Fröberg's conjecture for quintics and septics in four variables
Fröberg's predicted Hilbert series is proved for ideals generated by any number of general forms of degree five or seven in four variables.
- Lab
- Independent
- Model
- GPT-5.6 Sol; Claude Fable 5; Grok 4.6
- Field
- Mathematics
- Date
- 2026-08-25
- Human collaborators
- Qihang Wang, Dongming Zhang
- Problem posed
- 1985 · open 41 yrs
Sources
Original work
Independent commentary
What was found
Over a field of characteristic zero in four variables, the paper proves Fröberg's predicted Hilbert series for ideals generated by r general forms of equal degree d, for every r at least 1, in the two cases d = 5 and d = 7. Against the classical cases r at most 5 and the equal-degree theorem of Boij, Dannetun and Lundqvist through degree d + 2, the ranges needing new input are r between 6 and 11 for quintics and 6 and 21 for septics. Each slice reduces to finitely many endpoint ranks of Macaulay multiplication matrices: ten exact endpoint computations from twenty-one sparse forms for quintics, and fifteen from a nested family of 120 integral forms for septics, with every endpoint certificate recording an explicit maximal minor that is nonzero modulo 2 and hence a nonzero integer.
Novelty check
The paper states exactly which generator-count ranges were already covered and by what, the classical cases r at most 5 and the Boij-Dannetun-Lundqvist equal-degree theorem through degree d + 2, and therefore which ranges required new input. That makes the contribution a bounded and stated delta rather than a broad claim on Fröberg's conjecture, which remains open in general. No prior treatment of these ranges appears.
Caveats and known objections
A preprint, unrefereed, with no independent check on record, and vibemathed records the resolution as partial: this is two degrees in four variables, not Fröberg's conjecture. The certificates are explicit and the nonzero-modulo-2 minors make the endpoint computations checkable, but nothing is machine-checked, so the grade stays at author-verified. Autonomy is graded collaborative on a broad disclosure of automated assistance that spans the whole pipeline without separating human from model contribution: the workflow used GPT-5.6 Sol, Claude Fable 5 and Grok 4.6 for 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, and organization of arguments.
Also recorded at
vibemathedfroberg-s-conjecture-for-quintics-and-septics-in-four-variables
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and Collaborative.
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Graded author verified for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Fröberg's conjecture for quintics and septics in four variables. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-25-froberg-quintics-septicsBibTeX
@misc{whataifound-independent-2026-septics,
title = {Fröberg's conjecture for quintics and septics in four variables},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-08-25-froberg-quintics-septics}
}