Standard Relations of Multiple Polylogarithm Values at Roots of Unity

Details Standard Relations of Multiple Polylogarithm Values at Roots of Unity Standard Relations of Multiple Polylogarithm Values at Roots of Unity

2007-07-10

arxiv.org/abs/0707.1459v7

Documenta Mathematica 15 (2010), pp. 1-34

Mathematics

Number Theory

By numerical computation we propose more non-standard relations

Scientific paper

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 from lower weights, and seeded relations which are produced by relations of weight one MPVs. We call relations from the above four families \emph{standard}. Let $d(w,N)$ be the $\Q$-dimension of $\Q$-span of all MPVs of weight $w$ and level $N$. Then we obtain upper bound for $d(w,N)$ by the standard relations which in general are no worse or no better than the one given by Deligne and Goncharov depending on whether $N$ is a prime-power or not, respectively, except for 2- and 3-powers, in which case standard relations seem to be often incomplete whereas Deligne shows that their bound should be sharp by a variant of Grothedieck's period conjecture. This suggests that in general there should be other linear relations among MPVs besides the standard relations, some of which are written down in this paper explicitly with good numerical verification. We also provide a few conjectures which are supported by our computational evidence.

Affiliated with

Also associated with

No associations

Rating

Standard Relations of Multiple Polylogarithm Values at Roots of Unity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Standard Relations of Multiple Polylogarithm Values at Roots of Unity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Standard Relations of Multiple Polylogarithm Values at Roots of Unity will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-453790