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Math-2345 · mathematics · Algebra

The Lex-Plus-Powers Betti Number Conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $S=k[x_1,\ldots,x_n]$, let $F=(f_1,\ldots,f_r)$ be a homogeneous regular sequence with $2\leq e_1\leq\cdots\leq e_r$, and let $P=(x_1^{e_1},\ldots,x_r^{e_r})$. For every homogeneous ideal $I\supseteq F$, does there exist a unique lex-plus-$P$ ideal $L$ with the same Hilbert function, with $b_{i,j}(S/L)\geq b_{i,j}(S/I)$ for all $i,j$?

Formulation references: [S1], [S2].

Short English statement

Let $S=k[x_1,\ldots,x_n]$, let $F=(f_1,\ldots,f_r)$ be a homogeneous regular sequence with $2\leq e_1\leq\cdots\leq e_r$, and let $P=(x_1^{e_1},\ldots,x_r^{e_r})$. For every homogeneous ideal $I\supseteq F$, does there exist a unique lex-plus-$P$ ideal $L$ with the same Hilbert function, with $b_{i,j}(S/L)\geq b_{i,j}(S/I)$ for all $i,j$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2345-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2345

The first manuscript's characteristic-zero corollary constructs a lex-plus-powers ideal with the required Hilbert function for every regular-sequence length $1\le r\le n$. The second manuscript's characteristic-zero corollary also gives all the graded Betti inequalities for that same ideal, including $r<n$. Thus the selected claims cover the full characteristic-zero branch. The question allows fields of positive characteristic as well, which these corollaries do not treat; the overall match remains partial.

The Lex-Plus-Powers Betti Number Conjecture

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: 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; For characteristic-zero regular sequences, the proposed lex-plus-powers ideal simultaneously preserves the Hilbert function and bounds every graded Betti number above. Together they provide the following support within the scopes described below.

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 $\Delta[x]$.
Result 2.
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 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.

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

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