  We introduce a fragment of second-order unification, referred to as
\emph{Second-Order Ground Unification (SOGU)}, with the following properties:
(i) only one second-order variable is allowed, and (ii) first-order variables
do not occur. We study an equational variant of SOGU where the signature
contains \textit{associative} binary function symbols (ASOGU) and show that
Hilbert's 10$^{th}$ problem is reducible to ASOGU unifiability, thus proving
undecidability. Our reduction provides a new lower bound for the undecidability
of second-order unification, as previous results required first-order variable
occurrences, multiple second-order variables, and/or equational theories
involving \textit{length-reducing} rewrite systems. Furthermore, our reduction
holds even in the case when associativity of the binary function symbol is
restricted to \emph{power associative}, i.e. f(f(x,x),x)= f(x,f(x,x)), as our
construction requires a single constant.
