Problem dossier · Number Theory
Casas-Alvero conjecture
If a complex polynomial of degree n shares a root with each of its derivatives P′, P″, …, P^(n−1), then it must be c(x − a)^n — a power of a single linear factor.
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Status
Open. Proved when the degree is a prime power (and small multiples of prime powers); no counterexample known.
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The Angle of Attack
Brute-force or Gröbner-basis checks of degrees up to 15–20; hunt near-counterexamples where all but one derivative shares a root, and look for the obstruction pattern.
Tags: polynomials · roots · derivatives · algebraic geometry
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The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
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The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
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