Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

By an effective construction of pairs of bases in duality in shuffle and quasi-shuffle algebras, we identify the local coordinates of noncommutative generating series of polyzetas which are group-like. Algorithms lead to the ideal of homogeneous polynomial relations, in weight, among polyzetas and their explicit representation on irreducible elements.