Finiteness results for Abelian tree models

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:1103.533

Regular maps of high density

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a graph of density strictly exceeding 1/2, and we conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves x^n+y^n+z^n=0, excepting the one corresponding to the tetrahedron.Comment: 13 pages, 4 figure

Topological noetherianity for cubic polynomials

Let $P_3(\mathbf{C}^{\infty})$ be the space of complex cubic polynomials in infinitely many variables. We show that this space is $\mathbf{GL}_{\infty}$-noetherian, meaning that any $\mathbf{GL}_{\infty}$-stable Zariski closed subset is cut out by finitely many orbits of equations. Our method relies on a careful analysis of an invariant of cubics introduced here called q-rank. This result is motivated by recent work in representation stability, especially the theory of twisted commutative algebras. It is also connected to certain stability problems in commutative algebra, such as Stillman's conjecture.Comment: 13 page

Polynomials and tensors of bounded strength

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalises a theorem by Derksen-Eggermont-Snowden on cubic polynomials, as well as a theorem by Kazhdan-Ziegler which says that a polynomial all of whose directional derivatives have bounded strength must itself have bounded strength.Comment: Improved the bounds on strength as a function of the dimension of the space where one first sees nontrivial equations for the tensor property

Noetherianity for infinite-dimensional toric varieties

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.Comment: 20 page

Algebraic boundary of matrices of nonnegative rank at most three

The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Grobner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.Comment: 15 pages, 2 figure

Birth weight and early lung compliance as predictors of short-term outcome in premature infants with respiratory distress syndrome

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