Sharp bounds for the valence of certain logharmonic polynomials

The paper studies logharmonic polynomials of the form f(z) = p(z)·overline{q(z)} with q linear and proves that the maximal valence 3n−1 is achieved for every n>1, completing the Bshouty–Hengartner conjecture (upper bound from KLP24 and sharpness here). Their proof passes through anti-holomorphic dynamics for r(z)=c+1/overline{p(z)} and a harmonic argument principle count, yielding exactly 3n−1 solutions to p(z)·overline{q(z)}=w for suitable p,q,w . The model, however, solved a different (holomorphic) question about p(z)q(z)=w, where the Fundamental Theorem of Algebra bounds the number of distinct roots by n+1, and thus incorrectly rejected the paper’s claim. The mismatch is that the paper’s equation is logharmonic (involving conjugation), not a holomorphic product.