On cogrowth function of algebras and its logarithmical gap

Alexei Ya Kanel-Belov; Igor Melnikov; Ivan Mitrofanov

doi:10.5802/CRMATH.170

On cogrowth function of algebras and its logarithmical gap

Translated title of the contribution: On cogrowth function of algebras and its logarithmical gap

Alexei Ya Kanel-Belov
, Igor Melnikov
, Ivan Mitrofanov

Department of Mathematics

Moscow Institute of Physics and Technology
Université PSL

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let A ≅ k 〉X〈/I be an associative algebra. A finite word over alphabet X is I -reducible if its image in A is a k-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal I - reducibleword. If the number of obstructions is finite then I has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.

Translated title of the contribution	On cogrowth function of algebras and its logarithmical gap
Original language	English
Pages (from-to)	297-303
Number of pages	7
Journal	Comptes Rendus Mathematique
Volume	359
Issue number	3
DOIs	https://doi.org/10.5802/CRMATH.170
State	Published - 2021

Bibliographical note

Publisher Copyright:
© 2021 Académie des sciences, Paris and the authors.

Funding

Funding. The paper was supported by Russian Science Foundation (grant no. 17-11-01377).

Funders	Funder number
Russian Science Foundation	17-11-01377

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10.5802/CRMATH.170

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title = "On cogrowth function of algebras and its logarithmical gap",

abstract = "Let A ≅ k 〉X〈/I be an associative algebra. A finite word over alphabet X is I -reducible if its image in A is a k-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal I - reducibleword. If the number of obstructions is finite then I has a finite Gr{\"o}bner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.",

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N2 - Let A ≅ k 〉X〈/I be an associative algebra. A finite word over alphabet X is I -reducible if its image in A is a k-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal I - reducibleword. If the number of obstructions is finite then I has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.

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On cogrowth function of algebras and its logarithmical gap

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