1,267 research outputs found
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 be the space of complex cubic polynomials in
infinitely many variables. We show that this space is
-noetherian, meaning that any
-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
No Abstract
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