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

1. Local saddle geometry and the refuted tree estimate

The full-disc model $x^2-y^2$, an annulus in which lower branches rejoin, two saddles joined by a separatrix and simultaneous saddle levels are mandatory tests. Repeating the one-lower/two-upper assertion does not answer the problem.

2. Cutting a compact region into flow strips

Let $p$ be a polynomial with simple roots and simple critical points, and set $u=-\log |p|$ away from the roots. Simplicity of the roots makes every critical value nonzero; critical points themselves may lie at $0$. For regular values $c<T$, let $V$ be a component of $\{u>c\}$ and assume that $\overline{V\cap \{c<u<T\}}$ is a compact genus-zero surface, its lower boundary is one smooth Jordan curve, its upper boundary consists of $m$ smooth level curves each enclosing one root, all interior critical points are nondegenerate index-one saddles, the normalised gradient field, defined away from the saddles,

is transverse to the level boundaries, and no maximal $X$-trajectory has two saddle endpoints. The equality uses the identification of a planar vector with a complex number: $\mathrm{\nabla }u=-\overline{p^{\prime }/p}$. In particular, $du(X)=1$ wherever $X$ is defined.

No particular formula such as $2s+1$ is presumed. Boundary tangencies, simultaneous levels and branch reunion through an annulus require explicit cuts. The closure is taken in the cut-open surface, with its separate boundary copies, not after regluing them in the plane. The quadratic example following Problem 15.5 explains why replacing this construction by the ordinary orbit quotient would fail.

3. Length estimates for joining flow strips

For a strip $S$, write

and write

for its transverse flux. On a regular sub-band, $u$ is harmonic and the side boundaries are flow lines. The divergence theorem therefore makes ${\mathrm{\Phi }}_S(t)$ independent of $t$. Normalise the transverse measure by

The choice of measure is determined by the length calculation. Since $u=t$ along the normalised flow, its Euclidean speed is $1/|\mathrm{\nabla }u|$. Transporting the flux measure to level $t$ therefore gives

Here $0<{\mathrm{\Phi }}_S(t_0)<\mathrm{\infty }$ for a nonempty regular strip. Tonelli’s theorem applies to the nonnegative integrand. At a critical endpoint level, exhaustion through regular sub-bands gives the same identity, allowing the common value to be infinite. A finite right side gives a trajectory of length at most that mean; an identity with both sides infinite gives no length estimate. A strip flux need not be an integer multiple of $2\pi$. For $f(z)=z$ and $u=-\log |z|$, take an annular sector of angular width $\theta$. On a circular level arc, $|\mathrm{\nabla }u|=1/r$ and $ds=rd\varphi$, so its flux is ${\mathrm{\Phi }}_S=\theta$. For $f(z)=z^n$, the same calculation gives ${\mathrm{\Phi }}_S=n\theta$. A closed regular level curve enclosing $k$ roots has flux $2\pi k$ by the argument principle. Restricting it to a strip replaces that full winding by the strip’s share of the flux. The averaging measure therefore depends on the width of the strip. Summing separate strip estimates cannot replace those widths by the winding number of a closed level curve. They do not reopen (5.1). Cassini already excludes the printed coefficient $1/(2\pi )$ for any embedded tree that contains all $m$ roots: Problem 15.1 records that no later gluing argument can retain that estimate.

The restriction on $\alpha$ is essential in the first alternative. For a fixed $f(z)=z^2-a^2$, with $0<a<1$, choose $0<c<-2\log a$ and let $V$ be the component of $\{-\log |f|>c\}$ containing both roots. At sufficiently deep levels, the Cassini parametrisation used above gives

Consequently ${\int }_{2\alpha }^{\mathrm{\infty }}{P}_V(t)dt\to 0$ as $\alpha \to \mathrm{\infty }$, whereas every spanning tree has length at least $2a$. No fixed $C$ can cover arbitrary truncations. This is a limitation of the unrestricted formulation of (6.1), separate from the earlier counterexample to the particular coefficient $1/(2\pi )$.

The error term assigned to the collar in the printed argument

uses the same excluded coefficient. It cannot be treated as an unused error allowance in an otherwise complete spanning-tree estimate. Shortest trajectories chosen independently in different strips may have endpoints that do not meet; the lengths needed to join them must also be bounded. Even such a bound would not restore the printed $1/(2\pi )$ tree estimate.

