arXiv:1507.01089v1 [cs.SC] 4 Jul 2015 (Pure) transcendence bases in ϕ-deformed shuffle bialgebras ∗ January 10, 2019 V.C. Bui 2 , G.H.E. Duchamp 12 , Q.H. Ngô 2 , V. Hoang Ngoc Minh 32 , C. Tollu 12 1 Université Paris XIII, 1, 93430 Villetaneuse, France. 2 LIPN - UMR 7030, CNRS, 93430 Villetaneuse, France. 3 Université Lille II, 1, Place Déliot, 59024 Lille, France. Abstract 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 alge- bra. That gives an adequate framework for a computer-algebra implemen- tation via monoidal factorization, (pure) transcendence bases and Poincaré- Birkhoff-Witt bases. In this paper, we systematically study these deformations, obtaining nec- essary 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 con- struction of pairs of dual bases in duality in ϕ-deformed shuffle algebras. It is a sequel to [3] and its content was presented in several seminars and meetings, including the 74th Séminaire Lotharingien de Combinatoire. 1