An algorithm for finding symmetric Grobner bases in infinite dimensional rings

Aschenbrenner, Matthias; Hillar, Christopher J.

doi:10.1145/1390768.1390787

Cited by 5 publications

(11 citation statements)

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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: 97%

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

confidence: 97%

Equivariant Gröbner bases and the Gaussian two-factor model

Brouwer

Draisma

2011

Math. Comp.

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“…What follows is certainly not the most general setting, but it will suffice for our purposes. For much more on this theme see [Coh67,Coh87,AH07,AH08,HS12,HMdC13].…”

Section: Equivariant Gröbner Basesmentioning

confidence: 99%

Noetherianity up to Symmetry

Draisma

2014

Lecture Notes in Mathematics

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Abstract. 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.

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“…However, most of the common packages implementing noncommutative Gröbner bases do not support such cases [55,56]. For some recent advances, we refer the reader to [3,14,43,51] as well as Ufnarovski's extensive survey chapter [86].…”

Section: Reduction Rings and Gröbner Basesmentioning

confidence: 99%

“…We study first some direct consequences of the section axiom (3). For further details on linear left and right inverses, we refer for example to [13, p. 211] or to [63] in the context of generalized inverses.…”

Section: Definitionmentioning

confidence: 99%

“…Passing to the quotient yields the shuffle product · : Since the multiplication on F and the shuffle product are commutative, associative, and distributive over addition, the multiplication on [R] is well-defined and gives [R] the structure of a commutative K-algebra. The definition of a derivation ∂ on this algebra is straightforward, using the fact that it should respect K-linearity and the Leibniz rule (treating also the u(0) α as constants), that it should restrict to the derivation on differential polynomials (which in turn restricts to the derivation on F ), and finally that it should also satisfy the section axiom (3). Here is a sample computation:…”

Section: Canonical Forms For Integro-differentialmentioning

confidence: 99%

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Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases

Rosenkranz

Regensburger

Tec³

et al. 2011

Texts &Amp; Monographs in Symbolic Computation

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We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.

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An algorithm for finding symmetric Grobner bases in infinite dimensional rings

Cited by 5 publications

References 11 publications

Equivariant Gröbner bases and the Gaussian two-factor model

Equivariant Gröbner bases and the Gaussian two-factor model

Noetherianity up to Symmetry

Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases

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