Additive primitive length in relatively free algebras

Abstract

The additive primitive length of an element [Formula: see text] of a relatively free algebra [Formula: see text] in a variety of algebras [Formula: see text] is equal to the minimal number [Formula: see text] such that [Formula: see text] can be presented as a sum of [Formula: see text] primitive elements. We give an upper bound for the additive primitive length of the elements in the [Formula: see text]-generated polynomial algebra over a field of characteristic 0, [Formula: see text]. The bound depends on [Formula: see text] 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 [Formula: see text]-generated nilpotent-by-abelian Lie algebras is bounded by 5 for [Formula: see text] and by 6 for [Formula: see text]. If the field has two elements only, then our bounds are 6 for [Formula: see text] and 7 for [Formula: see text]. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.