Related papers: Decompositions of Binomial Ideals
We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest…
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. These mesoprimary…
We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies…
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and…
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…
The NumericalHilbert package for Macaulay2 includes algorithms for computing local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. These techniques are numerically stable, and can be used…
We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay…
We introduce a new Macaulay 2 package, SimplicialDecomposability, which works in conjunction with the extant package SimplicialComplexes in order to compute a shelling order, if one exists, of a specified simplicial complex. Further,…
This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps,…
In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.
The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this…
Let $I$ be an arbitrary ideal generated by binomials. We show that certain equivalence classes of fibers are associated to any minimal binomial generating set of $I$. We provide a simple and efficient algorithm to compute the indispensable…
Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first…
The Macaulay2 package GraphicalModels contains algorithms for the algebraic study of graphical models associated to undirected, directed and mixed graphs, and associated collections of conditional independence statements. Among the…
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…
This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose…
In this paper we describe the method which we applied to successfully compute the primary decomposition of a certain ideal coming from applications in combinatorial algebra and algebraic statistics regarding conditional independence…