this curve, investigate its analogies and differences with the most renowned conic…
We report about significant enhancements of the complex algebraic geometry theorem proving subsystem in GeoGebra for automated proofs in Euclidean geometry, concerning the extension of numerous GeoGebra tools with proof capabilities. As a…
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…
Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…
Automatic code generation for low-dimensional geometric algorithms is capable of producing efficient low-level software code through a high-level geometric domain specific language. Geometric Algebra (GA) is one of the most suitable…
We demonstrate how methods in Functional Programming can be used to implement a computer algebra system. As a proof-of-concept, we present the computational-algebra package. It is a computer algebra system implemented as an embedded…
We construct surface codes corresponding to genus greater than one in the context of quantum error correction. The architecture is inspired by the topology of invariant integral surfaces of certain non-integrable classical billiards.…
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's…
GeoGebra is an open source mathematics education software tool being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians…
Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying…
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in…
It is demonstrated how the software system polymake can be used for computations in toric geometry. More precisely, counter-examples to conjectures related to A-determinants and defect polytopes are constructed.
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the…
In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…