Torseurs associes a certaines relations algebriques entre polyzetas aux racines de l'unite
Résumé
We describe in this note a torsor structure arising on the affine scheme defined by a system of rationnal algebraic relations between polyzetas at roots of unity (values of hyperlogarithmic functions on a fixed finite group of complex roots of unity). When this group is reduced to 1, we call these numbers the polyzetas. They generalize the values of the Riemann zeta function at odd positive integers and are also called MZVs, multizetas, multiple harmonic series or Euler/Zagier sums. It is believed that the relations we consider here span all algebraic relations between them. The torsor structure underlying these relations should in that case be se same as Drinfeld's torsor made of the Grothendieck- Teichmueller group acting on associators. The formulas for the general case are derived from the action of the absolute Galois group on the projective line minus finitely many points. It is an easy consequence of the torsor structure that the algebra defined by these relations is a polynomial algebra. This was stated first by J. Ecalle in the polyzetas case.