On the Algebraic Bases of Polyzetas

Abstract

Two confluent rewriting systems in noncommutatives polynomials are constructed using the equations allowing the identification of the local coordinates (of second kind) of the graphs of the $\zeta$ polymorphism as being (shuffle or quasi-shuffle) characters and bridging two algebraic structures of polyzetas. In each system, the left side of each rewriting rule corresponds to the leading monomial of the associated homogeneous in weight polynomial while the right side is canonically represented on the Q-algebra generated by irreducible terms which encode an algebraic basis of the Q-algebra of polyzetas. These polynomials are totally lexicographically ordered and generate the kernels of the $\zeta$ polymorphism meaning that the Q-free algebra of polyzetas is graded and the irreducible polyzetas are transcendent numbers, Q-algebraically independent, and then $\pi$ 2 is Q-algebraically independent on odd zeta values (so does $\pi$).