Analytic Semiroots for Plane Branches and Singular Foliations

The paper’s Theorem 1.1 and its in-text version Theorem 8.8 assert that for each i and each free point P on the last divisor E, the semimodule of differential values of the analytic semiroot C^i_P equals Λ_{i−1}, and that the truncated list (ω_{−1}, ω_0, …, ω_{i−1}; ω_i) is an extended/enlarged standard basis; the proof proceeds via Delorme’s decomposition (Theorem 8.5) and a careful comparison of ν_E and ν_C, notably Lemma 8.9 and Corollary 8.10, together with the structural results on critical divisorial orders and axes (Theorem 7.13) and Appendix C’s raising-ν_E lemma. These results match the candidate’s conclusion exactly (Λ_{C^i_P} = Λ_{i−1} and truncated basis), though the model argues by a valuation-grading “ν_E-division” and invariance by ω_i rather than by Delorme’s decomposition. The model’s sketch is broadly correct but omits some technical lemmas used in the paper—e.g., the uniformity of leading-term cancellations and a formal division/raising-ν_E step (cf. Theorem 8.5 and Lemma C.1). Hence both are correct, with different proof styles and the model’s requiring minor completion. Key paper pointers: Theorem 1.1 (statement), Theorem 8.8 (main result), Lemma 8.9 (values along C^i_P), Theorem 8.5 (Delorme), and the critical order framework around Theorem 7.13 and Appendix C (raising ν_E) .