# Polynomial bound for the nilpotency index of finitely generated nil algebras

## TL;DR

This paper establishes a polynomial upper bound for the nilpotency index of finitely generated nil algebras over infinite fields of positive characteristic, linking it to invariants of matrix conjugation and extending previous bounds.

## Contribution

It provides a new polynomial bound for the nilpotency index based on invariants of matrix conjugation, improving understanding of finitely generated nil algebras.

## Key findings

- Polynomial bound for nilpotency index in terms of matrix invariants
- Connection between nilpotency index and degree bounds of matrix invariants
- Characteristic-free lower bound for nilpotency index

## Abstract

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of $n$ by $n$ matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin's lower bound for the nilpotency index.