Exponential map of germs of vector fields

The paper’s Example 1 constructs u(z)=e^{iπ/m}z+z^{2m+1} and rules out a formal half-time map g with g∘g=u by coefficient comparison: c1^2=e^{iπ/m}, c2=…=c2m=0, and the z^{2m+1}-coefficient forces c_{2m+1}c1(c1^{2m}+1)=a≠0, contradicting c1^{2m}+1=0; hence no flow with ϕ1=u exists, so u∉im(exp) (definition of exp and the flow problem appear earlier in the paper). The candidate solution reproduces the same coefficient-chasing argument, adding a clean minimal-order r analysis and the standard estimate |e^{iπ/m}−1|=2 sin(π/(2m))<1 for m>4. Aside from a minor numerical slip in the paper’s near-identity remark (it states “|e^{iπ/m}−1|=2−√2<1” instead of the correct 2 sin(π/(2m)) bound), the proofs coincide in substance and reach the same conclusions. See the paper’s setup of exp and problem (2) and the details of Example 1 for the coefficient computations and the near-identity comment.