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

Passing a length bound to a limit.

Let $f_j\to f$ in ${\mathcal{K}}_n$. If the lower limit of $\mathrm{\Lambda }(f_j)$ is finite, pass to a subsequence on which these infima converge to it and choose rectifiable curves ${\gamma }_j$ with lengths at most $\mathrm{\Lambda }(f_j)+1/j$. Every closed unit sublevel lies in $|z|\le 2$: for $|z|>2$, the root factorization gives $|f_j(z)|\ge (|z|-1{)}^n>1$. Constant-speed parametrisation on $[0,1]$ therefore gives uniformly bounded images and Lipschitz constants. Arzelà–Ascoli supplies a uniformly convergent subsequence.

To retain the endpoints with their multiplicities, list the two chosen occurrences first among the roots of $f_j$, and list the remaining roots arbitrarily. Compactness of the product of $n$ closed unit discs gives a further subsequence on which this entire list converges. The limiting factorization is $f$, so its first two entries remain two occurrences, even if their locations coincide. Uniform convergence of the curves and of the polynomials on $|z|\le 2$ gives $|f|\le 1$ on the limiting curve $\gamma$. For any partition $0=t_0<\cdots <t_M=1$,

Taking the supremum over partitions proves the required lower semicontinuity of length and hence $\mathrm{\Lambda }(f)\le \underset{j}{lim inf}\mathrm{\Lambda }(f_j)$. An infinite lower limit needs no argument. With $f$ fixed, the same compactness proof shows that a finite infimum is attained. Thus a bound $\mathrm{\Lambda }\le 2$ on a dense class gives an actual path of length at most $2$ throughout ${\mathcal{K}}_n$. For an open-disc polynomial, choose a containing disc of radius $0<R<1$ and scale: the resulting path has length at most $2R<2$ and lies in $\{|f|\le R^n\}\subset \{|f|<1\}$.

For $f(z)=z^n-1$, the inequality $|z^n-1|\le 1$ implies $2\mathrm{\Re }(z^n)\ge |z{|}^{2n}>0$ whenever $z\ne 0$. Thus the punctured sublevel lies in $n$ disjoint angular sectors, one per root. A path between different roots must pass through zero and has length at least $2$ by the triangle inequality. The two radial segments attain that length. The corresponding historical assertion on the closed class is

The lower-semicontinuity argument is proved in the compactness proof.

The displayed universal inequality is false in degree seven. Indeed, if it held throughout ${\mathcal{K}}_7$, the scaling step above would give a path of length less than $2$ inside the open lemniscate of the polynomial in Theorem 1.2. That theorem excludes such a path. Thus no estimate on a dense class can imply the displayed universal inequality by this compactness argument. The reduction remains valid for separately specified polynomial classes.