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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. © 2012 Elsevier Inc. All rights reserved.

- Jan Draisma, Emil Horobet, Giorgio Ottaviani, Bernd Sturmfels, Rekha R. Thomas
- Foundations of Computational Mathematics
- 2016

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… (More)

Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. 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; this… (More)

- Jan Draisma
- 2007

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 Shimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for… (More)

We prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue of Weyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take… (More)

- Jan Draisma, Jan Draisma
- 2008

This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n × n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality… (More)

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… (More)

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… (More)

Let G denote the collection of all undirected graphs, two of them being the same if they are isomorphic. In this paper, all graphs are finite and may have loops and multiple edges. Let k ∈ N and let F be a commutative ring. Following de la Harpe and Jones [4], call any function y : N → F a (k-color) vertex model (over F).6 The partition function of y is the… (More)

- Andries E. Brouwer, Jan Draisma
- Math. Comput.
- 2011

We show that the kernel I of the ring homomorphism R[yij | i, j ∈ N, i > j]→ R[si, ti | i ∈ N] determined by yij 7→ sisj +titj is 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. Our proof is computational: inspired by work of… (More)