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Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture
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Abstract
We prove the Eisenbud–Green–Harris conjecture over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. Thus every homogeneous ideal containing such a sequence has the Hilbert function of a monomial ideal containing the corresponding pure powers.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- mo271, Paper/EisenbudGreenHarris: the conjecture holds in characteristic zero for degrees at least two (google-deepmind/formal-conjectures issue #6913, 2026-10-07). Cites this paper and proposes adding a
char_zerovariant (characteristic zero, all degrees ≥ 2) tagged as solved, keeping the general statement open; no Lean proof, and the issue says its comparison was AI-assisted and needs checking. [AI-generated per source] - Formal Conjectures authors, FormalConjectures/Paper/EisenbudGreenHarris.lean (google-deepmind/formal-conjectures, accessed 2026-10-21). Lean 4 statement of the conjecture over an arbitrary field, tagged
research open, with no characteristic-zero variant and all proofssorryat the time of access; its docstring lists quadratic sequences with c ≤ 5 (Güntürkün–Hochster) as a known case.
Related prior literature mentioned by others
- S. Gunturkun and M. Hochster, The Eisenbud-Green-Harris Conjecture for Defect Two Quadratic Ideals (arXiv:1812.07539, 2018). Listed as a known case in the Formal Conjectures docstring and not in this paper's bibliography; proves EGH for five quadrics in five variables, related to the introduction's account of the quadratic cases.
Possibly relevant literature (found by the agent, not checked)
- S. Gunturkun, A Survey on The Eisenbud-Green-Harris Conjecture (arXiv:2103.14106, 2021). A survey of earlier EGH results grouped by approach, alongside the account in the "Earlier results on EGH" subsection.
- G. Caviglia and E. Sbarra, Distractions of Shakin rings (arXiv:1308.4659, 2013). Extends Abedelfatah's EGH results to a wider class of ideals using Hilbert-function embeddings, related to the special cases listed in the introduction.
- Benjamin P. Richert and Sindi Sabourin, The residuals of lex plus powers ideals and the Eisenbud-Green-Harris conjecture (arXiv:math/0508334, 17 August 2005). Relates the family's Hilbert-function and Betti-number claims by proving an equivalence between EGH and extremality of the last graded Betti numbers of lex-plus-powers ideals. [family-level]
Digests and expositions
- AI Proved What?, Count like monomials? (episode 200, October 2026). Explains both manuscripts' characteristic-zero claims, the division-algebra construction and the transfer of resolutions, with examples and a distinction between its small-case checks and the unchecked geometric construction. [AI-generated per source] [family-level]
- Orply, One Monomial Ideal Matches the Entire Hilbert Function (11 October 2026). Summarizes the commuting-form construction and least-exponent extraction of a Hilbert-matching monomial ideal, distinguishing the Artinian construction over the complex numbers from the extension to characteristic-zero fields. [AI-generated per source]
News and blogs
- Cyber, Cryptographic assumptions audit (public audit, 8 October 2026; release discussion dated 9 October 2026). Discusses family 200 as a possible source of degree-of-regularity bounds in Gröbner-basis cryptanalysis, noting that transfer from characteristic zero to finite-field systems would require an additional assumption. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 200
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026,
author = {{OpenAI}},
title = {{Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf}{OAI:Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026}},
year = {2026}
}Family 200: Eisenbud–Green–Harris and lex-plus-powers
Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.
Other papers in this family:
- The Artinian Lex-Plus-Powers Betti Theorem · September 23, 2026