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

TL;DR

This paper investigates the asymptotic optimality of bounds on the nilpotency step of quadratic algebras, demonstrating that certain estimates are tight and related to Fibonacci numbers, with implications for algebraic structure analysis.

Contribution

The paper proves the asymptotic optimality of Golod--Shafarevich bounds on k-step nilpotency for quadratic algebras and connects these results to Fibonacci numbers.

Findings

Bounds are asymptotically optimal for k-step nilpotency.

The estimates are attained infinitely often for various n.

Results relate algebraic properties to Fibonacci numbers.

Abstract

It follows from the Golod--Shafarevich theorem that if R is an associative algebra given by n generators and $d<\frac{n^2}{4}\cos^{-2}(\frac{\pi}{k+1})$ quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal, and establish number of related results. For example, we show that for any k this estimate is attained for ifinitely many n.