The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.Communicated by James Renegar.

We produce Brill-Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill-Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.

X 1 = λ 01 + ε 1 , X 2 = λ 02 + λ 12 X 1 + ε 2 , X 3 = λ 03 + λ 23 X 2 + ε 3 , with an error vector ε that has zero mean vector and covariance matrix Ω =   ω 11 0 0 0 ω 22 ω 23 0 ω 23 ω 33   . IDENTIFIABILITY OF LINEAR STRUCTURAL EQUATION MODELS 3The possibly nonzero entry ω 23 can absorb the effects that unobserved confounders (such as age, income, genetics, etc.) may have on both X 2 and X 3 ; compare Richardson and Spirtes (2002) and Wermuth (2011) for background on mixed graph representations of latent variable problems.Formally, a mixed graph is a triple G = (V, D, B), where V is a finite set of nodes and D, B ⊆ V × V are two sets of edges. In our context, the nodes correspond to the random variables X 1 , . . . , X m , and we simply let V = [m] := {1, . . . , m}. The pairs (v, w) in the set D represent directed edges and we will alwaysIf the directed part (V, D) does not contain directed cycles (i.e., no cycle v → · · · → v can be formed from the edges in D), then the mixed graph G is said to be acyclic.the remark after equation (2.3).] Similarly, let PD m be the cone of positive definite symmetric m × m-matrices Ω = (ω vw ) and define PD(B) ⊂ PD m to be the subcone of matrices with support B, that is, ω vw = 0 if v = w and v ↔ w / ∈ B.

Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre-Veronese embeddings. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; and indeed, no Segre-Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included.

We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based models such as the Jukes-Cantor and Kimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices. The main novelty is that our results yield generators of the full ideal rather than an ideal which only defines the model set-theoretically. Set-up and theoremsIn phylogenetics, tree models have been introduced to describe the evolution of a number of species from a distant common ancestor. Given suitably aligned strings of nucleotides of n species alive today, one assumes that the individual positions in J. Draisma has been supported by DIAMANT, an NWO mathematics cluster and J. Kuttler by an NSERC Discovery Grant.

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar $d$-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.Comment: Published in at http://dx.doi.org/10.1214/09-AOS760 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebrogeometric analysis of the set of orthogonally decomposable tensors.More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary.A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras-associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.Contents arXiv:1512.08031v1 [math.AG] 25

Abstract. Exploiting symmetry in Gröbner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Gröbner basis in a setting where a monoid acts by homomorphisms on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Gröbner bases.Using this algorithm and the monoid of strictly increasing functions N → N we prove that the kernel of the ring homomorphismis generated by two types of polynomials: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.