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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)
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 combinatorial-polyhedral 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;(More)
We introduce equivariant tree models in algebraic statistics, which unify and generalize 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)
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)
We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. The hidden functions of the generalized problem are not restricted to be linear but can also be m-variate polynomial functions of total degree n ≥ 2. For fixed m and(More)
We characterize which graph parameters are partition functions of a vertex model over an algebraically closed field of characteristic 0 (in the sense of de la Harpe and Jones [4]). We moreover characterize when the vertex model can be taken so that its moment matrix has finite rank. Let G denote the collection of all undirected graphs, two of them being the(More)