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

The two roots in the component

The local estimates identify a critical point, but we still have to find the two roots in its component. The connections from $c_s$ to $b_3,b_6$ use a small disc and four segments near the critical point, short segments to the root directions, and radial segments out to the roots. The following ordinary argument identifies the component directly; the Lean source uses barriers. Put

The elementary inequality $1-{\rho }^{14}\ge 14{\rho }^{14}{\epsilon }^8$ shows that

Thus ${\mathcal{H}}_0>0$ suffices for containment. To check a segment from $u$ to $v$, expand ${\mathcal{H}}_0(u+t(v-u))=\sum _{k=0}^{14}{a}_k{t}^k$. Its Bernstein coefficients of degree $14$ are

All weights on the right are nonnegative and sum to one for $0\le t\le 1$. Exact substitution gives the lower bounds

The formula for $T_k$ above, followed by the displayed formula for ${\beta }_j$, specifies these checks using rational arithmetic alone; no sampling between endpoints is involved. The same Taylor test at $w_q$, with radius ${10}^{-8}$, gives one zero of $Q^{\prime }$ in that disc and ${\mathcal{H}}_0>3556874/{10}^{12}$ throughout it. The disc is contained in $|w-v_2|<{10}^{-6}$, so this zero is $w_s$. For each $j=3,6$, we can therefore follow the segments from $w_s$ to $w_q$, then to $x_j$ and $m_j$, staying where ${\mathcal{H}}_0>0$. It remains to connect $m_j$ to the corresponding root.

For the remaining segments no large polynomial expansion is needed. The Cayley intervals give $|m_j/8-{\zeta }_j|<1/10$ and $|m_j/8|<101/100$, for $j=3,6$. If $x$ lies between $m_j/8$ and ${\zeta }_j$, then $|x|\le 101/100$ and $|x-{\zeta }_j|<1/10$. From $F({\zeta }_j)=0$ and $|{\zeta }_j|=1$ we have $|{\zeta }_j^7-1|<{10}^{-40}$. Telescoping $x^7-{\zeta }_j^7$ therefore gives

The coefficient bounds

imply

Hence ${\mathcal{H}}_0(8x)>0$. This connects $m_j$ to $8{\zeta }_j$. Finally, for $8\epsilon \le t\le 1$, the root equation gives

so

Scaling by $\rho$ joins $\rho \epsilon 8{\zeta }_j$ to $b_j$ within $\{|f|<1\}$. The concatenated segments connect $c_s$ to both roots.

Every component of a polynomial lemniscate at a regular level is simply connected. The component-wise Riemann–Hurwitz formula [8] says that a component containing $k$ zeros has $k-1$ critical points, counted with multiplicity. The critical point table therefore shows that the component $U$ of $c_s$ contains exactly $b_3,b_6$, and every other component contains one root.