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.
Abstract. Let R be the polynomial ring K[x i,j ] where 1 ≤ i ≤ r and j ∈ N, and let I be an ideal of R stable under the natural action of the infinite symmetric group S ∞ . NagelRömer recently defined a Hilbert series H I (s, t) of I and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
It has been shown recently that monomial maps in a large class respecting the
action of the infinite symmetric group have, up to symmetry, finitely generated
kernels. We study the simplest nontrivial family in this class: the maps given
by a single monomial. Considering the corresponding lattice map, we explicitly
construct an equivariant lattice generating set, whose width (the number of
variables necessary to write it down) depends linearly on the width of the map.
This result is sharp and improves dramatically the previously known upper bound
as it does not depend on the degree of the image monomial. In the case of of
width two, we construct an explicit finite set of binomials generating the
toric ideal up to symmetry. Both width and degree of this generating set are
sharply bounded by linear functions in the exponents of the monomial.Comment: 18 pages; v2: small improvements, added Corollary 2.7, Remark 4.10,
v3: final version, accepted at ISSAC 201
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However the usual standard basis algorithms are not numerically stable. A numerically stable approach to describing the ideal is by finding the space of dual functionals that annihilate it, which reduces the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zerodimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.
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