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This paper aims at presenting the first steps towards a formulation of theExact Renormalization Group Equation in the Hopf algebra setting of Connes andKreimer. It mostly deals with some algebraic preliminaries allowing toformulate perturbative renormalization within the theory of differentialequations. The relation between renormalization, formulated as a change ofboundary condition for a differential equation, and an algebraic Birkhoffdecomposition for rooted trees is explicited

An algebraic Birkhoff decomposition for the continuous renormalization group