The Eisenbud-Green-Harris conjecture over fields of characteristic zero
Let be a standard graded polynomial ring over a field, let , and let be a homogeneous ideal containing a regular sequence with . Eisenbud, Green and Harris, from their work on higher Castelnuovo theory, proposed that the pure powers can replace the regular sequence without changing the Hilbert function: there should be a monomial ideal containing , indeed a lex-plus-powers ideal, with the same Hilbert function as . Known cases required fast-growing degrees or regular sequences of special form. Does every such have the Hilbert function of a lex-plus-powers ideal?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Commutative algebra; Hilbert functions
- Posed by
- David Eisenbud, Mark Green and Joe Harris, Higher Castelnuovo theory, Asterisque 218 (1993), Section 4
- Year posed
- 1993
- Years open
- 33y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims the Eisenbud-Green-Harris conjecture over every field of characteristic zero, for homogeneous regular sequences of any positive length with all degrees at least two and with no growth condition on the degrees, via commuting division-coefficient linear forms whose ordered monomials form a basis of the whole polynomial algebra. It does NOT treat positive characteristic. The Betti-number strengthening (lex-plus-powers) is the companion's result, listed separately.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The principal manuscript constructs commuting linear forms with division-algebra coefficients and deduces the Hilbert-function statement; its companion of the same date uses that construction to prove the stronger lex-plus-powers Betti inequalities, listed as a separate entry.
Verification
No independent mathematician has checked this yet. Corollary 1.2 (Artinian case over ) and the characteristic-zero corollary of the principal manuscript were read against the conjecture: every homogeneous ideal containing a regular sequence of degrees at least two has the Hilbert function of a monomial ideal containing the corresponding pure powers, for regular sequences of any length. The passage from Artinian to general uses the published Caviglia-Maclagan reduction and a coefficient-field descent. Positive characteristic is not addressed. No Lean formalization.