ON THE CLASSIFICATION OF PLANAR CONSTRAINED DIFFERENTIAL SYSTEMS UNDER TOPOLOGICAL EQUIVALENCE

The paper establishes three main results that match the candidate’s Parts A–C: (i) Theorem A: if the strict transforms of two constrained systems share the same non-degenerated elementary singularity scheme, then the systems are orientation-preserving C0-orbitally equivalent near the singularities; this is implemented via a conjugacy near the exceptional divisors and a push-down construction (Proposition 6) . (ii) Theorem B: for ADE-type impasse singularities with X(0) ≠ 0, under explicit coefficient conditions (Table 2), the system is equivalent to a model with constant adjoint field and the same δ; the ADE menu is recalled in Table 1 and the proof proceeds by weighted blow-ups and comparison of the schemes . (iii) Theorem C: when the impasse set is regular and the origin is an isolated Newton non-degenerated singularity with a characteristic orbit, the system is equivalent to its principal part; the argument again runs by synchronizing blow-ups guided by the Newton polygon and showing the schemes agree . The candidate’s solution mirrors this structure: it uses resolution to reduce to elementary singularities, constructs/patches local orbital conjugacies along the divisor, and pushes down (Part A); reduces ADE cases to a constant field via weighted Newton-type desingularization with the same case splits as in the paper (Part B); and invokes the Newton polygon/principal part mechanism (Part C). Minor differences are methodological (e.g., citing Seidenberg for reduction rather than the paper’s tailored resolution theorem) and do not affect correctness. Overall, both arguments coincide in substance.