论文标题
通过Mordell-lang通过密集对的$（\ Mathbb {z}，+）$的DP-Minimal扩展
Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang
论文作者
论文摘要
这是对$（\ mathbb {z}，+）$的DP最低扩展的分类问题的贡献。令$ s $为$（\ mathbb {z}，+）$的密集循环组订单。我们在“密集对”上使用结果来构建$（\ mathbb {z}，+，s）$的许多DP最低扩展。这些构造是Mordell-Lang猜想的应用，是$（\ Mathbb {Z}，+）$的“非模块化” DP最低扩展的第一个示例。我们从典型地关联了$（\ Mathbb {r}，+，+，\ times）$的O-Minimal扩展$ \ Mathcal {r} $，$ \ Mathcal {r} $ - 定义圆圈组$ \ mathbbbb { $（\ mathbb {z}，+，s）$的DP-Minimal扩展。 We also construct a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,\mathrm{Val}_p)$ from the character $\mathbb{Z} \to \mathbb{Z}^\times_p$, $k \mapsto \mathrm{exp}(pk)$.
This is a contribution to the classification problem for dp-minimal expansions of $(\mathbb{Z},+)$. Let $S$ be a dense cyclic group order on $(\mathbb{Z},+)$. We use results on "dense pairs" to construct uncountably many dp-minimal expansions of $(\mathbb{Z},+,S)$. These constructions are applications of the Mordell-Lang conjecture and are the first examples of "non-modular" dp-minimal expansions of $(\mathbb{Z},+)$. We canonically associate an o-minimal expansion $\mathcal{R}$ of $(\mathbb{R},+,\times)$, an $\mathcal{R}$-definable circle group $\mathbb{H}$, and a character $\mathbb{Z} \to \mathbb{H}$ to a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,S)$. We also construct a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,\mathrm{Val}_p)$ from the character $\mathbb{Z} \to \mathbb{Z}^\times_p$, $k \mapsto \mathrm{exp}(pk)$.
