Summary: 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 $N$. 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 from lower weights, and weight one relations which are produced by relations of weight one MPVs. We call relations from the above four families standard. Let $d(w,N)$ be the dimension of the $\Q$-vector space generated by all MPVs of weight $w$ and level $N$. Recently Deligne and Goncharov were able to obtain some lower bound of $d(w,N)$ using the motivic mechanism. We call a level $N$ standard if $N=1,2,3$ or $N=p^n$ for prime $p\ge 5$. Our computation suggests the following dichotomy: If $N$ is standard then the standard relations should produce all the linear relations and if further $N>3$ then the bound of $d(w,N)$ by Deligne and Goncharov can be improved; otherwise there should be non-standard relations among MPVs for all sufficiently large weights (depending only on $N$) and the bound by Deligne and Goncharov may be sharp. We write down some of the non-standard relations explicitly with good numerical verification. In two instances ($N=4, w=3,4$) we can rigorously prove these relations by using the octahedral symmetry of ${0,\infty,\pm1,\pm\sqrt{-1}}$. Throughout the paper we provide many conjectures which are strongly supported by computational evidence.

11R42, 11Y40, 33B30

multiple polylogarithms, double shuffle relations, distribution relations, regularization, standard relations