1 · The essentials
These manuscripts study how collections of polynomial equations can be replaced by more specially arranged collections without changing certain counts. The counts record how much remains in each degree after the equations are imposed. The starting collection must contain a sequence of equations satisfying a condition called regularity, and each equation in that sequence must have degree at least two. The claims apply over every field of characteristic zero, but do not cover positive characteristic. The main manuscript claims a replacement that preserves those counts and has at least as many relations, including relations among relations, in every specified degree and level. The companion manuscript claims the count-preserving replacement without this additional comparison of relations. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine rearranging a storage room into a prescribed shelf layout while keeping the same number of items on each floor. The stronger claim also compares a second set of counts describing connections between items: the prescribed layout must have no fewer at any level. This image illustrates the two comparisons, not the mathematical conditions that make them possible.
Why it matters
The claims address both the Eisenbud–Green–Harris and lex-plus-powers conjectures under the stated characteristic-zero hypotheses. They would provide specially structured replacements that preserve Hilbert functions while also giving upper bounds on graded Betti numbers. The distinction matters: preserving the degree-by-degree counts alone is weaker than controlling the relations as well.
2 · Going further
The precise setting is a homogeneous ideal containing a homogeneous regular sequence of any positive length, with degrees , over a field of characteristic zero. The Eisenbud–Green–Harris claim asks for a monomial ideal containing the corresponding pure powers and having the same Hilbert function. The main manuscript states the stronger lex-plus-powers conclusion: the corresponding lex-plus-powers ideal has the same Hilbert function and a graded Betti number at least as large in every homological and internal degree.
The source identifies these statements as the Eisenbud–Green–Harris and lex-plus-powers conjectures. Their two comparison measures serve different roles: the Hilbert function records degree-by-degree dimensions, whereas graded Betti numbers record the numbers of generators and successive relations, separated by degree and homological level. The claimed Betti comparison is entry by entry, rather than merely a comparison of totals.
The companion manuscript, “Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture,” claims the Hilbert-function conclusion under the same hypotheses. The main manuscript, “The Artinian Lex-Plus-Powers Betti Theorem,” adds the lex-plus-powers replacement and the Betti-number bounds. Although the companion title mentions commuting division-coefficient forms, neither abstract explains the argument or how those forms are used; the supplied text therefore does not support a description of the proof mechanism.
Useful notions
- Polynomial rings over fields
- Homogeneous ideals and degree
- Regular sequences
- Hilbert functions
- Graded Betti numbers
What was known before / what is new
Before
The source names the Eisenbud–Green–Harris and lex-plus-powers conjectures as the problems being addressed. It gives no account of earlier partial results, their hypotheses, or the history of either conjecture.
New
The main manuscript claims both conjectures over every characteristic-zero field for regular sequences of arbitrary positive length and degrees at least two. Its replacement preserves the Hilbert function and has no smaller graded Betti number in any homological and internal degree. The companion manuscript claims the Hilbert-function conclusion with a monomial ideal containing the corresponding pure powers.
Limits and assumptions
The claims require a homogeneous ideal containing a homogeneous regular sequence of positive length, with every degree at least two, over a characteristic-zero field. Positive characteristic and sequences involving degree-one elements are outside the stated scope; the source does not say whether those cases remain open. Despite the Artinian wording in the titles, the abstracts explicitly state claims for sequences of any positive length. The supplied abstracts do not describe the argument, and the main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Homogeneous ideal —
- An ideal generated by polynomials whose terms each have the same total degree.
- Regular sequence —
- A sequence in which each element is a non-zero-divisor after the preceding elements have been set to zero, with the whole sequence generating a proper ideal.
- Hilbert function —
- The function recording the dimension of each degree component of the quotient by a homogeneous ideal.
- Graded Betti numbers —
- Numbers counting generators and successive relations in a minimal graded resolution, indexed by homological and internal degree.
- Pure power —
- A power of a single variable, such as .
- Characteristic zero —
- The property that no positive number of copies of the field element sums to zero.
Keywords : Eisenbud–Green–Harris · Lex-plus-powers · Graded Betti numbers
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
The Artinian Lex-Plus-Powers Betti Theorem
23/09/2026We 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.
Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture
23/09/2026We 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.