# Additive primitive length in relatively free algebras

## TL;DR

This paper establishes bounds on the additive primitive length of elements in various relatively free algebras, providing explicit bounds depending on algebra type, number of generators, and field characteristics, with efficient constructive methods.

## Contribution

It introduces new bounds for additive primitive length in polynomial and Lie algebras, extending previous results and providing effective algorithms for primitive element decompositions.

## Key findings

- Bound depends on algebra type, generators, and field characteristic.
- Primitive decompositions can be found efficiently in polynomial time.
- Results generalize recent findings for free metabelian Lie algebras.

## Abstract

The additive primitive length of an element $f$ of a relatively free algebra $F_d$ in a variety of algebras is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. We give an upper bound for the additive primitive length of the elements in the $d$-generated polynomial algebra over a field of characteristic 0, $d>1$. The bound depends on $d$ and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free $d$-generated nilpotent-by-abelian Lie algebras is bounded by 5 for $d=3$ and by 6 for $d>3$. If the field has two elements only, then our bound are 6 for $d=3$ and 7 for $d>3$. This generalizes a recent result of Ela Ayd{\i}n for two-generated free metabelian Lie algebras. In all cases considered in the paper the presentation of the elements as sums of primitive can be found effectively in polynomial time.