4. Coefficient perturbation and stability

Once a finite critical-value family is injective, Lean proves that an arbitrarily small translation makes every value nonzero and pairwise positive-ray separated. Lean also proves the explicit root-retention estimate: for a monic polynomial $f$ of degree $n\ge 1$, a constant shift of modulus below $\epsilon >0$ keeps all roots in the unit disc provided

Adding a constant leaves the difference of two critical values unchanged, so it cannot separate values that were equal to begin with.

For a constant shift alone, factorisation gives the sharper elementary bound

where the product counts roots with multiplicity. Hence $\underset{f(b)=0}{max}|b|+|\beta {|}^{1/n}<1$ already ensures retention in the open unit disc. The factor $n+1$ in the Lean estimate is not needed for this ordinary constant-shift argument. The coefficient one in the displacement bound is optimal, as $f(z)=z^n$ shows. The estimate does not match labelled root lists or bound the cost of transporting an existing curve.

Qualitatively, a small linear perturbation also supplies distinct critical values as well as simple critical points. For $\mathrm{deg}f\ge 2$, a multiple critical point of $f+\lambda z$ must satisfy $f^{″}(c)=0$ and $\lambda =-f^{\prime }(c)$; only finitely many parameters are excluded. Near any other parameter, label the distinct critical points by holomorphic functions $c_j(\lambda )$. Their values

Thus $v_i-v_j$ cannot vanish identically for $i\ne j$, and its zeros are isolated. Avoiding the finitely many pairs in a sufficiently small parameter neighbourhood gives simple critical points with distinct values. The neighbourhood can be chosen inside any prescribed disc about $\lambda =0$, proving the required arbitrarily small choice.

If distinct moduli are also needed for the excellent Morse function, the translation $\beta$ can avoid them at the same time as the ray collisions. For each distinct pair $v_i,v_j$, the equality $|v_i+\beta |=|v_j+\beta |$ is the real affine line

For ray separation it is enough to avoid, for each pair, the real line through $-v_i$ and $-v_j$: the ray-collision formula puts every forbidden translation on that line. Avoid also the finitely many points $-v_j$ to keep the translated values nonzero. Thus the full excluded set is contained in two real affine lines per pair of critical values, together with finitely many points. Its complement meets every open disc, so imposing distinct moduli costs no lower bound on the size of the perturbation. Root retention is also qualitative: choose a circle $|z|=R<1$ enclosing the original open-disc roots. If $|\lambda |R+|\beta |<\underset{|z|=R}{min}|f(z)|$, Rouché’s theorem keeps all roots of $f+\lambda z+\beta$ inside that circle. These ordinary arguments establish genericity and root retention; they do not estimate the collar or the cost of transferring a selected curve. The first two requirements below therefore have qualitative proofs, while their simultaneous quantitative use is the remaining task.

For the next problem, fix a specified valid replacement for the refuted metric inequality. Its slack is the difference between the permitted length and the bound supplied by that replacement. The symbols $q$ and $q_g$ refer to this same specified quantity for $f$ and $g$, respectively; they are not the historical collar integral with coefficient $1/(2\pi )$. Without a choice of replacement inequality, the requirement $q_g>q/2$ has no defined geometric content.

The cited declarations supply finite planar avoidance and the constant translation. The ordinary argument above additionally supplies a linear perturbation with simple critical points and distinct critical values. What remains for the stated problem is simultaneous control of the protected collar, the component, the quantitative slack and the root-to-root transfer cost. No formalisation of the new genericity argument or solution of those stability requirements is claimed.

5. Newton flow on a compact region between regular levels

Classical Newton-flow theory describes maximal trajectories and the graph they form. Away from the roots and critical points, every orbit satisfies

by the lemma of Shub, Tischler and Williams [7]. For an initial point that is neither a root nor a critical point, the maximal Newton trajectory has one of four endpoint patterns: each finite-time end is a critical point, an infinite forward end is a root, and an infinite backward end is the point at infinity [7]. Stationary trajectories at simple roots are not part of this classification. The maximal nonconstant trajectories with endpoints among the roots and critical points form a connected Newtonian graph with finitely many edges [7]. Critical-to-critical edges are included; pairwise distinct arguments exclude them by the value equation, not by the definition of the graph.

