∑math20

Algebra · Result 200

Eisenbud–Green–Harris and lex-plus-powers

2 papers

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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1

The problem

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▶Introduction to the problemin preparation
2

The solution proposed by OpenAI

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▶The proposed solutionin preparation

Papers 2

First page of The Artinian Lex-Plus-Powers Betti Theorem

The Artinian Lex-Plus-Powers Betti Theorem

OpenAI · September 23, 2026

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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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}
}
First page of Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture

Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture

OpenAI · September 23, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.