# Waring number for a word

## Definition

### In the group sense

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

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

### In the group sense

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

The **Waring number** in the monoid sense for is defined as the smallest such that every element of can be expressed as a product of length at most involving elements of . The key difference with the previous definition is that we do not include inverses. Note that if 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).