236 research outputs found
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
Finite Groebner bases in infinite dimensional polynomial rings and applications
We introduce the theory of monoidal Groebner bases, a concept which
generalizes the familiar notion in a polynomial ring and allows for a
description of Groebner bases of ideals that are stable under the action of a
monoid. The main motivation for developing this theory is to prove finiteness
theorems in commutative algebra and its applications. A major result of this
type is that ideals in infinitely many indeterminates stable under the action
of the symmetric group are finitely generated up to symmetry. We use this
machinery to give new proofs of some classical finiteness theorems in algebraic
statistics as well as a proof of the independent set conjecture of Hosten and
the second author.Comment: 24 pages, adds references to work of Cohen, adds more details in
Section
Constructive degree bounds for group-based models
Group-based models arise in algebraic statistics while studying evolution
processes. They are represented by embedded toric algebraic varieties. Both
from the theoretical and applied point of view one is interested in determining
the ideals defining the varieties. Conjectural bounds on the degree in which
these ideals are generated were given by Sturmfels and Sullivant. We prove that
for the 3-Kimura model, corresponding to the group G=Z2xZ2, the projective
scheme can be defined by an ideal generated in degree 4. In particular, it is
enough to consider degree 4 phylogenetic invariants to test if a given point
belongs to the variety. We also investigate G-models, a generalization of
abelian group-based models. For any G-model, we prove that there exists a
constant , such that for any tree, the associated projective scheme can be
defined by an ideal generated in degree at most d.Comment: Boundedness results for equations defining the projective scheme were
extended to G-models (including 2-Kimura and all JC
Equivariant lattice bases
We study lattices in free abelian groups of infinite rank that are invariant
under the action of the infinite symmetric group, with emphasis on finiteness
of their equivariant bases. Our framework provides a new method for proving
finiteness results in algebraic statistics. As an illustration, we show that
every invariant lattice in , where
, has a finite equivariant Graver basis. This result
generalizes and strengthens several finiteness results about Markov bases in
the literature.Comment: 31 page
Noetherianity up to symmetry
These lecture notes for the 2013 CIME/CIRM summer school Combinatorial
Algebraic Geometry deal with manifestly infinite-dimensional algebraic
varieties with large symmetry groups. So large, in fact, that subvarieties
stable under those symmetry groups are defined by finitely many orbits of
equations---whence the title Noetherianity up to symmetry. It is not the
purpose of these notes to give a systematic, exhaustive treatment of such
varieties, but rather to discuss a few "personal favourites": exciting examples
drawn from applications in algebraic statistics and multilinear algebra. My
hope is that these notes will attract other mathematicians to this vibrant area
at the crossroads of combinatorics, commutative algebra, algebraic geometry,
statistics, and other applications.Comment: To appear in Springer's LNM C.I.M.E. series; several typos fixe
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