# Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers

Natalia Iyudu, Stanislav Shkarin

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

## Abstract

It follows from the Golod–Shafarevich theorem that if $k\in{\mathbb N}$ and $R$ is an associative algebra given by $n$ generators and $d<n^2\cos^{-2}(\pi/(k+1))/4$ quadratic relations, then $R$ is not $k$-step nilpotent. We show that the above estimate is asymptotically optimal. Namely, for every $k$ there is a sequence of algebras $R_n$ given by $n$ generators and $d_n$ quadraticrelations such that $R_n$ is k-step nilpotent and $\lim_{n\to\infty d_n/n^2=\cos^{-2}(\pi/(k+1))/4$.
Original language English 465–479 15 Combinatorica 37 3 17 Oct 2016 https://doi.org/10.1007/s00493-016-3009-6 Published - Jun 2017

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