Related papers: SubalgebraBases in Macaulay2
The purpose of this paper is to characterize one-dimensional local domains, or more in general reduced, in terms of its Macaulay's inverse system. This leads to study almost finitely generated modules in the divided power ring. We…
The Macaulay2 package PHCpack.m2 provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The…
We describe a Macaulay2 package for computing Schur complexes. This package expands on the ChainComplexOperations package by David Eisenbud.
We introduce a tropical version of the Fukaya algebra of a Lagrangian submanifold. Tropical graphs arise as large-scale behavior of pseudoholomorphic disks under a multiple cut operation on a symplectic manifold that produces a collection…
We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if…
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…
The Sally module of a Rees algebra $\BB$ relative to one of its Rees subalgebras $\AA$ is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated…
Using discrete Morse theory, Batzies and Welker introduced Morse resolutions of monomial ideals. In this note, we present the {\it Macaulay2} package {\tt MorseResolutions} for working with two important classes of Morse resolutions:…
We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.
Gr\"obner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faug\`ere-type F4 and signature-based F5) reduce many…
Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a…
Let $f\_1,\ldots, f\_s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[ f1,\ldots, f\_s]] \subseteq K[[ x ]]$ (respectively the semigroup of…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of…
We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and $L_2$ loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables…
Following Kashiwara's algebraic approach, we construct crystal bases and canonical bases for quantum supergroups with no isotropic odd roots and for their integrable modules.
Schubert calculus provides algebraic tools to solve enumerative problems. There have been several applied problems in systems theory, linear algebra and physics which were studied by means of Schubert calculus. The method is most powerful…
We extend the work of M.Borovoi on the nonabelian Galois cohomology of linear reductive algebraic groups over number fields to a general base scheme. As an application, we obtain new results on the arithmetic of such groups over global…
We show that any Algebraic Branching Program (ABP) computing the polynomial $\sum_{i = 1}^n x_i^n$ has at least $\Omega(n^2)$ vertices. This improves upon the lower bound of $\Omega(n\log n)$, which follows from the classical result of Baur…
This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over…