ON LINEARIZATION OF BIHOLOMORPHISM WITH NON-SEMI-SIMPLE LINEAR PART AT A FIXED POINT

The paper proves holomorphic linearization for non-semisimple linear parts under nonresonance plus Diophantine and QR–Diophantine conditions, via an (S,P)-decomposition of monomials, sharp bounds for solutions of the homological equation, and a Zehnder-type Newton scheme; see the statement of Theorem 1.1 and the precise arithmetic hypotheses (1.9)–(1.11) along with Λ_ε = Λ + εN and the modulus stratification (1.5)–(1.6) . The key technical tools are the homological operator L(ϕ)=ϕ∘Λ_ε−Λ_εϕ, the exact finite expansion for (Ω_α−εN)^{-1} using commutation and nilpotency, and uniform control of the remainder R_ε via the (S,P)-decomposition, culminating in Proposition 5.1 and Lemma 6.1 which drive the Newton iteration . By contrast, the candidate solution’s Step (3) relies on a uniform small-perturbation resolvent estimate for L_ε^{(d)} relative to L_0^{(d)} whose operator-norm gap is assumed independent of the degree d. Since ∥(L_0^{(d)})^{-1}∥ grows polynomially like (2+d)^σ due to small divisors, the product ∥L_ε^{(d)}−L_0^{(d)}∥·∥(L_0^{(d)})^{-1}∥ cannot be made uniformly <1 across all degrees for fixed weights and ε. This invalidates the claimed bound ∥(L_ε^{(d)})^{-1}∥≲(2+d)^σ via that resolvent argument. The paper avoids this pitfall by using the exact finite-series inversion (Ω_α−εN)^{-1} and a careful (S,P)-analysis of R_ε .