Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7

Explicit formulas for Grassmannian polylogarithms

We give new explicit formulas for Grassmannian and Aomoto polylogarithms in terms of iterated integrals, for arbitrary weight. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of $\mathrm{GL}_4$ in terms of $\mathrm{Li}_4$.

权重6多重多对数函数的对称性与Goncharov深度猜想

Symmetries of weight 6 multiple polylogarithms and Goncharov's Depth Conjecture

本文证明了权重6深度3的多重多对数函数满足二对数函数的非调和对称性，从而确立了Goncharov深度猜想在权重6深度3情形下的成立。

On the Goncharov Depth Conjecture and polylogarithms of depth two

We prove the surjectivity part of Goncharov's depth conjecture. We also show that the depth conjecture implies that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function $\mathrm{Li}_{n-d+1,1,\dots,1}(a_1,a_2,\dots,a_d)$, and we prove this latter statement for $d=2$.

Sum formulas for double polylogarithms with a shifting parameter and their derivatives

We prove sum formulas for double polylogarithms of Hurwitz type, that is, involving a shifting parameter $b$ in the denominator. These formulas especially imply well-known sum formulas for double zeta values, and sum formulas for double $L$-values. Further, differentiating in $b$, we obtain a kind of weighted sum formula for double polylogarithms and double $L$-values. We also give sum formulas for partial double zeta values with some congruence conditions. Our proofs of those sum formulas are based on certain functional relations for double polylogarithms of Hurwitz type.

Standard Relations of Multiple Polylogarithm Values at Roots of Unity

Let $N$ be a positive integer. In this paper we shall study the special values of multiple polylogarithms at $N$th roots of unity, called multiple polylogarithm values (MPVs) of level $N$. These objects are generalizations of multiple zeta values and alternating Euler sums, which was studied by Euler, and more recently, many mathematicians and theoretical physicists.. Our primary goal in this paper is to investigate the relations among the MPVs of the same weight and level by using the regularized double shuffle relations, regularized distribution relations, lifted versions of such relations f…

关于多重polylogarithms的Landen公式及其ℓ-adic伽罗瓦类比

On the Landen formula for multiple polylogarithms and its $\ell$-adic Galois analogue

本文通过代数和几何方法证明了复数多重polylogarithms的Landen公式，并将其推广到ℓ-adic伽罗瓦设定，引入了特定于该设定的低阶项。

Special Values of Generalized Polylogarithms

We study values of generalized polylogarithms at various points and relationships among them. Polylogarithms of small weight at the points 1/2 and -1 are completely investigated. We formulate a conjecture about the structure of the linear space generated by values of generalized polylogarithms.

Multiple polylogarithms in weight 4

We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,...,n_d}(z_1,...,z_d), in depth d>1 and weight \sum_i n_i=4 in terms of the classical tetralogarithm Li_4. In the process, we prove a statement conjectured by Goncharov which can be rephrased as writing the sum of iterated integrals I_{3,1}(V(x,y),z), where V(x,y) denotes a formal version of the five term relation for the dilogarithm, in terms of Li_4-terms (we need 122 such).

Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six

We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form $G(a_1,\ldots,a_w;1)$ where the indices $a_i$ are equal to zero or a sixth root of unity, with $a_1\neq 1$. For $w\leq 6$, we present bases of the linear spaces generated by the real and imaginary parts of $G(a_1,\ldots,a_w;1)$ and present a table for expressing them as linear combinations of the elements of the bases.

Variations of mixed Hodge structures of multiple polylogarithms

It's well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications: First we study of the limit mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}logarithms. Then following Deligne and Beilinson we …

多重多对数的李余代数

The Lie coalgebra of multiple polylogarithms

本文利用Goncharov的余积在任意域上构造多重多对数的李余代数，通过归纳定义的反演与洗牌关系生成符号，并将其与Bloch群及权小于5的具体模型建立联系。

A review of Dan's reduction method for multiple polylogarithms

In this paper we will give an account of Dan's reduction method for reducing the weight $ n $ multiple logarithm $ I_{1,1,\ldots,1}(x_1, x_2, \ldots, x_n) $ to an explicit sum of lower depth multiple polylogarithms in $ \leq n - 2 $ variables. We provide a detailed explanation of the method Dan outlines, and we fill in the missing proofs for Dan's claims. This establishes the validity of the method itself, and allows us to produce a corrected version of Dan's reduction of $ I_{1,1,1,1} $ to $ I_{3,1} $'s and $ I_4 $'s. We then use the symbol of multiple polylogarithms to answer Dan's question …

A Weight-Depth Theorem for a Class of Multiple L-values

An arbitrary-depth reduction theorem for the `convolution' multiple L-values of Euler-Zagier type is proven by an analytic method. To this end, generalized polylogarithms associated to Dirichlet characters are defined. The proof uses the monodromies of these functions and the parities of generating functions of generalized Bernoulli numbers. Some general formulas for resulting reductions in the depth 2 case are provided.

某些多重对数的伽罗瓦下降

Galois descents of certain multiple polylogarithms

该研究证明了K. C. Au关于多重对数和割圆级下降的猜想，还确认了Broadhurst关于偶权荣誉多重zeta值的猜想。

多重对数函数与Steinberg模

Multiple polylogarithms and the Steinberg module

本文建立环面上多重对数函数与$\mathbb{Q}$的Steinberg模的联系，给出特定深度权重多重对数的单函数表示形式，据此简化Bykovskiĭ定理证明、解释相关对偶性并为两个猜想提供多重对数解释。

Dickman多重多重对数函数与Lindemann-Furry通信

Dickman multiple polylogarithms and the Lindemann-Furry letters

本研究用可快速计算的多重多重对数函数展开Dickman函数，求解Buchstab微分延迟方程，梳理二战期间Lindemann与Furry的相关通信，并将Furry的分析拓展至权重最高200的多重多重对数函数。

迭代积分与代数参数的多重polylogarithm

Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments

本文通过迭代积分与广义多重ζ值理论，揭示了代数参数下多重polylogarithm与cyclotomic 多重ζ值之间的等价关系，并提出Galois下降猜想及Apéry型级数的应用。

多重非正指数多项对数与Magnus多项式的组合学

Multiple polylogarithms at non-positive indices and combinatorics of Magnus polynomials

本文通过Magnus多项式与非正多重多项对数的对应关系，揭示了非正MPLs的组合结构及其与非交换代数的联系。

Multiple polylogarithm values at roots of unity

For any positive integer $N$ let $μ_N$ be the group of the $N$th roots of unity. In this note we shall study the $\Q$-linear relations among values of multiple polylogarithms evaluated at $\mmu_N$. We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N=4, weight $w=3$ or 4, and (ii) $w=2$, $7<N<50$, and $N$ is a power of 2 or 3, or $N$ has at least two prime factors. We further find some (presumably all) of the missing relations in (i) by using the octahedral symmetry of $¶^1-(\{0,\infty\}\cup μ_4)$. We also prove…

Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analog…

On the Goncharov depth conjecture and a formula for volumes of orthoschemes

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 p…

Multiple zeta value cycles in low weight

In a recent work, the author has constructed two families of algebraic cycles in Bloch cycle algebra over the prjective line minus 3 points that are expected to correspond to multiple polylogarithms in one variable and have a good specialization at 1 related to multiple zeta values. This is a short presentation, by the way of toy examples in low weight (5), of this contruc- tion and could serve as an introduction to the general setting. Working in low weight also makes it possible to push ("by hand") the construction further. In particular, we will not only detail the construction of the cycle…

Cycle complex over the projective line minus three points : toward multiple zeta values cycles

In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for…