A topological structure on certain initial algebras

Anderson, Sarah E.; Smith, Andrew G.; Stewart, Peter; Tesemma, Mohammed; Usatine, Jeremy

doi:10.1016/j.topol.2014.11.008

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…Moreover, #{in k[x] G | ∈ Ω 0 } is equal to |G| if G is generated by transpositions, and #R otherwise, where Ω 0 is the set of monomial orders on k[x] (cf. Kuroda's papers cited above, Tesemma [17] and Anderson et al [1]). This result is interesting because #{in [15]).…”

Section: Introductionmentioning

confidence: 99%

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A new class of finitely generated polynomial subalgebras without finite SAGBI bases

Kuroda¹

2021

Preprint

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The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known.The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is continuum, as well as a subalgebra with finitely many distinct initial algebras.

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Section: Introductionmentioning

confidence: 99%

“…This result is interesting because #{in [15]). We mention that [6], [7], [11], [3], [17] and [1] treated more general classes of invariant rings than (1.1).…”

Section: Introductionmentioning

confidence: 99%

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

Kuroda¹

2021

Preprint

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A new class of finitely generated polynomial subalgebras without finite SAGBI bases

Kuroda

2022

Proc. Amer. Math. Soc.

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The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras.

show abstract

A topological structure on certain initial algebras

Cited by 2 publications

References 6 publications

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

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