Waring number for a word

Definition

In the group sense

Suppose $G$ is a group, $w$ is a word in $n$ letters, and $H$ is the verbal subgroup of $G$ generated by the image of the word map $w$ in $G$, i.e., $H$ has as a generating set the set:

$A=\{w(g_{1},g_{2},\dots ,g_{n})\mid (g_{1},g_{2},\dots ,g_{n})\in G^{n}\}$

The Waring number in the group sense for $w$ is defined as the diameter for the generating set $A$ of $H$, i.e., it is the smallest $k$ such that every element of $H$ can be expressed as a product of length at most $k$ involving elements of $A$ and their inverses.

In the group sense

Suppose $G$ is a group, $w$ is a word in $n$ letters, and $H$ is the verbal subgroup of $G$ generated by the image of the word map $w$ in $G$, i.e., $H$ has as a generating set the set:

$A=\{w(g_{1},g_{2},\dots ,g_{n})\mid (g_{1},g_{2},\dots ,g_{n})\in G^{n}\}$

The Waring number in the monoid sense for $w$ is defined as the smallest $k$ such that every element of $H$ can be expressed as a product of length at most $k$ involving elements of $A$. The key difference with the previous definition is that we do not include inverses. Note that if $A$ is a symmetric subset (as is the case with the images of most word maps of interest to us) then these two Waring numbers are the same).