Ghosh 2024 number components polynomial lemniscates problem erdos

../

---

Subhajit Ghosh and Koushik Ramachandran, Number of components of polynomial lemniscates: a problem of Erdös, Herzog, and Piranian. J. Math. Anal. Appl. 540 (2024), no. 1, Paper No. 128571, 21 pp. DOI: https://doi.org/10.1016/j.jmaa.2024.128571.

For compact K in C of positive logarithmic capacity c(K), P_n(K) the monic degree-n polynomials with zeros in K, and C_n(K) the maximal number of components of the filled lemniscate {|p| < 1}, the paper studies M(K) = limsup C_n(K)/n and m(K) = liminf C_n(K)/n. Theorem 2.1 proves M(K) < 1 when 0 < c(K) < 1, with m(K) > 0 when c(K) in (1/2,1) and K is a bounded Jordan domain closure or a C^2 Jordan arc, and C_n(K) = 1 for all n when K is connected with c(K) <= 1/4; parts (b) and (c) are shown sharp (the disc of capacity 1/2 has m(K) = 0, and disconnected sets of small capacity can have m(K) > 0). Theorem 2.2 proves that if c(K) > 1 and either the equilibrium measure satisfies nu(B(z,r)) <= C r^eps or K is connected, then C_n(K) = n for all large n, so M(K) = m(K) = 1; in the capacity-one case, Proposition 2.3 gives M(K) = 1 for period-m sets and closed lemniscates (with C_n(K) = n along a subsequence for a closed lemniscate), and Theorem 2.4 gives 1/2 <= m(K) <= M(K) = 1 for the closure of a bounded Jordan domain with C^2 boundary. The methods are potential-theoretic, using equilibrium measure and capacity estimates. This answers the 1958 question of Erdos, Herzog and Piranian recorded as Erdos problem 1042 (Question 1.2, p. 3): below capacity 1 always M(K) < 1, and above it M(K) = 1 under the hypotheses of Theorem 2.2, while linearly many components (m(K) > 0) already occur for suitable sets of capacity above 1/2.

Source: https://arxiv.org/abs/2312.13673. The copy read for this card is the held arXiv:2312.13673v1 PDF (21 December 2023, 20 pages); theorem numbers and pages refer to it, not to the journal version. The arXiv record (https://arxiv.org/abs/2312.13673, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #1042

Results to transcribe.

• Theorem 2.1(a): If 0 < c(K) < 1 then M(K) = limsup C_n(K)/n < 1.
• Theorem 2.1(b): If c(K) in (1/2,1) and K is a Jordan domain closure or C^2 Jordan arc, then m(K) > 0; sharp at capacity 1/2.
• Theorem 2.1(c): If K is connected with c(K) <= 1/4 then C_n(K) = 1 for all n, so m(K) = 0; connectedness is essential.
• Theorem 2.2: If c(K) > 1 and K is connected or its equilibrium measure satisfies nu(B(z,r)) <= C r^eps, then C_n(K) = n for large n.
• Proposition 2.3: If K is a closed lemniscate or a period-m set (capacity 1), then M(K) = 1; for a closed lemniscate C_n(K) = n along a subsequence.
• Theorem 2.4: If c(K) = 1 and K is the closure of a bounded Jordan domain with C^2 boundary, then 1/2 <= m(K) <= M(K) = 1.