Stuffle product of multiple zeta values and regularizations

[videlec]

First of all, it should be possible to have access to both shuffle and stuffle product in the motivic algebra of MZV.

Next, the evaluation Z: y_s -> ζ(s) extends in two distinct algebra morphisms to divergent MZV, eg including ζ(1). In other words, these two extensions do not satisfy in general the relation

Z_stuffle(y_s1 stuffle y_s2) = Z_shuffle(y_s1 shuffle y_s2)

(which is nevertheless valid on convergent symbols s1 and s2). However, there is a simple way to go around that as explained in Ihara-Kaneko-Zagier (2006).

CC: @fchapoton

Component: algebra

Issue created by migration from https://trac.sagemath.org/ticket/30992

[videlec]

Description changed:

--- 
+++ 
@@ -1,8 +1,8 @@
 First of all, it should be possible to have access to both shuffle and stuffle product in the motivic algebra of MZV.

-Next, the evaluation `Z: y_s -> ζ(s)` extends in two distinct algebra morphisms to divergent MZV, eg including ζ(1). In other words, these two extensions do not satisfy the relation
+Next, the evaluation `Z: y_s -> ζ(s)` extends in two distinct algebra morphisms to divergent MZV, eg including ζ(1). In other words, these two extensions do not satisfy in general the relation

Z_stuffle(y_s1 stuffle y_s2) = Z_shuffle(y_s1 shuffle y_s2)

-which is valid on convergent symbols `s1` and `s2`. However, there is a simple way to go around that as explained in [Ihara-Kaneko-Zagier (2006)](https://www.cambridge.org/core/journals/compositio-mathematica/article/derivation-and-double-shuffle-relations-for-multiple-zeta-values/5DD6B349C751EE191011123287B3C031).
+(which is nevertheless valid on convergent symbols `s1` and `s2`). However, there is a simple way to go around that as explained in [Ihara-Kaneko-Zagier (2006)](https://www.cambridge.org/core/journals/compositio-mathematica/article/derivation-and-double-shuffle-relations-for-multiple-zeta-values/5DD6B349C751EE191011123287B3C031).

[fchapoton]

comment:2

BEWARE : I am not sure if it is known whether the motivic MZV satisfy the double shuffle relations. What is known is that their images by the period map do.

I remember some work by Soudères on the question, that one should check.

[fchapoton]

comment:3

Apparemment le travail de Soudères porte sur les MZV motiviques à la Goncharov, moins fines que celles à la Brown.

https://mathscinet-ams-org.scd-rproxy.u-strasbg.fr/mathscinet-getitem?mr=2646761

Donc on ne peut pas conclure, sauf si quelqu'un d'autre a fait le boulot depuis.

[mkoeppe]

comment:4

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.