(Pure) transcendence bases in $\phi$-deformed shuffle bialgebras

Computations with integro-differential operators are often carried out in an associative algebra with unit and they are essentially non-commutative computations. By adjoining a cocommutative co-product, one can have those operators perform on a bialgebra isomorphic to an enveloping algebra. That gives an adequate framework for a computer-algebra implementation via monoidal factorization, (pure) transcendence bases and Poincar\'e-Birkhoff-Witt bases. In this paper, we systematically study these deformations, obtaining necessary and sufficient conditions for the operators to exist, and we give the most general cocommutative deformations of the shuffle co-product and an effective construction of pairs of bases in duality. The paper ends by the combinatorial setting of systems of local systems of coordinates on the group of group-like series. * The present work is part of a series of papers devoted to the study of the renormalization of divergent polyzetas (at positive and at non-positive indices) via the factorization of the non commutative generating series of polylogarithms and of harmonic sums and via the effective construction of pairs of dual bases in duality in $\phi$-deformed shuffle algebras. It is a sequel to [3] and its content was presented in several seminars and meetings, including the 74th S\'eminaire Lotharingien de Combinatoire.