We consider the fragment of Second-Order unification, referred to as \emph{Second-Order Ground Unification (SOGU)}, with the following properties: (i) only one second-order variable allowed, (ii) first-order variables do not occur. We show that Hilbert's 10$^{th}$ problem is reducible to a \emph{necessary condition} for SOGU unifiability if the signature contains a binary function symbol and two constants, thus proving undecidability. This generalizes known undecidability results, as either first-order variable occurrences or multiple second-order variables were required for the reductions. Furthermore, we show that adding the following restriction: (i) the second-order variable has arity 1, (ii) the signature is finite, and (iii) the problem has \emph{bounded congruence}, results in a decidable fragment. The latter fragment is related to \emph{bounded second-order unification} in the sense that the number of bound variable occurrences is a function of the problem structure. We conclude with a discussion concerning the removal of the \emph{bounded congruence} restriction.

翻译：我们研究了二阶合一的子片段，称为*二阶基合一（SOGU）*，其性质如下：（i）仅允许一个二阶变量，（ii）不出现一阶变量。我们证明了如果签名包含一个二元函数符号和两个常量，则希尔伯特第10问题可归约至SOGU可合一的*必要条件*，从而证明其不可判定性。这一结果推广了已知的不可判定性结论——此前归约需要一阶变量出现或多个二阶变量。此外，我们证明添加以下限制：（i）二阶变量的元数为1，（ii）签名为有限，（iii）问题具有*有界同余*，将得到一个可判定片段。后者片段与*有界二阶合一*相关，因为约束变量出现次数是问题结构的函数。最后，我们讨论了移除*有界同余*限制的可能性。