Classification of systems of Dyson–Schwinger equations in the Hopf algebra of decorated rooted trees

We consider systems of combinatorial Dyson–Schwinger equations (briefly, SDSE) , … , in the Connes–Kreimer Hopf algebra HI of rooted trees decorated by I={1,…,N}, where is the operator of grafting on a root decorated by i, and F1,…,FN are non-constant formal series. The unique solution X=(X1,…,XN) of this equation generates a graded subalgebra H(S) of HI. We characterise here all the families of formal series (F1,…,FN) such that H(S) is a Hopf subalgebra. More precisely, we define three operations on SDSE (change of variables, dilatation and extension) and give two families of SDSE (cyclic and fundamental systems), and prove that any SDSE (S) such that H(S) is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions.