On the Hilbert Number for Piecewise Linear Vector Fields with Algebraic Discontinuity Set

The paper proves L(2k) ≥ k^2+2k+1 and L(2k+1) ≥ k^2+2k+3 via a second-order Melnikov analysis plus an ECT-system argument and a pseudo-Hopf bifurcation, with careful exponent bookkeeping for the nonlinear switching curve. The candidate solution reaches the same numerical bounds but its derivation has multiple errors: it (i) miscounts the independent functions by including j=0 terms that violate the positivity condition on exponents, (ii) claims only two (rather than three) additional basis functions from the linear perturbations, and (iii) claims an ECT-system can realize as many simple zeros as its dimension (instead of dimension minus one). Hence, the paper’s argument is correct and complete, while the model’s proof is flawed in several key steps, even though the final bounds match.