Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the generic completion rank. Completions over the reals may generically have multiple completion ranks, called typical ranks. We demonstrate techniques for proving that many sets of specified entries have only one typical rank, and show other families with two typical ranks, specifically focusing on entry sets represented by circulant graphs. This generalizes the results of Bernstein, Blekherman, and Sinn. In particular, we provide a complete characterization of the set of unspecified entries of an $n\times n$ matrix such that $n-1$ is a typical rank and fully determine the typical ranks for entry set $G(n,1)$ for $n<9$. Moreover, we study the asymptotic behaviour of typical ranks and present results regarding unique matrix completions.

Phylogenetic networks represent evolutionary histories of sets of taxa where horizontal evolution or hybridization has occurred. Placing a Markov model of evolution on a phylogenetic network gives a model that is particularly amenable to algebraic study by representing it as an algebraic variety. In this paper, we give a formula for the dimension of the variety corresponding to a triangle-free level-1 phylogenetic network under a group-based evolutionary model. On our way to this, we give a dimension formula for codimension zero toric fiber products. We conclude by illustrating applications to identifiability.

Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition.

A primary ideal in a polynomial ring can be described by the variety it defines and a finite set of Noetherian operators, which are differential operators with polynomial coefficients. We implement both symbolic and numerical algorithms to produce such a description in various scenarios as well as routines for studying affine schemes through the prism of Noetherian operators and Macaulay dual spaces.

We introduce the package PhylogeneticTrees for Macaulay2 which allows users to compute phylogenetic invariants for group-based tree models. We provide some background information on phylogenetic algebraic geometry and show how the package PhylogeneticTrees can be used to calculate a generating set for a phylogenetic ideal as well as a lower bound for its dimension. Finally, we show how methods within the package can be used to compute a generating set for the join of any two ideals.

Nonnegative matrix factorizations are often encountered in data mining applications where they are used to explain datasets by a small number of parts. For many of these applications it is desirable that there exists a unique nonnegative matrix factorization up to trivial modifications given by scalings and permutations. This means that model parameters are uniquely identifiable from the data. Rigidity theory of bar and joint frameworks is a field that studies uniqueness of point configurations given some of the pairwise distances. The goal of this paper is to use ideas from rigidity theory to study uniqueness of nonnegative matrix factorizations in the case when nonnegative rank of a matrix is equal to its rank. We characterize infinitesimally rigid nonnegative factorizations, prove that a nonnegative factorization is infinitesimally rigid if and only if it is locally rigid and a certain matrix achieves its maximal possible Kruskal rank, and show that locally rigid nonnegative factorizations can be extended to globally rigid nonnegative factorizations. These results give so far the strongest necessary condition for the uniqueness of a nonnegative factorization. We also explore connections between rigidity of nonnegative factorizations and boundaries of the set of matrices of fixed nonnegative rank. Finally we extend these results from nonnegative factorizations to completely positive factorizations.

The Cayley-Menger variety is the Zariski closure of the set of vectors specifying the pairwise squared distances between $n$ points in $\mathbb{R}^d$. This variety is fundamental to algebraic approaches in rigidity theory. We study the tropicalization of the Cayley-Menger variety. In particular, when $d = 2$, we show that it is the Minkowski sum of the set of ultrametrics on $n$ leaves with itself, and we describe its polyhedral structure. We then give a new, tropical, proof of Laman's theorem.

We show that, under a natural probability distribution, random monomial ideals will almost always have minimal free resolutions of maximal length; that is, the projective dimension will almost always be n n , where n n is the number of variables in the polynomial ring. As a consequence we prove that Cohen–Macaulayness is a rare property. We characterize when a random monomial ideal is generic/strongly generic, and when it is Scarf—i.e., when the algebraic Scarf complex of M ⊂ S = k [ x 1 , … , x n ] M\subset S=k[x_1,\ldots ,x_n] gives a minimal free resolution of S / M S/M . It turns out, outside of a very specific ratio of model parameters, random monomial ideals are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers.

Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when \(n \ge 2r+5\). We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

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 Grobner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.

Let R be the polynomial ring K[xi,j] where 1≤i≤r and j∈N, and let I be an ideal of R stable under the natural action of the infinite symmetric group S∞. Nagel–Römer recently defined a Hilbert series HI(s,t) of I and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.

We provide a closed formula for the degree of $\text{SO}(n)$ over an algebraically closed field of characteristic zero. In addition, we describe symbolic and numerical techniques which can also be used to compute the degree of $\text{SO}(n)$ for small values of $n$. As an application of our results, we give a formula for the number of critical points of a low-rank semidefinite programming optimization problem. Finally, we provide some evidence for a conjecture regarding the real locus of $\text{SO}(n)$.

We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.

We provide a closed formula for the degree of SO(n) over an algebraically closed field of characteristic zero. In addition, we describe symbolic and numerical techniques which can also be used to compute the degree of SO(n) for small values of n. As an application of our results, we give a formula for the number of critical points of a low-rank semidefinite programming optimization problem. Finally, we provide some evidence for a conjecture regarding the real locus of SO(n).

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 dual spaces of quotient ideals, and develop an algorithm for detection of embedded points on an algebraic curve.

It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have finite Gröbner bases up to symmetry for certain monomial orders. An algorithm is presented for computing equivariant Gröbner bases that terminates whenever a finite basis exists, improving on previous algorithms that only guaranteed termination in rings Noetherian up to symmetry. This algorithm can be used to compute equivariant Gröbner bases of the above toric ideals, given the monomial map.

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a more general equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: recent work by Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an explicit degree bound for the kernel of the first part, while for the second part the finiteness follows from the Noetherianity of the matching monoid ring.