Related papers: Cylindrical Algebraic Decomposition in Macaulay2
We introduce the package \texttt{EliminationTemplates} for the Macaulay2 computer algebra system, which provides tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent…
In this article we show how to compute a matrix representation and the implicit equation by means of the method developed in [Botbol: arXiv:1007.3437], using the computer algebra system Macaulay2 \cite{M2}. As it is probably the most…
Macaulay dual spaces provide a local description of an affine scheme and give rise to computational machinery that is compatible with the methods of numerical algebraic geometry. We introduce eliminating dual spaces, use them for computing…
Algebraic methods have a long history in statistics. The most prominent manifestation of modern algebra in statistics can be seen in the field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
In this Macaulay2 \cite{M2} package we define an object called {\it linear code}. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code,…
We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
We present a new algorithm for determining the satisfiability of conjunctions of non-linear polynomial constraints over the reals, which can be used as a theory solver for satisfiability modulo theory (SMT) solving for non-linear real…
We introduce the Macaulay2 package $\mathtt{LinearTruncations}$ for finding and studying the truncations of a multigraded module over a standard multigraded ring that have linear resolutions.
{\tt AbstractSimplicialComplexes.m2} is a computer algebra package written for the computer algebra system {\tt Macaulay2} \cite{M2}. It provides new infrastructure to work with abstract simplicial complexes and related homological…
We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows…
We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and…
The cylindrical algebraic covering method was originally proposed to decide the satisfiability of a set of non-linear real arithmetic constraints. We reformulate and extend the cylindrical algebraic covering method to allow for checking the…
We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in…
We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in…
In the last chapter of his book "The Algebraic Theory of Modular Systems " published in 1916, F. S. Macaulay developped specific techniques for dealing with " unmixed polynomial ideals " by introducing what he called " inverse systems ".…
We explain a Macaulay2 implementation of a construction, which appeared in [Holweck-Oeding arXiv:2206.13662], of a graded algebra structure on the direct sum of a Lie algebra $\mathfrak{g}$ (typically $\mathfrak{sl}_n$) and a…