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Level A · Machine-checkable Hard Algebra P-paper-casas-alvero

Casas-Alvero Conjecture

The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-paper-casas-alvero,
  title        = {Casas-Alvero Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-casas-alvero}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.CasasAlvero.

theorem casas_alvero_conjecture (hP' : HasCasasAlveroProp P) :
    ∃ α : K, P = (X - C α) ^ P.natDegree

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

The Casas-Alvero conjecture states that if a univariate polynomial P of degree d over a field of characteristic zero shares a non-trivial factor with its Hasse derivatives up to order d-1, then P must be of the form (X - α)ᵈ for some α in the field.

The conjecture has been proven for:

  • Degrees d ≤ 8
  • Degrees of the form p^k where p is prime
  • Degrees of the form 2p^k where p is prime

The conjecture is false in positive characteristic p for polynomials of degree p+1.

The conjecture is now claimed to be proven in this paper:

Source and licence

Imported from Formal Conjectures (research papers), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.