Complex analysis and polynomial geometry

Sendov’s conjecture

Collaboration beta

Every zero of a degree-at-least-two complex polynomial with all zeros in the closed unit disk should have a nearby critical point, at distance at most one.

\deg p \geq 2, p (a) = 0 \Rightarrow \exists w, p' (w) = 0 \land | w - a | \leq 1

Known results and sources

A geometric view of Sendov’s distance-one question for a zero of a degree-at-least-two polynomial and its nearby critical points.

Research problem

Exact mathematical statement

Let $p\in\mathbb C[z]$ be a polynomial of degree at least $2$ whose zeros lie in the closed unit disk. For every zero $a$ of $p$ , the conjecture asserts that there is a zero $w$ of $p'$ with

|w-a|\le 1.

Problem infographic

Problem at a glance

A forest-green mathematical plate states degree at least two and shows polynomial zeros and critical points inside the unit disk, with a dashed measurement segment between a distinguished zero a and an illustrative critical point w; two insets show the centered-zero case and the regular-polygon equality model, while the general Sendov conjecture is marked open. — Sendov’s conjecture asks whether every zero a of a degree-at-least-two polynomial whose zeros lie in the closed unit disk has a critical point w within distance one. The centered-zero case follows from Gauss–Lucas, and p(z) = zⁿ − 1 realizes equality at the boundary; the general case remains open.

Current mathematical picture

Where work on Sendov’s conjecture stands

Open conjecture

The current work develops a barrier normalization, reciprocal and polar reconstruction dictionaries, exact mean and product constraints, two coupled Schur budgets, a division-free double-Schur cancellation law, and a finite-population coherence estimate. It reports the real-coefficient regime and degrees at most five as completed in the current work, and reports an increasing-degree exclusion that still requires priority audit. The current frontier is a corrected global coverage theorem plus a directed finite certificate that must explicitly retain coherence-ineligible, nonpositive-lower-bound, zero-functional, multiplicity, and collision branches. No complete proof or counterexample is claimed.

Strongest supported footholdFinite-population coherence mechanism

A sampling-without-replacement comparison supplied an explicit lower bound for the constant-term functional on a certified coherence domain.

Evidence posture · Reported result

Leading routeCorrected two-sieve coverage

Combine polar mean forcing, polynomial bulk exclusion, and full complex coherence-Schur exclusion while keeping every domain and zero branch explicit.

Route status · Active route

Useful failureZeroth signed Clark obstruction

The zeroth quadrature is termwise tautological after conversion and supplies no independent obstruction.

Route status · Eliminated route

Main reductionTwo explicit scalar sieves assembled

The bulk and full-complex coherence margins were made explicit, with numerical reconnaissance reporting overlap but no proof of coverage.

Evidence posture · Reported reduction

Completed special caseDegree-five endpoint reached in the current work

The exact polar-product integral and factorizations report the conjecture for every degree at most five.

Evidence posture · Reported special case

Priority open bridgeAudit the exact latest-route core

Independently verify the polar mean, constant-term product, two first Schur inequalities, division-free double-Schur law, integration-by-parts identity, and finite-population transfer on symbolic and high-precision examples.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Sendov’s conjecture in numbers

3.6kretained lines of mathematical investigation3,555 in the current working snapshot

Argument development: 3,135 · 88%
Explored or eliminated routes: 71 · 2%
Computational analysis: 69 · 2%
Open obligations: 121 · 3%
Definitions and setup: 159 · 4%

20selected mapped statements9routes investigated9reported milestones6open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byBlagovest SendovProposed1958Status checkedAug 2, 2026

Source recordedAug 2, 2026

Recommended next task

Audit the exact latest-route core

Suggested move: Recheck Sections 84–87 and 99 with exact symbolic examples while preserving the division-free form at J=0.

What would count as progress

Every displayed identity and inequality is independently derived with assumptions and cancellation rules explicit.
The checks include J=0, active unit-circle factors, multiplicities, and representative collision limits.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

24 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeCorrected two-sieve coverage

Combine polar mean forcing, polynomial bulk exclusion, and full complex coherence-Schur exclusion while keeping every domain and zero branch explicit.

Route status · Active route

Active routeExact core audit

Independently audit Sections 84–87 and 99 before relying on their combination or asymptotic applications.

Route status · Active route

Explored alternatives

Other routes

7 recorded

Narrowed routeIncreasing-degree finite reduction

The reported three-regime argument would leave only finitely many degrees, but it remains narrowed to independent checking reduction with no explicit threshold.

Route status · Narrowed route

Route held in reservePick and boundary-Schur residual analysis

Reserve the root-defect two-jet Pick inequality, active boundary Schur transform, equal-Clark, mirror-polar, and weighted-collision frameworks for certified residual boxes.

