The largest relation counts come from a lex-plus-powers ideal

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Family 200 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

A collection of homogeneous polynomial equations determines how many independent polynomials remain in each degree. It also determines relations among the equations, relations among those relations, and so on. Can a simple monomial model reproduce the first kind of data while maximizing the second?

The Eisenbud–Green–Harris conjecture predicts such a model when the equations contain a regular sequence. Replace that sequence by pure powers of the variables, then arrange the remaining monomials in lexicographic order. The stronger lex-plus-powers conjecture predicts that this same model has at least as many graded syzygies in every position.

Family 200 claims both statements over every characteristic-zero field, for regular sequences of any positive length and arbitrary degrees at least two. One manuscript supplies a division-algebra construction and the Hilbert-function theorem. Its companion transfers entire resolutions to obtain the stronger Betti-number comparison.

2. Background and essential definitions

Count what survives in each degree

Let $S=F[x_1,\ldots,x_n]$, with each variable of degree one. A homogeneous ideal $I$ is generated by polynomials whose terms have equal total degree. The quotient splits into finite-dimensional graded pieces, and its Hilbert function is

$$H_{S/I}(d)=\dim_F(S/I)_d.$$

It counts independent degree-$d$ polynomial classes after imposing the equations.

A sequence $f_1,\ldots,f_c$ is regular if multiplication by $f_i$ is injective modulo its predecessors. Write their degrees in increasing order as $a_1\le\cdots\le a_c$. Regularity means the equations impose independent codimension in the homological sense; merely having the right number of nonzero equations would not suffice.

When $c=n$, the quotient by the sequence is finite-dimensional, also called Artinian, with Hilbert series

$$\prod_{i=1}^{n}(1+z+\cdots+z^{a_i-1}).$$

The corresponding pure-power ideal is

$$P=(x_1^{a_1},\ldots,x_c^{a_c}).$$

For full length, monomials outside $P$ have exponents in the box $0\le u_i<a_i$. This turns polynomial algebra into a finite counting problem.

Lexicographic order and the prescribed model

Fix $x_1>\cdots>x_n$. Among monomials of equal total degree, lexicographic order compares exponents at the first variable where they differ. In two variables, degree-three monomials are ordered

$$x_1^3>x_1^2x_2>x_1x_2^2>x_2^3.$$

A lex-plus-powers ideal contains $P$ and, in each degree, includes the required number of lexicographically greatest monomials outside $P$. The difficult point is that these degree-by-degree choices must fit together under multiplication to form one ideal.

For example, outside $(x^3,y^3)$, the allowed degree-three monomials are $x^2y$ and $xy^2$. If exactly one of them must be removed, the lex rule removes $x^2y$. The theorem concerns consistency across all degrees, not merely this isolated choice.

Count relations as well as dimensions

A minimal graded free resolution expresses the quotient using free modules with generators of specified degrees. Its graded Betti number

$$\beta_{p,j}^S(S/I)=\dim_F\operatorname{Tor}_p^S(S/I,F)_j$$

counts generators of internal degree $j$ at homological position $p$. Position one records minimal equations; later positions record higher relations. The coefficient field on the right is $S/(x_1,\ldots,x_n)$.

The Hilbert function constrains alternating sums of these numbers but does not determine them individually. Matching Hilbert functions therefore cannot by itself prove the stronger Betti inequalities.

3. Development history and previous partial results

Eisenbud, Green, and Harris proposed the Hilbert-function comparison in work extending Castelnuovo theory, with connections to Cayley–Bacharach questions about how many intersection points force a hypersurface to contain all of them.

Without a prescribed regular sequence, lex ideals already maximize Betti numbers at a fixed Hilbert function by the theorems of Bigatti, Hulett, and Pardue. Prescribed pure powers encode extra information about the equations and should yield a sharper extremal model.

For ideals already containing those powers, Clements–Lindström compression gives the Hilbert-function comparison. The corresponding Betti comparison developed through work of Mermin, Peeva, and Stillman for variable squares, Murai for stable ideals plus powers, and Mermin and Murai for monomial regular sequences over every field. The new manuscripts explicitly credit that monomial comparison as established work.

