We present a survey on the developments related to Gröbner bases, and show explicit examples in CoCoA.

The CoCoA project dates back to 1987: its aim was to create a “mathematician”-friendly computational laboratory for studying Commutative Algebra, most especially Gröbner bases. Always maintaining this “friendly” tradition, the project has grown and evolved, and the software has been completely rewritten.

CoCoA offers Gröbner bases for all levels of interest: from the basic, explicit call in the interactive system CoCoA-5 [5], to problem-specific optimized implementations, to the computer–computer communication with the open source C++ software library, CoCoALib [4], or the prototype OpenMath-based server.

The openness and clean design of CoCoALib and CoCoA-5 are intended to offer different levels of usage, and to encourage external contributions.

In my PhD thesis 1965 and the subsequent publication 1970 in aequationes mathematicae, I introduced the notion of Gröbner bases and proved a characterization theorem for Gröbner bases on which an algorithm for constructing Gröbner bases can be based. The main idea for the theorem and the algorithm was the notion of "S-polynomials". Most of the subsequent work on the algorithmic theory of Gröbner bases, including the implementation of the Gröbner bases technology in mathematical software systems like Mathematica, Maple, etc. was based on this approach.

In the early eighties, I proposed a completely different strategy for computing Gröbner bases proceeding by the following three steps:

1. Produce the set of all multiples $u.f$ of the polynomials $f$ in the initial basis $F$ by all power products $u$ (which we call "generalized Sylvester matrix" or "Macaulay matrix" of $F$).

2. Triangularize this matrix.

3. Take the "contour" in the diagonal of the matrix, i.e. the set of all those polynomials in the diagonal whose leading power product is not a multiple of the leading power product of any other polynomial in the diagonal.

It is easy to prove that the above procedure yields a Gröbner basis if one starts with the infinite matrix of all shifts $u.f$. However, of course, this is not an algorithm. So I posed the question whether one can give an a-priori bound on $D$ so that, if one puts all shifts $u.f$ with degree of $u$ smaller than $D$ to the initial matrix, the matrix resulting by triangularization and contour formation will be a Gröbner basis for $F$. Over the years, several people tried to find such a bound but only recently (2014) my PhD student Manuela Wiesinger-Widi was able to establish such a bound combining known bounds by G. Hermann and T. W. Dubé for the ideal membership and the Gröbner bases degree problem, respectively.

The paper gives an overview of recent advances in structural equation modeling. A structural equation model is a multivariate statistical model that is determined by a mixed graph, also known as a path diagram. Our focus is on the covariance matrices of linear structural equation models. In the linear case, each covariance is a rational function of parameters that are associated to the edges and nodes of the graph. We statistically motivate algebraic problems concerning the rational map that parametrizes the covariance matrix. We review combinatorial tools such as the trek rule, projection to ancestral sets, and a graph decomposition due to Jin Tian. Building on these tools, we discuss advances in parameter identification, i.e., the study of (generic) injectivity of the parametrization, and explain recent results on determinantal relations among the covariances. The paper is based on lectures given at the 8th Mathematical Society of Japan Seasonal Institute.

In this paper we consider the problem of finding a set of monomials $\mathcal O$ and a polynomial $f$ whose support is contained in $\mathcal O$, such that (1) $f$ is almost vanishing at a set of points $\mathbb X$ whose coordinates are not known exactly and (2) $\mathcal O$ exhibits structural stability, that is the model/design matrix associated to $\mathcal O$ is full rank for each set of points differing only slightly from $\mathbb X$. We review some numerical versions of the Buchberger-Möller (BM) algorithm for computing the set $\mathcal O$ and the polynomial $f$ and we present a variant, called LDP-LP, which integrates one of these methods with a classical statistical least squares algorithm for implicit regression from [1]. To illustrate the usefulness of these numerical BM algorithms, we review some of their application in the analyses of data sets for which standard techniques did not yield satisfactory results.

Multiple root estimation problems in statistical inference arise in many contexts in the literature. In maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inference.

In this paper, we study the multiple roots phenomenon in maximum likelihood estimation for factor analysis. We prove that the corresponding likelihood equations have uncountably many feasible solutions even in the simplest cases. For the case in which the observed data are two-dimensional and the unobserved factor scores are one-dimensional, we prove that the solutions to the likelihood equations form a one-dimensional real curve.

Let $G$ be a finite simple connected graph. We give a necessary condition for the edge ring of $G$ to satisfy Serre's condition $(S_2)$ in terms of a graph.

Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible the development of effective routines. Ability to compute relies on finite generation up to symmetry for ideals invariant under a large group or monoid action, such as the permutations of the natural numbers. We summarize the current state of theory and applications for equivariant Gröbner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.

The question whether there exists an integral solution to the system of linear equations with non-negativity constraints, $A\mathbf{x} = \mathbf{b}, \, \mathbf{x} \ge 0$, where $A \in \mathbb{Z}^{m\times n}$ and ${\mathbf b} \in \mathbb{Z}^m$, finds its applications in many areas such as operations research, number theory, combinatorics, and statistics. In order to solve this problem, we have to understand the semigroup generated by the columns of the matrix $A$ and the structure of the “holes” which are the difference between the semigroup and its saturation. In this paper, we discuss the implementation of an algorithm by Hemmecke, Takemura, and Yoshida that computes the set of holes of a semigroup and we discuss applications to problems in combinatorics. Moreover, we compute the set of holes for the common diagonal effect model and we show that the $n^\text{th}$ linear ordering polytope has the integer-decomposition property for $n\leq 7$. The software is available at http://ehrhart.math.fu-berlin.de/People/fkohl/HASE/.

We present algebraic techniques to analyze state space models in the areas of structural identifiability, observability, and indistinguishability. While the emphasis is on surveying existing algebraic tools for studying ODE systems, we also present a variety of new results. In particular: on structural identifiability, we present a method using linear algebra to find identifiable functions of the parameters of a model for unidentifiable models. On observability, we present techniques using Gröbner bases and algebraic matroids to test algebraic observability of state space models. On indistinguishability, we present a sufficient condition for distinguishability using computational algebra and demonstrate testing indistinguishability.

This article is based on a series of lectures on toric varieties given at RIMS, Kyoto. We start by introducing toric varieties, their basic properties and later pass to more advanced topics relating mostly to combinatorics.

This is an enlarged and revised version of the slides presented in a series of survey lectures given by the present author at MSJ SI 2015 in Osaka. The goal is to introduce an algorithm for computing a holonomic system of linear (ordinary or partial) differential equations for the integral of a holonomic function over the domain defined by polynomial inequalities. It applies to the cumulative function of a polynomial of several independent random variables with e.g., a normal distribution or a gamma distribution. Our method consists in Gröbner basis computation in the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients. In the algorithm, generalized functions are inevitably involved even if the integrand is a usual function. Hence we need to make sure to what extent purely algebraic method of Gröbner basis applies to generalized functions which are based on real analysis.

Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are closely related to the localization and the local cohomology of $M$. We show that the $b$-function, if it exists, controls these modules and present general algorithms for computing these modules and the $b$-function if it exists without any further assumptions. We also give some examples of multiplicity computation of such $D$-modules including a possibly well-known explicit formula for the localization of the polynomial ring by a hyperplane arrangement.

We graph-theoretically classify all Gorenstein planar graphs.

A holonomic system for the probability density function of the largest eigenvalue of a non-central complex Wishart distribution with identity covariance matrix is derived. Furthermore a new determinantal formula for the probability density function is derived (for the dimensions $m=2,3$) or conjectured.