The study of Locally Checkable Labelings (LCLs) has led to a remarkably precise characterization of the distributed time complexities that can occur on bounded-degree trees. A central feature of this complexity landscape is the existence of strong gap results, which rule out large ranges of intermediate complexities. While it was initially hoped that these gaps might extend to more general graph classes, this has turned out not to be the case. In this work, we investigate a different direction: we remain in the class of trees, but allow arbitrarily large degrees. We focus on the polynomial regime ($Θ(n^{1/k} \mid k \in \mathbb{N})$) and show that whether polynomial gap results persist in the unbounded-degree setting crucially depends on how LCLs are generalized beyond bounded degrees. We first demonstrate that if one allows LCLs to be defined using infinitely many local configurations, then the polynomial gaps disappear entirely: for every real exponent $0 < r \leq 1$, there exists a locally checkable problem on trees with deterministic LOCAL complexity $Θ(n^r)$. Rather than stopping at this negative result, we identify a natural class of problems for which polynomial gap results can still be recovered. We introduce Locally Finite Labelings (LFLs), which formalize the intuition that ''every node must fall into one of finitely many local cases'', even in the presence of unbounded degrees. Our main result shows that this restriction is sufficient to restore the polynomial gaps: for any LFL $Π$ on trees with unbounded degrees, the deterministic LOCAL complexity of $Π$ is either - $Θ(n^{1/k})$ for some integer $k \geq 1$, or - $O(\log n)$. Moreover, which case applies, and the corresponding value of $k$, can be determined solely from the description of $Π$.

翻译：局部可检查标记（LCL）的研究已对有界度树上的分布式时间复杂性给出了极为精确的表征。这一复杂性图景的核心特征是存在强间隙性结果，排除了大范围的中间复杂性。尽管最初希望这些间隙能够推广到更一般的图类，但事实并非如此。在本工作中，我们探索了一个不同的方向：我们仍停留在树类结构中，但允许任意大的节点度。我们聚焦于多项式区域（$Θ(n^{1/k} \mid k \in \mathbb{N})$），并证明多项式间隙结果在无界度设置下是否持续存在，关键取决于LCL如何推广到有界度之外。我们首先证明，如果允许LCL使用无限多个局部配置来定义，则多项式间隙完全消失：对于每个实数指数$0 < r \leq 1$，都存在一个树上的局部可检查问题，其确定性LOCAL复杂度为$Θ(n^r)$。我们并未止步于这一否定性结果，而是识别出了一类自然问题，其多项式间隙结果仍可恢复。我们引入了局部有限标记（LFL），该形式化定义捕捉了“每个节点必须属于有限多个局部情形之一”的直观思想，即使在无界度条件下亦然。我们的主要结果表明，这一限制足以恢复多项式间隙：对于无界度树上的任意LFL问题$Π$，其确定性LOCAL复杂度要么是$Θ(n^{1/k})$（其中$k \geq 1$为整数），要么是$O(\log n)$。此外，哪种情况适用以及对应的$k$值，均可仅从$Π$的形式描述中确定。