In the local setting, Gr\"obner cells are affine spaces that parametrize ideals in $\mathbf{k}[\![x,y]\!]$ that share the same leading term ideal with respect to a local term ordering. In particular, all ideals in a cell have the same Hilbert function, so they provide a cellular decomposition of the punctual Hilbert scheme compatible with its Hilbert function stratification. We exploit the parametrization given in \cite{HW21} via Hilbert-Burch matrices to compute the Betti strata, with hands-on examples of deformations that preserve the Hilbert function, and revisit some classical results along the way. Moreover, we move towards an explicit parametrization of all local Gr\"obner cells.

The recently introduced graphical continuous Lyapunov models provide a new approach to statistical modeling of correlated multivariate data. The models view each observation as a one-time cross-sectional snapshot of a multivariate dynamic process in equilibrium. The covariance matrix for the data is obtained by solving a continuous Lyapunov equation that is parametrized by the drift matrix of the dynamic process. In this context, different statistical models postulate different sparsity patterns in the drift matrix, and it becomes a crucial problem to clarify whether a given sparsity assumption allows one to uniquely recover the drift matrix parameters from the covariance matrix of the data. We study this identifiability problem by representing sparsity patterns by directed graphs. Our main result proves that the drift matrix is globally identifiable if and only if the graph for the sparsity pattern is simple (i.e., does not contain directed two-cycles). Moreover, we present a necessary condition for generic identifiability and provide a computational classification of small graphs with up to 5 nodes.

In this paper we study linear non-Gaussian graphical models from the perspective of algebraic statistics. These are acyclic causal models in which each variable is a linear combination of its direct causes and independent noise. The underlying directed causal graph can be identified uniquely via the set of second and third order moments of all random vectors that lie in the corresponding model. Our focus is on finding the algebraic relations among these moments for a given graph. We show that when the graph is a polytree these relations form a toric ideal. We construct explicit trek-matrices associated to 2-treks and 3-treks in the graph. Their entries are covariances and third order moments and their 2-minors define our model set-theoretically. Furthermore, we prove that their 2-minors also generate the vanishing ideal of the model. Finally, we describe the polytopes of third order moments and the ideals for models with hidden variables.

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound $2.431$ on $\omega$. We also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating. Our results form possible paths in the search for valuable tensors for the laser method away from Coppersmith-Winograd tensors.

We introduce the package "GraphicalModelsMLE" for computing the maximum likelihood estimates (MLEs) of a Gaussian graphical model in the computer algebra system Macaulay2. This package allows the computation of MLEs for the class of loopless mixed graphs. Additional functionality allows the user to explore the underlying algebraic structure of the model, such as its maximum likelihood degree and the ideal of score equations.

We study the reciprocal variety to the LSSM of catalecticant matrices associated with ternary quartics. With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM. We provide a geometric explanation to why equality between these two invariants is not reached, as opposed to the case of binary forms, by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci. Moreover, we prove that only the rank-1 locus, namely the Veronese surface v(4)(P-2), contributes to the degree of the reciprocal variety.

Sets of zero-dimensional ideals in the polynomial ring k[x, y] that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Grobner cells. Conca-Valla and Constantinescu parametrize such Grobner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of (x, y)-primary ideals in Grobner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring k [x, y] with a given leading term ideal with respect to a local term ordering. (C) 2021 Elsevier Inc. All rights reserved.

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 a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations.

We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra A=k〚x1,…,xn〛/I, compute an Artin Gorenstein k-algebra G=k〚x1,…,xn〛/J such that ℓ(G)−ℓ(A) is minimal. We approach the problem by using Macaulay's inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.

We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra A = k[[x1, . . . , xn]]/I, compute an Artin Gorenstein k-algebra G = k[[x1, . . . , xn]]/J such that l(G) − l(A) is minimal. We approach the problem by using Macaulay’s inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.

The aim of this paper is to extend the characterization of Teter rings of [9] to Artin rings of Gorenstein colength two and to improve and complete the result [1, Theorem 5.5] on such rings by using Macaulay's inverse system. We provide several explicit examples and families of Gorenstein colength two rings taking into account their analytic type.

In this paper we give a complete analytic classification of Artin Gorenstein almost stretched algebras, i.e., Artin Gorenstein algebras with Hilbert function {1, n, 2,..., 2, 1,..., 1}.