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The lex-plus-powers conjecture over fields of characteristic zero

With S=k[x1,…,xn]S=k[x_1,\ldots,x_n], 2≤a1≤⋯≤ac2\le a_1\le\cdots\le a_c and P=(x1a1,…,xcac)P=(x_1^{a_1},\ldots,x_c^{a_c}), let II be a homogeneous ideal containing a regular sequence of degrees a1,…,aca_1,\ldots,a_c, and let LL be the lex-plus-powers ideal (lex segment plus PP) with the same Hilbert function, which exists if the Eisenbud-Green-Harris conjecture holds. The lex-plus-powers conjecture, attributed to Charalambous and Evans, strengthens Bigatti-Hulett-Pardue extremality of lex ideals: LL should have the largest graded Betti numbers. It was known for ideals containing PP itself (Mermin-Murai) and for fast-growing degrees in characteristic zero (Caviglia-Sammartano). Is βp,j(S/I)≤βp,j(S/L)\beta_{p,j}(S/I)\le\beta_{p,j}(S/L) for all p,jp,j?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Commutative algebra; graded Betti numbers and syzygies
Posed by
Attributed to Hara Charalambous and E. Graham Evans; first printed in work of Evans and Richert per Francisco-Richert (2007); public formulations in Richert (2004) and Caviglia-Sammartano (2018)
Year posed
—
Years open
—
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
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims the lex-plus-powers conjecture over every characteristic-zero field, for regular sequences of any length with degrees at least two and no growth condition, with no restriction on the other generators of II; this contains the Eisenbud-Green-Harris statement. The key new step is a Hilbert-matching monomial ideal with no smaller Betti numbers (Mermin-Murai Conjecture 10.2). It does NOT treat positive characteristic.

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. This manuscript takes the division-coefficient construction from its same-day companion (the Eisenbud-Green-Harris entry) and adds a resolution transfer to a monomial ideal; the final monomial comparison follows Mermin-Murai and is included for completeness.

Verification

No independent mathematician has checked this yet. Theorem 1.1 (Artinian, over C\mathbb C) and Corollary 1.2 (every characteristic-zero field, regular sequences of any length) were read against the conjecture: the lex-plus-powers ideal exists, has the same Hilbert function and has graded Betti numbers at least those of S/IS/I in every homological and internal degree. The non-Artinian case uses the published Caviglia-Kummini reduction. Positive characteristic is not addressed. No Lean formalization.

Sources

Changelog1 change

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