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We present Binomials, a package for the computer algebra system Macaulay2, which specializes well known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that…
A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. The operational rules are described and the method is…
Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers…
Over the last few years lattice techniques have been used to investigate candidate theories of new physics beyond the Standard Model. This review gives a survey of results from these studies. Most of these investigations have been of…
Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct…
Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of…
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…
The main purpose of this paper is to investigate prime, primary, and maximal ideals of semirings. The localization and primary decomposition of ideals in semirings are also studied.
This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial…
A constructive procedure is given to determine all ideals of a solvable Lie algebra. This is used in determining algorithmically all conjugacy classes of subalgebras of a given solvable Lie algebra.
A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…
In this paper we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers $\mathbb{N}$, describe the lattice of its ideals for arbitrary field $K$ and study its derivations over any commutative, unital…
We investigate the defining ideal of the algebra over a field generated by the join-meet binomials coming from a finite distributive lattice. In the frame of algebras with straightening laws, the problem when the defining ideal is generated…
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…
Bipartite determinantal ideals are introduced by Illian and the author as a vast generalization of the classical determinantal ideals intensively studied in commutative algebra, algebraic geometry, representation theory and combinatorics.…
We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of…
In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence…
We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…
We characterize all graphs whose binomial edge ideals have pure resolutions. Moreover, we introduce a new switching of graphs which does not change some algebraic invariants of graphs, and using this, we study the linear strand of the…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…