Equivariant Gröbner bases and the Gaussian two-factor model

Brouwer, AE Andries; Draisma, Jan

doi:10.1090/s0025-5718-2010-02415-9

Cited by 31 publications

(50 citation statements)

References 14 publications

(27 reference statements)

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“…Thus, if h k is contained in the Sym(N)-invariant ideal of R p generated by the set {h ℓ | l ∈ N, ℓ = k} then, by (2), h k belongs to the Sym(N)-invariant ideal of R p,Fp generated by {h ℓ | l ∈ N, ℓ = k}. Hence, one may assume that K = F p , as claimed.…”

Section: Proof Of Theoremmentioning

confidence: 88%

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Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

Costa¹,

Krasilnikov²

2015

J Math Sci

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Abstract. Let K be a field and let N = {1, 2, . . . }. Let Rn = K[xij | 1 ≤ i ≤ n, j ∈ N] be the ring of polynomials in xij (1 ≤ i ≤ n, j ∈ N) over K. Let Sn = Sym({1, 2, . . . , n}) and Sym(N) be the groups of the permutations of the sets {1, 2, . . . , n} and N, respectively. Then Sn and Sym(N) act on Rn in a natural way: τ (xij) = x τ (i)j and σ(xij) = x iσ(j) for all τ ∈ Sn and σ ∈ Sym(N). Let Rn be the subalgebra of the symmetric polynomials in Rn,In 1992 the second author proved that if char(K) = 0 or char(K) = p > n then every Sym(N)-invariant ideal in Rn is finitely generated (as such). In this note we prove that this is not the case if char(K) = p ≤ n.We also survey some results about Sym(N)-invariant ideals in polynomial algebras and some related results.

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Section: Proof Of Theoremmentioning

confidence: 88%

“…1. For other recent results about Sym(N)-Noetherian polynomial algebras see, for instance, articles [2,8,10,11,12] and a survey [7]. 2.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

Costa¹,

Krasilnikov²

2015

J Math Sci

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“…Note that since Π orbits of G are a Gröbner basis of I, we naturally have G Π = I. Proposition 2.9 (Remark 2.1 of [7]). Let Π be a group which acts nontrivially on M .…”

Section: Preliminariesmentioning

confidence: 99%

“…At its essence, the strategy is to find a collection of elementary moves through model space that preserves the sufficient statistics of the data. The idea then is to consider growing families of model classes and show that, up to obvious symmetries, only a finite set of moves suffices for all infinite numbers of models (e.g., [2,46,35,21,14,7,19,33,16]). Typically, these moves correspond to elements of a Gröbner basis or at least a generating set for some ideal.…”

Section: Toric Algebra and Algebraic Statisticsmentioning

confidence: 99%

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Equivariant Gröbner bases

Hillar¹,

Krone²,

Leykin³

Advanced Studies in Pure Mathematics

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Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible the development of effective routines. Ability to compute relies on finite generation up to symmetry for ideals invariant under a large group or monoid action, such as the permutations of the natural numbers. We summarize the current state of theory and applications for equivariant Gröbner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.

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Codimension and projective dimension up to symmetry

Nagel

Nguyen

et al. 2019

Mathematische Nachrichten

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Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen-Macaulayness obstructions. K E Y W O R D S invariant ideal, monoid, polynomial ring, symmetric group M S C ( 2 0 1 0 ) 13A50, 13C15, 13D02, 13F20, 16P70, 16W22 Let Sym( ) denote the symmetric group on {1, … , }. Considering it as stabilizer of + 1 in Sym( + 1), similarly one gets an ascending chain of symmetric groups. Define an action of Sym( ) on induced by ⋅ , = , ( ) for every ∈ Sym( ), 1 ≤ ≤ , 1 ≤ ≤ . 346

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Equivariant Gröbner bases and the Gaussian two-factor model

Cited by 31 publications

References 14 publications

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

Equivariant Gröbner bases

Codimension and projective dimension up to symmetry

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