The harder step is passing from arbitrary polynomial regular sequences to a monomial ideal containing the prescribed powers while retaining the desired invariants. Mermin and Murai isolated a reduction that would preserve the Hilbert function and not decrease any Betti number. M1 claims to obtain precisely that reduction.

Earlier nonmonomial cases include Abedelfatah’s theorem when the forms split into linear factors. Caviglia and Maclagan established Eisenbud–Green–Harris for sufficiently fast-growing degrees, and Caviglia and De Stefani weakened the growth condition. These Hilbert-function results hold over arbitrary fields.

Caviglia and Sammartano obtained characteristic-zero Betti bounds under a related, slightly stronger degree-growth condition. Results for small numbers of quadrics, special liaison configurations, and sufficiently large degrees in bounded comparison ranges supplied further partial progress. Kuzmanovski’s recent asymptotic Betti estimates use a generic initial ideal associated with a lex-plus-powers ideal, rather than asserting the full comparison with that ideal in every degree.

The claimed advance removes the degree-growth, additional-generator, and bounded-degree-range restrictions in characteristic zero. It does not claim the unrestricted positive-characteristic theorem.

4. What the new manuscripts claim

Both manuscripts are dated 23 September 2026. Let $F$ be any characteristic-zero field, $1\le c\le n$, and

$$2\le a_1\le\cdots\le a_c.$$

Suppose a homogeneous ideal $I\subseteq S$ contains a homogeneous regular sequence of these degrees. Form $P=(x_1^{a_1},\ldots,x_c^{a_c})$.

M2 claims that the Hilbert function of $S/I$ is realized by a monomial ideal containing $P$, and that the ideal can be chosen lex-plus-powers. M1 claims that this single prescribed ideal $J$ also satisfies

$$H_{S/I}(d)=H_{S/J}(d)\quad(d\ge0),$$

and

$$\beta_{p,j}^S(S/I)\le\beta_{p,j}^S(S/J) \quad(p\ge0,\ j\in\mathbb Z).$$

No bound is placed on the number or degrees of additional generators of $I$. The result includes positive-dimensional quotients when $c<n$ and fields that are not algebraically closed.

The foundational construction itself is stated for full regular sequences over $\mathbb C$. Established Artinian reductions of Caviglia–Maclagan and Caviglia–Kummini, together with coefficient-field descent, extend the Hilbert and Betti conclusions to the broader scope above.

Further stated consequences

M2 applies a conditional theorem of Caviglia and Sbarra to bound every graded local-cohomology dimension of $S/I$ by the corresponding one for $S/J$. Local cohomology measures information supported near the homogeneous maximal ideal. In the Artinian case this reduces to Hilbert-function equality; its additional content concerns positive-dimensional quotients.

It also gives a scheme-theoretic quadratic Cayley–Bacharach consequence. If $\Omega\subseteq\mathbb P^r_F$ is a zero-dimensional complete intersection of $r$ quadrics, its degree is $2^r$. For a degree-$k$ hypersurface $X$, with $1\le k\le r$,

$$\Omega\not\subseteq X \quad\Longrightarrow\quad \deg(\Omega\cap X)\le2^r-2^{r-k}.$$

Degrees mean scheme lengths, so repeated or nonreduced intersection structure is included. Containment of a subscheme of strictly greater length forces containment of all of $\Omega$.

5. Why the advance matters

Provisional importance: 5/5. If correct, the family resolves two central extremal predictions in graded commutative algebra in characteristic zero.

The Hilbert-function theorem supplies a concrete combinatorial model for every allowed ideal. The stronger theorem says that one model simultaneously bounds all equation and syzygy counts, rather than giving different extremizers for different resolution positions.

This is useful because arbitrary polynomial equations can be difficult to analyze, while monomials in a bounded exponent box are explicit. The model retains the degrees of a regular sequence, giving more information than a general lex bound that ignores those degrees. The geometric corollary shows how such algebraic control returns to questions about forced containment of intersections.

6. The main idea

Enlarge the coefficients to reorganize the equations

M2 constructs an extension field $k/\mathbb C$ and a finite-dimensional central division algebra $\Delta$ over $k$. A division algebra is allowed to be noncommutative, but every nonzero element is invertible. The polynomial variables themselves remain central.

