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Math-2508 · mathematics · Algebra

Pure-power Hilbert functions for complex Artinian complete intersections

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If $I\subset C[x_{1},\dots ,x_{n}]$ is homogeneous and contains a regular sequence of degrees $2\leq a_{1}\leq \cdots \leq a_{n}$, does some homogeneous ideal J containing $x_{1}^{a_{1}},\dots ,x_{n}^{a_{n}}$ have the same quotient Hilbert function as I?

Short English statement

If $I\subset C[x_{1},\dots ,x_{n}]$ is homogeneous and contains a regular sequence of degrees $2\leq a_{1}\leq \cdots \leq a_{n}$, does some homogeneous ideal J containing $x_{1}^{a_{1}},\dots ,x_{n}^{a_{n}}$ have the same quotient Hilbert function as I?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2508-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2508

Specializing the Artinian Hilbert-function theorem to the complex field gives a homogeneous ideal containing the prescribed pure powers and preserving every graded quotient dimension. This is exactly the stated Artinian EGH assertion.

Pure-power Hilbert functions for complex Artinian complete intersections

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. In characteristic zero, the Eisenbud--Green--Harris Hilbert-function comparison is claimed for ideals containing a regular sequence, using a new commuting-form model over division coefficients.

Mathematical statement of the claimed result

Result 1.
Let $F$ have characteristic zero, $S_F=F[x_1,\ldots,x_n]$, and $I$ contain a homogeneous regular sequence of length $1\le c\le n$ with degrees $2\le a_1\le\cdots\le a_c$. The claimed monomial ideal $J$ contains $P_F=(x_1^{a_1},\ldots,x_c^{a_c})$ and satisfies \[ \dim_F(S_F/J)_d=\dim_F(S_F/I)_d\qquad(d\ge0). \] It may be selected as lex-plus-powers: outside $P_F$, each degree uses the lexicographically greatest $\dim I_d-\dim(P_F)_d$ monomials, with $x_1>\cdots>x_n$. The construction theorem over $\mathbb C$, for a full regular sequence $f_i$, additionally supplies a finite-dimensional central division algebra $\Delta$ over a finitely generated extension $k/\mathbb C$ and a commutative graded algebra $k[x]\subseteq A\subseteq\Delta[x]$, finite over $k[x]$, generated by commuting linear forms. Its distinguished forms satisfy \[ t_i^{a_i}=c_if_i+\sum_{\ell<i}B_{i\ell}f_\ell+\sum_{d>i}C_{id}t_d, \] with $c_i\in k^\times$, $B_{i\ell}\in A_{a_i-a_\ell}$, $C_{id}\in A_{a_i-1}$; all monomials in the $t_i$ form a left $\Delta$-basis of $\Delta[x]$.

Outline of the argument

The central construction replaces variables by commuting linear forms with coefficients in a division algebra while retaining an ordered polynomial basis. Triangular identities model the regular sequence by powers; monomial compression gives the Artinian comparison, and reduction plus coefficient descent handles shorter sequences.

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

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