no. 3
Algèbre
On cogrowth function of algebras and its logarithmical gap
[Sur la fonction de co-croissance des algèbres et son écart logarithmique]
Kanel-Belov, Alexei Ya.1 ; Melnikov, Igor2 ; Mitrofanov, Ivan3 
1 Bar Ilan University, Ramat-Gan, Israel
2 Moscow Institute of Physics and Technology, Dolgoprudny, Russia
3 C.N.R.S., École Normale Superieur, PSL Research University, France
Comptes Rendus. Mathématique, Tome 359 (2021) no. 3, pp. 297-303.

Soit A≅k〈X〉/I une algèbre associative. Un mot fini sur l’alphabet X est I-réductible si son image dans A est une combinaison linéaire k de mots de longueur lexicographiquement moindre. Une obstruction dans un mot minimal I -réductible. Si le nombre d’obstructions est fini, alors I a une base finie Gröbner, et le mot problème pour l’algèbre est décidable. Une fonction co-croissance est le nombre d’obstructions de longueur ≤n. Nous montrons que la fonction de co-croissance d’une algèbre finement présentée est soit bornée, soit au moins logarithmique. Nous montrons également qu’un mot uniformément récurrent a au moins une co-croissance logarithmique.

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-reducible word. 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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DOI : 10.5802/crmath.170
Kanel-Belov, Alexei Ya. 1 ; Melnikov, Igor 2 ; Mitrofanov, Ivan 3

1 Bar Ilan University, Ramat-Gan, Israel
2 Moscow Institute of Physics and Technology, Dolgoprudny, Russia
3 C.N.R.S., École Normale Superieur, PSL Research University, France 
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     title = {On cogrowth function of algebras and its logarithmical gap},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {297--303},
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     volume = {359},
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     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.170/}
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Kanel-Belov, Alexei Ya.; Melnikov, Igor; Mitrofanov, Ivan. On cogrowth function of algebras and its logarithmical gap. Comptes Rendus. Mathématique, Tome 359 (2021) no. 3, pp. 297-303. doi : 10.5802/crmath.170. http://www.numdam.org/articles/10.5802/crmath.170/

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