AI 中文总结
受Bugeaud - Corvaja - Zannier启发，对数域整数环上乘法群幂中的非平凡交集提出猜想。先获平凡交集结果作基准并推广前人工作，证明维数1时猜想，给出维数2部分结果及相关开放问题。
AI 中文摘要
受Bugeaud - Corvaja - Zannier工作的启发，我们针对数域整数环上乘法群幂中的非平凡交集提出一个猜想。大致而言，若因维数原因与子群概型的交集是非平凡的，其‘大小’相较于子群概型的‘复杂度’不应太大。我们首先得到一些关于平凡交集的结果作为非平凡情形的基准，并推广了Barroero - Capuano - Mérai - Ostafe - Sha的工作。接着表明维数1时的猜想可由Corvaja - Zannier的工作推出，在维数2得到一些部分结果，还给出了该猜想的一些特殊情况的开放问题。
英文摘要
Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its ``size" should not be too big compared to the ``complexity" of the subgroup scheme. We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-Mérai-Ostafe-Sha. We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.
CommentsAppendix written in collaboration with Robert Wilms. The preprint consists of 31 pages and 2 figures
