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

Erdős #1041 By 29 September 2026 Authorship, AI use and citation
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Summary. The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.

This paper covers the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository.

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References

  1. G. Pólya, Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1928), printed pp. 228–232 and 280–282. https://archive.org/details/sitzungsbericht1928preu.

  2. E. Crane, The areas of polynomial images and pre-images, Bull. London Math. Soc. 36 (2004), no. 6, 786–792, doi:10.1112/S0024609304003509; preprint arXiv:math/0302189v1, whose statement numbers are cited.

  3. T. F. Bloom, Erdős Problems, problem 1041. https://www.erdosproblems.com/1041

  4. P. Erdős, F. Herzog, and G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125–148, doi:10.1007/BF02790232.

  5. S. Ghosh and K. Ramachandran, Number of Components of Polynomial Lemniscates: A Problem of Erdős, Herzog, and Piranian, J. Math. Anal. Appl. 540 (2024), no. 1, 128571, doi:10.1016/j.jmaa.2024.128571; preprint arXiv:2312.13673v1, whose statement numbers are cited.

  6. S. Sutherland, Bad Polynomials for Newton’s Method, in B. Bielefeld and M. Lyubich (eds.), Conformal Dynamics Problem List, Stony Brook IMS preprint (1992), 42–44. https://www.math.stonybrook.edu/preprints/ims92-7.pdf

  7. D. Kozen and K. Stefánsson, Computing the Newtonian graph, J. Symbolic Comput. 24 (1997), no. 2, 125–136, doi:10.1006/jsco.1997.0118; authors’ copy https://www.cs.cornell.edu/kozen/Papers/newton.pdf.

  8. P. Ebenfelt, D. Khavinson, and H. S. Shapiro, Two-dimensional shapes and lemniscates, in Complex Analysis and Dynamical Systems IV, Part 1, Contemp. Math. 553, Amer. Math. Soc., Providence, RI, 2011, 45–59, doi:10.1090/conm/553/10931; preprint arXiv:1003.4567v1, whose statement numbers are cited.

  9. V. N. Dubinin, Lemniscates and inequalities for the logarithmic capacities of continua, Mat. Zametki 80 (2006), no. 1, 33–37, doi:10.4213/mzm2777; English translation, Math. Notes 80 (2006), no. 1, 31–35, doi:10.1007/s11006-006-0105-8.

  10. shtuka, A Short Path Joining Two Zeros Inside a Polynomial Lemniscate, manuscript posted 24 March 2026, 48 pp. https://shtuka123.github.io/1041/main.pdf. The file at this URL has since been replaced by a shorter partial version that no longer contains Proposition 12; the durable public record of the March version, its defect, and the author’s 26 March 2026 concession is the discussion thread at https://www.erdosproblems.com/forum/thread/1041.

  11. V. S. Pendyala, A Degree-Four Lemniscate Path Theorem, arXiv:2606.24875v1 (2026), doi:10.48550/arXiv.2606.24875.

  12. V. N. Dubinin, Some inequalities for polynomials and rational functions associated with lemniscates, Zap. Nauchn. Sem. POMI 404 (2012), 83–99; English translation, J. Math. Sci. 193 (2013), no. 1, 45–54, doi:10.1007/s10958-013-1432-4.

  13. P. Borwein, The arc length of the lemniscate {|p(z)|=1}, Proc. Amer. Math. Soc. 123 (1995), no. 3, 797–799, doi:10.1090/S0002-9939-1995-1223265-3.

  14. A. Eremenko and W. Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), no. 2, 409–415, doi:10.1307/mmj/1030132418; preprint arXiv:0805.2295.

  15. A. Fryntov and F. Nazarov, New estimates for the length of the Erdős–Herzog–Piranian lemniscate, in Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2 226, Amer. Math. Soc., Providence, RI, 2009, 49–60, doi:10.1090/trans2/226/05; preprint arXiv:0808.0717v1.

  16. T. Tao, The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree, arXiv:2512.12455v1 (2025).

  17. A. Eremenko and P. Yuditskii, Comb functions, Contemp. Math. 578 (2012), 99–118, doi:10.1090/conm/578/11472; preprint arXiv:1109.1464v1.

  18. A. Eremenko and L. Lempert, An extremal problem for polynomials, Proc. Amer. Math. Soc. 122 (1994), no. 1, 191–193, doi:10.1090/S0002-9939-1994-1207536-1.

  19. A. Eremenko, A Markov-type inequality for arbitrary plane continua, Proc. Amer. Math. Soc. 135 (2007), no. 5, 1505–1510, doi:10.1090/S0002-9939-06-08640-0; preprint arXiv:math/0606745v1.

  20. C. J. Bishop, A. Eremenko, and K. Lazebnik, On the shapes of rational lemniscates, Geom. Funct. Anal. 35 (2025), no. 2, 359–407, doi:10.1007/s00039-025-00704-2; preprint arXiv:2407.14610v1.

  21. V. N. Dubinin, Inequalities for critical values of polynomials, Sb. Math. 197 (2006), no. 8, 1167–1176, doi:10.1070/SM2006v197n08ABEH003793.

  22. E. Crane, A bound for Smale’s mean value conjecture for complex polynomials, Bull. London Math. Soc. 39 (2007), no. 5, 781–791, doi:10.1112/blms/bdm063; author preprint https://people.maths.bris.ac.uk/~maetc/SMVCbound.pdf.

  23. V. N. Dubinin, Four-point distortion theorem for complex polynomials, arXiv:1301.3985v1 (2013).

  24. O. S. Kuznetsova and V. G. Tkachev, Length functions of lemniscates, Manuscripta Math. 112 (2003), 519–538, doi:10.1007/s00229-003-0411-3; preprint arXiv:math/0306327.

  25. D. Tischler, Critical points and values of complex polynomials, J. Complexity 5 (1989), no. 4, 438–456.

  26. W. Cook / Plectis, Three refinements for the lemniscate-path programme, 16 September 2026, unreviewed research note, Sections 1–3, together with the earlier note Structural obstructions.

  27. I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 1 (1918), no. 4, 377–402.

  28. T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995, doi:10.1017/CBO9780511623776.

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Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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