Petal rigidity and access forcing in polynomial lemniscates: an exponent-sharp rigidity threshold for a path problem of Erdős, Herzog, and Piranian

Abstract

Let f be a monic polynomial of degree n ≥ 2 whose roots all lie in the open unit disk, let Λ = {z : |f(z)| < 1}, let c = |f(0)|, and let β = Σ_j |ζ_j| be the ℓ¹-mass of the critical points of f. Erdős, Herzog, and Piranian proved that some connected component of Λ contains at least two roots, and asked whether two roots can always be joined inside Λ by a path of length less than 2 (Problem 5 of their 1958 paper; Erdős problem #1041). The problem is open; the degree-four case was settled recently by Pendyala. We prove two theorems in the deviation coordinate β. First, if β ≤ ((1−c)n)^(1/n)/25, then every radial segment [0, z_k] lies in Λ; hence all roots lie in a single component and every two roots are joined by a path of length |z_i| + |z_j| < 2. Equivalently, a counterexample must satisfy β > ((1−c)n)^(1/n)/25, a threshold tending to 1/25 for fixed c < 1; an explicit family shows the exponent 1/n in 1−c is sharp at every degree n ≥ 3 for this radial-segment threshold, so only the constant is open. Second, in the regime β ≤ εn, and when ρ := (1−c)^(1/n) − β/(n−1) is positive, a counterexample is forced into a rim-maze shape: a single component contains all but (ε/ρ)n + O(1) of the roots; for every δ ∈ (0,1) and ρ′ < ρ, all but πρ′/δ + O(1) of these lie at intrinsic distance (shortest path inside Λ) at least 1 − δ from the closed disk of radius ρ′ about the origin; and the total wet area is at most π (as for every monic polynomial, by Pólya). We also record a sharp transfer inequality for logarithmic kernels on the far lune, with best constant log(1+w)/log(1+u), and close with a map of the obstructions that mark the boundary of these methods.