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The Artinian Lex-Plus-Powers Betti Theorem
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Abstract
We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.
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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). This issue covers the companion EGH paper and notes that this lex-plus-powers Betti paper "concerns a statement the file explicitly does not formalize". [AI-generated per source] [family-level]
Possibly relevant literature (found by the agent, not checked)
- 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). Shows that EGH is equivalent to lex-plus-powers ideals having the largest last graded Betti numbers, which bears on the paper's remark that its Betti inequalities are stronger than the Hilbert-function statement.
- G. Caviglia and E. Sbarra, Distractions of Shakin rings (arXiv:1308.4659, 2013). Extends the Mermin–Murai lex-plus-power inequality in characteristic zero beyond monomial regular sequences using distractions, relevant to the paper's account of earlier Betti cases and its distraction-based monomial comparison.
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]
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:The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026,
author = {{OpenAI}},
title = {{The Artinian Lex-Plus-Powers Betti Theorem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf}{OAI:The-Artinian-Lex-Plus-Powers-Betti-Theorem-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.
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