论文标题
在Goncharov深度的猜想和矫正量的公式
On the Goncharov depth conjecture and a formula for volumes of orthoschemes
论文作者
论文摘要
我们证明了Goncharov的猜想，该猜想说，任何多个多毛体可以通过深度的各个重量来表达。我们给出了此演示文稿的明确公式，涉及对对应于多边形分解为四边形的分解的树的求和。 我们的第二个结果是夸张矫正器量的公式，将lobachevsky的公式在尺寸$ 3 $中概括为任意维度。我们在两个结果之间显示出令人惊讶的关系，这是由于双曲线矫正器是通过$ \ mathbb {p}^^1的点配置参数化的事实。
We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension $3$ to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on $\mathbb{P}^1.$ In particular, we derive both formulas from their common generalization.
