Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of [Formula: see text]. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim’s conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim’s conjecture.

中文翻译：

---

计算 Chabauty–Kim 理论 I 中的 polylog 商和 Goncharov 商

多对数是通过被称为多对数商的三次穿刺线的 de Rham 基本组的某个商的多个多对数。在 Dan-Cohen、Wewers 和 Brown 的工作的基础上，我们将 Kim 方法的显式动机版本的计算边界推到了 [公式：见文本] 的开放子方案上的三次穿刺线的情况下。为此，我们开发了一个专门针对这种情况量身定制的 Dan-Cohen 算法的改进版本，并将注意力集中在多对数商上。这使我们能够将具有动机迭代积分的微积分限制在 Goncharov 广泛研究的混合 Tate Galois 群的所谓深度一部分。我们还发现了多对数商的对称性破坏性质的一个有趣结果，它迫使我们对称化我们对金猜想的多对数版本。在这个由两部分组成的系列的第一部分中，我们专注于一个具体的例子，它使我们能够验证金猜想的一个有趣的新案例。