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

Numerical tests of selection by the smallest critical value

We now ask whether a small critical value selects a short pair of inverse arms. The following records test that selection rule and a related sum over critical points. For a critical point $c$ whose two inverse arms over $[0,f(c)]$ reach roots, write $L_f(c)$ for their total length. The first computation tests the rule of choosing the critical point with uniquely smallest value modulus. A computational record gives an unnormalised quintic $f_0$, a selected critical point $c_0$, and the numerical values

where $R$ is the radius of a smallest disc containing the roots of $f_0$. The first number is not the length after normalisation to the unit disc. If $h$ is the centre of that disc, set

This preserves monicity, sends the roots into the closed unit disc, and divides every corresponding curve length by $R$. The reported normalised values are therefore

A further contraction uses $F_s(z)=s^{5}F(z/s)$, with $0<s<1$: roots and curve lengths are multiplied by $s$, and critical values by $s^5$. Thus the reported excess would persist for $s$ sufficiently close to $1$, with roots in the open unit disc.

These observations do not yet constitute a certified counterexample. A small residual at a computed fibre root does not enclose that root or establish its membership in the tracked inverse branch. The inscribed-polyline comparison needs both facts before it gives a rigorous lower bound for the intended arms. Another critical point supplies a shorter pair in the reported computation, which establishes no universal selection rule.

Second, a second computational record reports numerical violations of $\sum _c{L}_f(c)\le 2(n-1)R$ in degrees four and five. Here $R$ is the radius of a smallest disc containing the roots. The quartic computation gives $\sum _c{L}_f(c)=6.006352157\dots$ against the proposed bound $6$. Several differential-equation solvers agree, and an inscribed-polyline calculation using numerically computed fibre roots gives a similar excess. These checks do not provide rigorous enclosures for those roots, their branch assignment or the resulting length lower bound. They suggest failure of the aggregate inequality, but do not certify an example or an open set of violating polynomials. Its algebraic factor

is proved for every $n\ge 2$ in Theorem 10.2. The value estimate alone supplies no bound for the associated path-length sum. The algebraic theorem therefore remains useful independently of how the numerical aggregate examples are resolved.

Third, all five origin segments can escape even when the roots are arbitrarily close to a regular pentagon. The two-scale calculation above proves this for the family $F_{ϵ}$ attributed to the earlier note [26]; an isolated numerical example would not prove the assertion for arbitrarily small defects. Failure for every critical joining point is a different, stronger claim, reported in the addendum but not established by the origin calculation. No conclusion about curved inverse-ray paths or an arbitrary connecting curve follows from either straight-segment assertion.