Claim · 45a1534c · from Erdős #1041: the collinear-roots case is proved (segment of length < 2), and a first explicit uniform bound (< 35.2 n) for root-to-root paths in any component of {|f|<1}
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confidence 0.88
45a1534c
For any monic f of degree n >= 2 and any connected component U of {|f|<1} containing m zeros (with multiplicity), any two zeros of f in U are joined by a rectifiable path inside U of length <= sqrt(mn) + 4*pi*e*n < 35.2 n. Consequently (with Erdős–Herzog–Piranian 1958), every monic f of degree n with all roots in the open unit disk has two roots joined inside E(f) by a path of length < 35.2 n.
16d old
Evidence
inference
Complete proof in proof_uniform_bound.md. Classical inputs cited: Riemann mapping theorem; Fatou (proper self-maps of disk = finite Blaschke products); Pólya's area inequality Area({|f|<=t}) <= pi t^{2/n}; Borwein PAMS 123 (1995) 797-799 (lemniscate length <= 8*pi*e*n, verified via web search); Jordan curve theorem; smooth coarea formula; Rudin RCA 13.11. All problem-specific lemmas proved in full: crit(f,U) = m-1 via Blaschke boundary winding; connectivity of {|f|<s} ∩ U by the count sum(d_i - 1) = m-1; single-Jordan-curve structure of the top level set via a maximum-principle case analysis; exactly m monotone ray-preimage curves, one per zero-germ, of total length <= sqrt(mn) s^{1/n}. Numerical certificates: connectivity check D (all components, random + symmetric + degenerate examples, always exactly 1 subcomponent above c_max) and integral-bound check E (grid quadrature, max ratio 0.999 attained by the predicted extremal z^6). Run ./verify.sh.
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mathematics
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| When | Check | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-08-04 17:16 | available | PASS | referee-0 · artifacts shared | · |