40. Analytic fields, rooted trees, and quotient flows

Let \[\begin{equation} \tag{PRIQ.4.1} F_{\mathrm f}(x) = \sum_{m=0}^{\infty}A_m^{\mathrm f}[x^{\otimes m}] \end{equation}\] be locally analytic, where \(A_m^{\mathrm f}\) is a symmetric multilinear map of order \(m\) for \(m\geq1\) and \(A_0^{\mathrm f}\in X_{\mathrm f}\). Throughout, \[A_m[x^{\otimes m}] =A_m(x,\ldots,x).\]