we-solve (beta)

Math-2509-A1 · revision r1 · on Math-2509

OpenAI claim summary for Math-2509

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Over $\mathbb C$, the lex-plus-powers ideal in the theorem has the specified Hilbert function and bounds every Betti number of $S/I$ in its homological and internal bidegree. Thus the same single ideal gives all inequalities requested in the target.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Lex-plus-powers Betti bounds for Artinian quotients over C
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Lex-plus-powers Betti bounds for Artinian quotients over C

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For characteristic-zero regular sequences, the proposed lex-plus-powers ideal simultaneously preserves the Hilbert function and bounds every graded Betti number above.

Mathematical statement of the claimed result

Result 1.
For a characteristic-zero field $F$, a homogeneous ideal $I\subseteq S_F=F[x_1,\ldots,x_n]$ containing a regular sequence of degrees $2\le a_1\le\cdots\le a_c$, $1\le c\le n$, put $P_F=(x_1^{a_1},\ldots,x_c^{a_c})$. In degree $d$, define $J_d$ by adjoining to $(P_F)_d$ the lexicographically greatest $q_d=\dim I_d-\dim(P_F)_d$ monomials outside $P_F$. The source claims $J=\bigoplus_dJ_d$ is an ideal with \[ \dim_F(S_F/I)_d=\dim_F(S_F/J)_d\quad(d\ge0),\qquad \beta^{S_F}_{u,j}(S_F/I)\le\beta^{S_F}_{u,j}(S_F/J) \quad(u\ge0,\ j\in\mathbb Z). \] Here $\beta_{u,j}$ are graded Betti numbers. The additional generators of $I$ have no degree or number restriction. The main complex-field theorem is the full-length Artinian case $c=n$.

Outline of the argument

The division-coefficient commuting forms retain enough filtration information to compare minimal free resolutions, beyond merely counting graded dimensions. The Artinian Betti comparison is then transported through the established regular-sequence reduction and descent to arbitrary characteristic-zero coefficient fields.

Formal verification

Family 200 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources