Waring number for a word

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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).