Antichains of monomial ideals are finite

Maclagan, Diane

doi:10.1090/s0002-9939-00-05816-0

Cited by 32 publications

(21 citation statements)

References 5 publications

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“…Let be the Hilbert function of . There is only a finite number of monomial ideals in with Hilbert function , as the same is true in the polynomial ring for any field ; see, for example, [Mac01, Corollary 2.2]. Let be the maximum degree of any generator of a monomial ideal with Hilbert function .…”

Section: Gröbner Complex For Tropical Idealsmentioning

confidence: 99%

Tropical ideals

Maclagan¹,

Rincón²

2018

Compositio Math.

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We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379-3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

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Section: Gröbner Complex For Tropical Idealsmentioning

confidence: 99%

Tropical ideals

Maclagan¹,

Rincón²

2018

Compositio Math.

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“…. , x n ] has no antichain, which has been proved in [Mac01], but probably much before in an order theoretic setting: in this setting, the result states that the set of upward closed subsets of (N m 0 , ≤) has no antichain. A direct proof is given at the end of Section 4.…”

Section: Introductionmentioning

confidence: 83%

Dickson's Lemma, Higman's Theorem and Beyond: a survey of some basic results in order theory

Aichinger¹,

Aichinger²

2018

Preprint

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“…The fact that Sym(N)-stable monomial ideals are finitely generated up to the action of Sym(N) boils down to the statement that Young diagrams are well-quasi-ordered by inclusion. This, in turn, is a special case of the theorem in [10] that antichains of monomial ideals are finite. Remark 4.3.…”

Section: Scheme-theoretic G-noetherianitymentioning

confidence: 89%

Finiteness for the k-factor model and chirality varieties

Draisma

2010

Advances in Mathematics

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This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n × n covariance matrices of n observed Gaussian random variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all k-subsets among an n-set of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows.

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Antichains of monomial ideals are finite

Cited by 32 publications

References 5 publications

Tropical ideals

Tropical ideals

Dickson's Lemma, Higman's Theorem and Beyond: a survey of some basic results in order theory

Finiteness for the k-factor model and chirality varieties

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