Route status · Route held in reserve

Useful but insufficientOne-point Grace-Walsh closure

Retain the scalar cutoff as an audit tool, but do not use it alone because it loses simultaneous root reconstruction.

Route status · Useful but insufficient

Browse 4 more explored routes

Useful but insufficientPositive harmonic tests alone

The Poisson hierarchy may provide a partial cutoff but cannot finish without antiderivative and root-reconstruction information.

Route status · Useful but insufficient

Eliminated routeZeroth signed Clark obstruction

The zeroth quadrature is termwise tautological after conversion and supplies no independent obstruction.

Route status · Eliminated route

Not yet justifiedImaginary-only endpoint mismatch

The imaginary component degenerates near the almost-real branch; retain the full real-plus-imaginary modulus.

Route status · Not yet justified

Not yet justifiedUncertified numerical mesh as proof

Use the reported scan only as route guidance until exact code, directed logs, hashes, and an independent certificate checker are retained.

Route status · Not yet justified

Route statements and reductions

Statements the next route can inspect and build on

Route statementCorrected global coverage theorem

For every n>=6 and a in (0,1), the polar-mean exclusion, polynomial bulk margin, and full complex coherence margin should jointly exclude every barrier, with explicit treatment of alpha>=1/2, j_-<=0, J=0, and collision branches.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Audit the exact latest-route core

Suggested move: Recheck Sections 84–87 and 99 with exact symbolic examples while preserving the division-free form at J=0.

Ready to work on

Certify the coherence domain implications

Prove or interval-certify that every box assigned to coherence has alpha<1/2 and j_->0, and explicitly return every other box to a polynomial bulk or exact fallback test.

Suggested move: Test whether bulk failure implies x_*>1/2 and j_->0 on the intended coherence region; isolate a residual branch if the implication is false.

Ready to work on

Make the large-degree c=4 overlap uniform

Turn the a=1-c/n^2, alpha=y/n asymptotic guide into a uniform analytic coverage result with explicit remainders and an explicit large-degree threshold.

Suggested move: Bound the remainder terms uniformly on compact c,y ranges and prove overlap across the transition c=4.

Ready to work on

Preserve zero, multiplicity, and collision branches

Keep J=0 in the division-free polynomial branch and use cancelled or confluent Schur/Pick forms for repeated, boundary, and colliding factors.

Suggested move: Add explicit interval branches for J=0 and verify that every quotient-based step has a polynomial or confluent replacement.

Ready to work on

Certify the finite remainder

Partition the remaining finite parameter range into rational interval boxes and certify one exact exclusion test for each box with directed rounding and an independent checker.

Suggested move: Implement the Section 102 stable interval formulas and archive the partition, per-box exclusion record, empty residual list, hashes, and checker.

Prerequisites still open

Reproduce the exploratory scan

Regenerate the reported n<=1000 reconnaissance from the retained formulas before using it for route selection, then replace mesh evidence with a directed certificate for any proof claim.

Suggested move: Write and archive the missing source, dependency lock, interval log, and hashes; do not infer certification from a nondirected mesh.

Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusOpen conjecture

The full all-degree conjecture remains open. It is proved for every degree n < 9 and for all sufficiently large degrees, leaving an unspecified finite intermediate range.

[4][9]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2022
Peer reviewedTao proved Sendov's conjecture for every sufficiently large degree, with no effective threshold supplied by the compactness proof.[4]
2014
PreprintDégot proved a high-degree result for a fixed distinguished zero away from the center and boundary; Chalebgwa later made a substantial range explicit.[2]
1999
Peer reviewedBrown and Xiang proved the conjecture for polynomials of degree at most eight.[10][4]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionAdds the necessary degree-at-least-two hypothesis to the reader-facing Sendov statement. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Statement scope clarified

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

12 mapped milestonesretained argument map

Browse all 12 mapped stages

stage 1Barrier and reconstruction normalization
stage 2Real-coefficient regime completed in the current work
stage 3Scalar-only routes bounded
stage 4Degree-at-most-five regime completed in the current work
stage 5Root and critical Schur structures aligned
stage 6Division-free double-Schur law
stage 7Finite-population coherence bound
stage 8Bulk and full-complex sieves assembled
stage 9Increasing-degree exclusion reported
stage 10Uncertified two-sieve reconnaissance retained
stage 11Zeroth Clark and imaginary-only closures rejected
stage 12Global coverage obligation corrected

Barrier and reconstruction normalizationThe current work reduces strict counterexamples to barrier configurations and reconstructs roots from capped critical reciprocals.