Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Unformalised remark: Blaschke products and varying degree

Powers of a Blaschke product give critical values close to one another while also admitting short connectors in a suitable asymptotic regime. The finite certificates and the ordinary limiting arguments have different scopes, which we keep separate below.

A research note on powers of Blaschke products uses the degree-$2N$ polynomials

and their radial contractions. Its finite rational certificates and its asymptotic proofs have different roles. The degree-eight certificate excludes an unconditional critical-value exponent $8/7$. It does not contradict the ordinary exponent $4/7$ proved by the Poisson argument above, or the smaller exponent $2/7$ in Theorem 10.2. For $N\ge 2$, the root centroid of $F_N$ is $-b/2$, as its $z^{2N-1}$ coefficient is $Nb$. A radial contraction by $r>0$ changes this to $-rb/2$, not zero. Thus the degree-eight example does not satisfy the additional centroid condition that permits numerator $8$ above. The degree-twenty-four certificate gives critical values in $|v+1|<1/12$, so $\mu >11/12$ while every radius-$4/3$ separation test fails. These fixed-degree examples do not establish optimality uniformly in degree. That conclusion uses the limiting measures in Sections 2–3 of the note.

For $b=\lambda /N$ with fixed $0<\lambda \le 1$, Section 4 constructs connectors of length at most $[\log (2/\lambda )+\pi +o(1)]/N$. The two radial pieces reduce the oscillatory term; a short circular arc then joins them in a sector where the remaining factor has modulus below one. The unspecified starting degree depends on $\lambda$. For $b=e^{-\alpha N}/N$ with fixed $\alpha >0$, the same section instead proves $\mathrm{\Lambda }(F_N)\to 2(1-e^{-\alpha /2})$ for the uncontracted polynomials $F_N$ displayed above and the closed-level functional defined earlier. The second assertion needs its own argument: the first does not allow a varying parameter to be substituted. Both use ordinary asymptotic arguments. Neither follows from the finite certificates, and neither has a formally checked analytic proof. They concern varying degree and do not contradict fixed-degree compactness or prove the unrestricted assertion.