On composite formal power series (Q2701676)

Let \(F : (\mathbb C^n, 0) \to (\mathbb C^n,0)\) be the germ of an analytic map and \(J(F)\) its Jacobian determinant. Let \(\{M_p\}\) be a sequence of strictly positive real numbers with the following properties: NEWLINENEWLINENEWLINE\(M_0 = 1\) and \(\{M_p\}_{p\geq 0}\) is logarithmically convex; NEWLINENEWLINENEWLINE\(\lim_{p\to \infty} \root p\of {M_p}= \infty\). NEWLINENEWLINENEWLINELet \(C\) be a positive real number. It is proved that the following properties are equivalent: NEWLINENEWLINENEWLINE\(\text{ord}(J(F))< \infty\); NEWLINENEWLINENEWLINEthere exist \(D > 0\) and an integer \(d \geq 1\) such that for any \(G \in \mathbb C[[x_1, \dots, x_n]]\): NEWLINE\[NEWLINE\|G\|_{D,M^{(d)}} \leq D\|G \circ F\|_{C,M}.NEWLINE\]NEWLINE Here \(\|G\|_{C,M} = \sup_{J \in \mathbb N^n}\frac{|a_J|}{C^jM_j}\) if \(G = \sum_{J \in \mathbb N^n} a_J x^J\) and \((M^{(d)})_p = M_{dp}\).