Finite Gröbner bases in infinite dimensional polynomial rings and applications

Hillar, Christopher J.; Sullivant, Seth

doi:10.1016/j.aim.2011.08.009

Cited by 77 publications

(80 citation statements)

References 21 publications

(66 reference statements)

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“…If M is noetherian, we show that lim M is Sym(N)-noetherian, that is, every Sym(N)invariant submodule of lim M is generated by finitely many Sym(N)-orbits (see Theorem 4.6). Since, as a special case of our results, X is a noetherian FI-algebra over K, this implies in particular [14,Theorems 1.1], that is, Sym(N)-invariant ideals of K[X] are generated by finitely many Sym(N)-orbits (see Corollary 6.21).…”

Section: Introductionmentioning

confidence: 73%

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FI- and OI-modules with varying coefficients

Nagel

Römer

2019

Journal of Algebra

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We introduce FI-algebras over a commutative ring K and the category of FI-modules over an FI-algebra. Such a module may be considered as a family of invariant modules over compatible varying K-algebras. FI-modules over K correspond to the well studied constant coefficient case where every algebra equals K. We show that a finitely generated FI-module over a noetherian polynomial FI-algebra is a noetherian module. This is established by introducing OI-modules. We prove that every submodule of a finitely generated free OI-module over a noetherian polynomial OI-algebra has a finite Gröbner basis. Applying our noetherianity results to a family of free resolutions, finite generation translates into stabilization of syzygies in any fixed homological degree. In particular, in the graded case this gives uniformity results on degrees of minimal syzygies. 38,

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Section: Introductionmentioning

confidence: 73%

“…Moreover, an Inc(N)-invariant filtration I = (I n ) n∈N of ideals I n ⊂ P n (see, e.g., [14] or [18, Definition 5.1]) corresponds to an OI-algebra A over K, where A n = P n /I n .…”

Section: Definition 24mentioning

confidence: 99%

FI- and OI-modules with varying coefficients

Nagel

Römer

2019

Journal of Algebra

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“…For n = 1 this theorem was proved by Cohen [3] in 1967 and rediscovered independently by Aschenbrenner and Hillar [1] in 2007; for an arbitrary positive n this was proved by Cohen [4] in 1987 and rediscovered independently by Hillar and Sullivant [13] in 2012. Cohen's results were motivated by the finite basis problem for identities of metabelian groups and the results of Aschenbrenner, Hillar and Sullivant by applications to chemistry and algebraic statistics.…”

Section: Introductionmentioning

confidence: 93%

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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“…There is some overlap in definitions and results by Aschenbrenner, Hillar, and Sullivant [1,2,11], by Cohen and his student Emmott [3,4], and by ourselves; but the three groups pursued distinct goals. In particular, in Section 2 we derive the algorithm from Theorem 1.6 in the non-Noetherian situation where the partial order is not necessarily a well-quasi-order.…”

Section: Theorem 16 Under Conditions Egb1 Egb2 Egb3 and Egb4 Thementioning

confidence: 99%

“…So far, we could have taken Mon as any commutative monoid equipped with EGB1 and EGB2. This viewpoint, and a generalisation thereof, is adopted in [11]. However, for doing computations we need that Mon has more structure; see conditions EGB3 and EGB4 below.…”

Section: Definition 12 (Equivariant Gröbner Basis) Letmentioning

confidence: 99%

Equivariant Gröbner bases and the Gaussian two-factor model

Brouwer

Draisma

2011

Math. Comp.

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Abstract. Exploiting symmetry in Gröbner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Gröbner basis in a setting where a monoid acts by homomorphisms on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Gröbner bases.Using this algorithm and the monoid of strictly increasing functions N → N we prove that the kernel of the ring homomorphismis generated by two types of polynomials: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.

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Finite Gröbner bases in infinite dimensional polynomial rings and applications

Cited by 77 publications

References 21 publications

FI- and OI-modules with varying coefficients

FI- and OI-modules with varying coefficients

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

Equivariant Gröbner bases and the Gaussian two-factor model

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