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Fröberg’s conjecture for quintics and septics in four variables

Let kk be a field of characteristic zero and let S=k[x1,x2,x3,x4]S=k[x_1,x_2,x_3,x_4]. We prove Fröberg's predicted Hilbert series for ideals generated by rr general forms of equal degree dd for every r≥1r\geq1 in each of the two cases d=5d=5 and d=7d=7. Relative to the classical cases r≤5r\leq5 and the equal-degree theorem through degree d+2d+2 of Boij--Dannetun--Lundqvist, the generator-count ranges requiring new input are 6≤r≤116\leq r\leq11 for quintics and 6≤r≤216\leq r\leq21 for septics. The proof reduces each slice to finitely many endpoint ranks of Macaulay multiplication matrices. For quintics, ten exact endpoint computations based on twenty-one sparse forms suffice. For septics, a nested family of 120 integral forms supplies fifteen endpoint computations. In every endpoint certificate for these new ranges, an explicitly recorded maximal minor is nonzero modulo 22, hence is a nonzero integer. The case r=5r=5 is the classical strong Lefschetz instance; for quintics we also record a matching modular rank and Koszul bound. Zariski openness then gives the result over every characteristic-zero field. The unrestricted Fröberg conjecture remains outside the scope of the paper.

Result
Proved(see note)
Status
Partial result
AI contribution
AI co-developed
Method
Computation
Field
Commutative algebra
Posed by
Ralf Fröberg
Year posed
1985
Years open
41y
Solved
2026-08-25
Model
GPT-5.6 Sol; Claude Fable 5; Grok 4.6
Vendor
OpenAI; Anthropic; xAI
Collaborators
Qihang Wang, Dongming Zhang
Verification
Site-confirmed
Publication
Preprint
Significance
14 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Let S=k[x1,x2,x3,x4]S=k[x_1,x_2,x_3,x_4] over any characteristic-zero field. For each d∈{5,7}d\in\{5,7\} and every r≥1r\ge1, the paper proves that rr general degree-dd forms satisfy Fröberg’s predicted Hilbert series
HS⁡S/(F1,…,Fr)(t)=[(1−td)r(1−t)4]+. \operatorname{HS}_{S/(F_1,\ldots,F_r)}(t) = \left[\frac{(1-t^d)^r}{(1-t)^4}\right]_+.
The genuinely new ranges are 6≤r≤116\le r\le11 for quintics and 6≤r≤216\le r\le21 for septics. These are reduced to finitely many exact rank tests of Macaulay multiplication matrices; explicit maximal minors are nonzero mod 22, so the required ranks hold in characteristic zero. Thus the conjecture is completely settled for the equal-degree four-variable slices d=5d=5 and d=7d=7, but not in general.

What the AI did

Disclosed in the abstract and in a closing section, "Disclosure of automated assistance". The abstract states that "The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and Grok 4.6", and the disclosure section lists what the workflow did: "the formulation of mathematical ideas, generation of conjectures and proof strategies, derivation and checking of intermediate steps, construction of examples and exact certificates, comparison of literature and candidate proof approaches, organization of arguments, LaTeX drafting, and revision of the final text". It adds that the workflow "decomposed the problem into smaller subproblems and used repeated self-critique and alternative derivations to test the quantifiers, the characteristic-zero scope, and the endpoint-rank reductions".

That credits the mathematics rather than tooling, which is what puts this in scope, and it goes well beyond editing: proof strategies and the exact certificates are the substance of this paper. It is filed as co-developed rather than AI-discovered because no single step is attributed to a named model, the account is of a workflow rather than of a system solving a stated subproblem, and two human authors direct it throughout.

Verification

Site-confirmed: this site reproduced the computation, twice over, on 28 August 2026.

First, independently of the authors' code. The paper prints all 21 quintic forms with every coefficient +1+1 and states the monomial order, so the Macaulay matrices can be rebuilt from the text alone. I did that and ran my own elimination: all ten rows of Table 2 reproduce exactly, in shape and rank, over F2\mathbb F_2, F101\mathbb F_{101} and F1009\mathbb F_{1009}. I then checked the step the paper leaves implicit - that each rank gives exactly Fröberg's predicted dimension. Recomputing the prediction myself, with dim⁡(S/I)j=(j+33)−rank\dim(S/I)_j=\binom{j+3}{3}-\mathrm{rank}, every endpoint agrees: (6,9)→10(6,9)\to10, (8,8)→5(8,8)\to5, (11,7)→10(11,7)\to10, (20,6)→4(20,6)\to4, and 00 at each surjective endpoint.

Second, the authors' verifiers, published as arXiv ancillary files, both pass here: quintics 1.2 s, septics 0.9 s reporting 15 endpoints, 120 forms and 15 maximal minors. Neither uses floating point or randomness. The septic verifier genuinely recomputes rather than trusting stored numbers, which I confirmed by altering one exponent of one form: the run then failed at the matrix digest instead of passing.

Not established here: the reduction of each rr-range to its endpoints is a mathematical argument that was not checked, and the 120 septic forms exist only inside the certificate, so for septics I ran their verifier rather than rebuilding independently. Unrefereed, with no human review.

Sources

Submitted by VibeGene on

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