标题： 高亏格多重ζ值 显示英文标题

标题： Higher-genus multiple zeta values

摘要： 多重泽塔值作为定义在不同亏格黎曼曲面上的多对数函数的特殊值出现。 在已有经典和椭圆多重泽塔值的大量知识基础上，我们探讨了将形式扩展到更高亏格黎曼曲面的规范方法，这产生了高亏格多重泽塔值。 我们提供了高亏格多对数函数的正则化方案，并将其扩展到高亏格多重泽塔值。 我们的正则化使用施托克周期参数化，将高亏格端点正则化追溯到一亏格已知的正则化。 此外，我们评论了由底层几何退化引起的高亏格多重泽塔值之间的关系，其中我们区分了分离型和非分离型退化两种类型。 最后，利用施托克周期参数化中高亏格多对数函数的功能关系，我们探讨了高亏格多重泽塔值之间的关系，并将其与我们的数值测试设置进行对比。 我们发现了超越多对数恒等式所暗示的高亏格多重泽塔值的关系，从而与零亏格和一亏格的情况相匹配。 虽然我们发现了一些已知的椭圆多重泽塔值结构可以推广到高亏格多重泽塔值的关系，但还有其他类型的relations来自于不同循环之间的相互作用和组合学。 显示英文摘要

摘要： Multiple zeta values arise as special values of polylogarithms defined on Riemann surfaces of various genera. Building on the vast knowledge for classical and elliptic multiple zeta values, we explore a canonical extension of the formalism to Riemann surfaces of higher genera, which yields higher-genus multiple zeta values. We provide a regularization prescription for higher-genus polylogarithms, which we extend to higher-genus multiple zeta values. Our regularization uses the Schottky uniformization to trace back higher-genus endpoint regularization to known regularization at genus one. Additionally, we are commenting on relations among higher-genus multiple zeta values implied by degeneration of the underlying geometry, where we distinguish between the two types of separating and non-separating degeneration. Finally, employing functional relations for higher-genus polylogarithms in the Schottky uniformization, we explore relations among higher-genus multiple zeta values and check them against our numerical testing setup. We identify relations for higher-genus multiple zeta values beyond those implied by polylogarithm identities, thereby matching the situation for genus zero and genus one. While we find several known structures for elliptic multiple zeta values to generalize to relations for higher-genus multiple zeta values, there are further classes of relations arising from the interplay and combinatorics of different cycles.