Inside $\Delta[x_1,\ldots,x_n]$, it constructs commuting degree-one forms $t_1,\ldots,t_n$ satisfying triangular power identities adapted to the original $f_i$. Their powers reproduce $f_i$, up to earlier equations and later forms. Crucially, all ordered monomials in the $t_i$ form a left-$\Delta$ basis of the entire polynomial algebra, not just its finite complete-intersection quotient.

The forms commute with one another, although their individual coefficients need not. Left coefficients and right multiplication by the forms are therefore kept distinct throughout the argument.

The construction combines powers of linear forms in stages. Matrix pencils arising from curves implement binary power addition. Equivariant descent first yields a central simple algebra; an additional index-divisibility argument is needed to ensure it is division. A matrix representation alone would not provide the required coefficient property.

Extract an ordinary monomial ideal

Extend $I$ to the division-coefficient algebra and expand in the ordered $t$-basis. At each leading monomial position, the allowable coefficients form a left ideal of $\Delta$. Since $\Delta$ is division, that coefficient space is either zero or all of $\Delta$.

The nonzero positions therefore define an ordinary monomial ideal. The triangular identities ensure that it contains the prescribed pure powers. Counting basis positions gives the Hilbert-function match. Compression then supplies the lex-plus-powers arrangement.

This explains the role of division coefficients: they convert potentially complicated leading coefficient spaces into an all-or-nothing monomial pattern.

Carry the whole resolution through the same construction

M1 uses the stronger full-basis assertion. Introduce a polynomial ring whose variables act on the right by the $t_i$. The large coefficient algebra is free both over the original polynomial ring and over this new ring, with a common finite multiplicity factor $\dim_k\Delta$.

Transferring a minimal resolution multiplies every Betti number by that same factor. A filtered Koszul-complex comparison then passes to the leading monomial ideal without decreasing the relevant homology dimensions. Cancelling the common factor gives a Hilbert-matching monomial ideal with no smaller Betti numbers.

The remaining comparison is the established monomial lex-plus-powers step, which M1 also presents. It stabilizes monomials within the exponent box using allowed transfers from later variables to earlier ones. Degreewise lex compression controls bounded shadows, the multiples that stay inside the box, ensuring the result remains an ideal.

For Betti numbers, splitting according to the last variable leaves only endpoint slices contributing nonzero differential ranks. Their rank comparison, together with equal dimensions of the Koszul terms, gives the desired homology inequalities. This keeps internal degrees separate and yields one ideal bounding every $\beta_{p,j}$.

7. Scope, limitations, and open questions

Characteristic zero is part of the general claim. Some earlier monomial and fast-degree results hold over every field, but that does not extend these new arbitrary-sequence conclusions to positive characteristic.

The regular sequence has positive length, degrees at least two, and the ideal is homogeneous in a standard graded polynomial ring. The no-prescribed-sequence case is already the classical lex theorem. The manuscripts do not claim that the original ideal is isomorphic to, or has the same resolution as, its lex-plus-powers model.

Equality of Hilbert functions and upper bounds on Betti numbers are different assertions. The full coefficient basis and resolution transfer are necessary parts of the stronger claim; a basis of the Artinian quotient alone would not justify it.

The two manuscripts are dependent: M1 imports M2’s commuting-form construction, then supplies the resolution-level argument. Questions about characteristic-positive analogues, equality cases, and effective computation of the auxiliary division construction remain separate from the stated extremal theorem.

8. Sources and further reading

Eisenbud, Green, and Harris’s Higher Castelnuovo theory and Cayley–Bacharach account give the geometric origins. The Clements–Lindström theorem supplies the monomial compression background.

Mermin and Murai’s pure-powers Betti theorem establishes the earlier monomial comparison. Caviglia and De Stefani’s degree-growth result and Caviglia–Sammartano’s lex-plus-powers theorem describe important nonmonomial cases.

Caviglia and Kummini’s Hilbert and Betti embedding work provides the reduction used to pass beyond Artinian quotients. The two pinned sources below give the claimed division-coefficient construction and its stronger resolution consequence.

Source manuscripts

• M1. OpenAI, The Artinian Lex-Plus-Powers Betti Theorem (September 23, 2026).
• M2. OpenAI, Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture (September 23, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.