The smooth field needed for Morse theory is not $-p/p^{\prime }$, which is undefined at a critical point with nonzero value. On the root-free band the Euclidean gradient satisfies

It has the same oriented nonstationary trajectories as the Newton field, with a different time parameter. At a simple critical point $c$, its derivative is the real Hessian of $u$, whose eigenvalues are $\pm |p^{″}(c)/p(c)|$. Thus it extends as a hyperbolic saddle there. A finite critical endpoint in Newton time is not a finite arrival at an equilibrium of the smooth field. These two parametrisations must be distinguished in a statement about a Morse–Smale flow.

The endpoint classification on this band follows from the cited Newton classification and the time change. Each nonconstant trajectory runs from the lower boundary to the upper boundary, from the lower boundary to a saddle, or from a saddle to the upper boundary. Indeed, Newton time equals the change in $u$, so the part of a Newton trajectory in the band has bounded Newton-time interval. At either end it must meet a regular boundary or tend to a critical point. Ray separation excludes two different critical endpoints, and strict increase of $u$ excludes two ends at the same critical point. In smooth-gradient time, regular boundary crossings occur in finite time, whereas convergence to a saddle takes infinite time by uniqueness for the smooth differential equation. The remaining trajectories are the stationary saddles. This is an ordinary consequence of the classical classification, not an additional checked Lean theorem.

The planar transversality condition also follows from the hypotheses. Along a nonstationary gradient trajectory, the value of $p$ has constant argument, so a trajectory between two different saddles would contradict ray separation. A homoclinic trajectory is excluded by the strict increase of $u$. The stable-manifold theorem gives local stable and unstable curves tangent to the Hessian eigenspaces. Their global branches for different saddles therefore do not meet; at one saddle they meet transversely, since their tangent lines are the two distinct Hessian eigenspaces. The regular lower boundary is an entrance and the regular upper boundary an exit. With this boundary convention, the gradient satisfies the Morse–Smale transversality condition. This does not make the quotient by entire trajectories Hausdorff.

For an explicit obstruction, take $p(z)=z^2-1/4$ and the compact band

Its only critical point is the simple saddle at $0$, whose value is $-1/4$; both boundary levels are regular and ray separation is vacuous. The positive real separatrix, including its upper-boundary endpoint, is

Integration gives $\tau -{\tau }_0=\frac{1}{8}\log (x/x_0)-(x^2-x_0^2)/4$, so this trajectory tends to $0$ as smooth-gradient time $\tau \to -\mathrm{\infty }$. Thus this orbit is not closed in the band: its closure also contains the stationary orbit $\{0\}$. In the quotient topology its image is a nonclosed singleton, so the orbit quotient is not even $T_1$, hence not Hausdorff. Reparametrising the nonstationary trajectories leaves this quotient unchanged.

On an actual cut-open flow rectangle, by contrast, collapsing the trajectories gives the transverse interval. Identifying boundary copies is a further quotient and requires a separate separation argument. The Reeb graph uses yet another equivalence relation: it identifies connected components of level sets, not complete trajectories. Neither that graph nor the embedded Newtonian graph can be substituted for the orbit quotient without specifying a new equivalence relation.

Global solution theory and the finite graph are the classical results cited above. The relative endpoint classification and transversality are proved above. They do not construct the required finite cut-open space, justify its boundary identifications or control the length needed to join the pieces. The quadratic example settles the uncut-orbit-space question negatively; it does not obstruct a construction that retains the required separate boundary copies.

The unrestricted assertion in Erdős #1041 is false by Theorem 1.2. The positive results do not give a classification of the polynomials that admit a short connection. Their hypotheses and containment levels differ: low critical value, separated simple critical value, and the individually proved polynomial families treat separate sufficient conditions. In particular, the scaled $(5/2){\mu }^{1/n}$ bound is at level $(25/13)\mu$, which is smaller than $2\mu$; it therefore improves both the constant and the level of the $71/10$ construction. The latter is retained for its independent proof and component-sensitive estimates. The critical-value moment does not choose a contained pair. The ordinary generic slit-sheet topology has classical antecedents; the earlier proposed sufficient estimate concerned the lengths of particular inverse-ray curves. Their minimum need not equal the infimum over all contained curves. The degree-seven counterexample above rules out the corresponding universal estimate for these particular curves too, without invalidating conditional estimates on smaller classes.