  Let $S(X,B)$ be a symmetric (``palindromic'') word in two letters $X$ and
$B$. A theorem due to Hillar and Johnson states that for each pair of positive
definite matrices $B$ and $P$, there is a positive definite solution $X$ to the
word equation $S(X,B)=P$. They also conjectured that these solutions are finite
and unique. In this paper, we resolve a modified version of this conjecture by
showing that the Brouwer degree of such an equation is equal to 1 (in the case
of real matrices). It follows that, generically, the number of solutions is odd
(and thus finite) in the real case. Our approach allows us to address the more
subtle question of uniqueness by exhibiting equations with multiple real
solutions, as well as providing a second proof of the result of Hillar and
Johnson in the real case.
