Result 200, Algebra

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.

Proof

The bigger picture

Why it matters

Complicated systems of polynomial equations can be compared with models built from individual monomials. The claimed result preserves how many independent expressions survive in each degree while bounding the complexity of relations among the equations.

What changes?

The unreviewed manuscripts claim the Eisenbud-Green-Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous ideals, polynomial constraints grouped by degree, containing a regular sequence of any positive length and degrees at least two. Regularity means each constraint is a non-zero-divisor after imposing earlier ones. The claimed replacement is a lex-plus-powers ideal, built from corresponding pure variable powers and lexicographically selected monomials. It preserves the Hilbert function and bounds every graded Betti number from above, in every homological and internal degree.

What does that help mathematicians do?

The Hilbert function counts independent polynomial expressions remaining in each degree after the constraints are imposed. Graded Betti numbers count generators, relations, and successive relations among relations, separated by degree. The reported result therefore gives a structured comparison model that preserves the first set of counts while bounding the second. Researchers could use it to rule out proposed relation counts exceeding those of the corresponding lex-plus-powers ideal.

Are there practical applications?

Its immediate value is foundational: it connects arbitrary homogeneous ideals satisfying the stated assumptions to specially organized monomial ideals. This would let researchers study possible degree counts and limits on relation complexity through a more explicit class of objects. The abstracts do not establish a practical computational speedup or an application outside mathematics.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The Artinian Lex-Plus-Powers Betti Theorem

September 23, 2026 22 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

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

September 23, 2026 43 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.