Multiple zeta algebra

[rwst]

Old description:

There is code in Pynac's zeta2_eval allowing lst arguments. Let's check if that can be interfaced to Python. Is there numerics available in mpmath or arb?

https://en.wikipedia.org/wiki/Multiple_zeta_function

New description:

This ticket is now used to implement the algebra of MZV, using algorithms of Francis Brown.

Sample computation:

sage: Z = Multizeta
sage: M = matrix(2,2,[Z(2),Z(3),Z(4),Z(5)])
sage: M.det()
-10*ζ(1,6) - 5*ζ(2,5) - ζ(3,4) + ζ(4,3) + ζ(5,2)

CC: @videlec @tscrim @kedlaya @dwbump @JohnCremona

Component: number theory

Keywords: motives

Author: Frédéric Chapoton

Branch/Commit: c9e3b4c

Reviewer: Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/22713

[fchapoton]

comment:1

see also #18010

[fchapoton]

Branch: u/chapoton/27713

[fchapoton]

Commit: 67d78cc

[fchapoton]

New commits:

67d78cc

first sketch of MZV function

[sagetrac-git]

Changed commit from 67d78cc to 4e22cc3

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4e22cc3

more work on multiple zeta values

[sagetrac-git]

Changed commit from 4e22cc3 to 513688c

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

513688c

trying to use precision

[fchapoton]

Changed commit from 513688c to 63096a3

[fchapoton]

comment:6

Here is a first advanced sketch for an implementation of the ring of Multiple Zeta values. This in fact implements the motivic MZV, in some sense, following the algorithm given by Francis Brown.

I would appreciate comments and suggestions.

---

New commits:

63096a3

first sketch for a ring of multiple zeta values

[fchapoton]

Changed branch from u/chapoton/27713 to u/chapoton/ticket22713

[sagetrac-git]

Changed commit from 63096a3 to d825c5a

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d825c5a

details

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

47d0d72	first sketch for a ring of multiple zeta values
b66a693	more doc for MZV

[sagetrac-git]

Changed commit from d825c5a to b66a693

[fchapoton]

comment:10

Do you know anybody that could be interested in this stuff, and would review ?

[tscrim]

comment:11

Replying to @fchapoton:

Do you know anybody that could be interested in this stuff, and would review ?

I am not sure. I have cc-ed Dan Bump, who might be willing to or know someone who could. I can do the technical review and would be willing to set a positive review if nobody else is willing to check the mathematics. However, I would rather have a mathematical review.

[videlec]

comment:12

Actually, I thought about implementing the same thing! I can start the review next week (or maybe even this week-end).

[fchapoton]

Changed branch from u/chapoton/ticket22713 to public/22713

[fchapoton]

Changed commit from d9ce5ae to be3691c

[sagetrac-git]

Changed commit from be3691c to 1203ebf

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

a60c688	trac 22713 adding a simplify procedure using duality
37a5667	remove automatic simplification
7887eb2	have .simplify available
d9ce5ae	Merge branch 'public/22713' of ssh://trac.sagemath.org:22/sage into mzv
4c2cf94	some fixes and documentation to numerical MZV
1203ebf	100% doctest coverage

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b295b27

trac 22713, tiny details, one more reference

[sagetrac-git]

Changed commit from 1203ebf to b295b27

[fchapoton]

comment:90

Coverage is ok.

So, what remains to be done ?

[videlec]

comment:91

One feature which is not advertized in the main docstring of the module is the numerical approximation. I think it deserves a mention.

Otherwise I believe it is ready for inclusion.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f6e69bf	Merge branch 'public/22713' of ssh://trac.sagemath.org:22/sage into mzv
08c2b81	trac 22713 adding more doc about numerical approximation aka period map

[sagetrac-git]

Changed commit from b295b27 to 08c2b81

[fchapoton]

comment:93

I have added a little more doc about numerical approximation.

• Should I move references to master ref file ?

• I am tempted to squash all commits. May I ?

• Anything else ?

[videlec]

comment:94

Replying to @fchapoton:

I have added a little more doc about numerical approximation.

• I am tempted to squash all commits. May I ?

Sure

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

c9e3b4c

implementation of the ring of motivic multiple zeta values

[sagetrac-git]

Changed commit from 08c2b81 to c9e3b4c

[fchapoton]

comment:96

all commits squashed, and the patchbot is green

[videlec]

comment:97

Replying to @fchapoton:

all commits squashed, and the patchbot is green

Let us get that in!

Est-ce que tu es dispo mardi 05.05 14h-17h? Bill (Allombert) organise une hacking session PARI/GP et j'aimerais travailler sur les multizeta colorés (évaluation des polylogarithmes aux racines de l'unité). Henri (Cohen) a déjà programmé quelques trucs numériques. Si jamais ça t'intéresse.

[vbraun]

comment:98

Not python2-compatible

[videlec]

comment:99

Replying to @vbraun:

Not python2-compatible

Why is that a problem? The milestone is set to 9.2 which is supposed to be Python3 only. Or did I misunderstand something?

[fchapoton]

comment:100

Probably the merging tools do not respect the milestone..

[vbraun]

Changed branch from public/22713 to c9e3b4c