On systems of word equations with simple loop sets

contains one occurrence of p. There are nine words α ∈ H ∗ satisfying ϕ(α) =
aa, namely
α 1 = x 1 x 1 , α 2 = x 1 y 1 , α 3 = x 1 z 2 ,
α 4 = y 1 x 1 , α 5 = y 1 y 1 , α 6 = y 1 z 2 ,
α 7 = z 2 x 1 , α 8 = z 2 y 1 , α 9 = z 2 z 2 .
Therefore (13) has the form
9∑
9∑
|ψ(yzxy)| αi = |ψ(xyyz)| αi . (14)
i=1
i=1
The equality holds, since |ψ(yzxy)| αi is equal to one for i = 7, and is zero
otherwise, while |ψ(xyyz)| αi is one just for i = 5.
Informally, we can say that the factor aa comes on the left side of the equation
from a different source than on the right side. The formalism of the characteristic
equation is designed to express and exploit that fact.
5 One loop systems
We are now ready to prove Theorem 2. The theorem deals with the systems
S and T k when n = 1, hence we define
X = {u 1 ,u 2 , . . .,u m ,v 1 ,x 0 ,x 1 , . . .,x m ,y 0 ,y 1 }.
Fix k ≥ 2, and a morphism ϕ, which solves the system T k . Define H, morphisms
ψ and ϕ as in the previous section. Our task is to show that ϕ
solves S as well. It will be done by showing that the primitive roots of all
ϕ(u 1 ), ϕ(u 2 ), . . .,ϕ(u m ) are conjugate. Theorem 1 then applies.
Denote
l i = x 0 u i 1x 1 u i 2x 2 · · ·u i mx m ,
r i = y 0 v i y 1
for i = k, k + 1, k + 2. Recall that, for each i,
(l i , r i ) = (ψ(l i ), ψ(r i ))
is the characteristic equation of (l i , r i ) with respect to ϕ.
Define the word p, whose number of occurrences will be counted. Let t be the
shortest among the primitive roots of words ϕ(u 1 ), ϕ(u 2 ), . . . , ϕ(u m ). Then p
is defined by the following conditions:
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